Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials
SSparse Interpolation in Terms of Multivariate ChebyshevPolynomials
Evelyne Hubert ∗ Michael F. Singer † Abstract
Sparse interpolation refers to the exact recovery of a function as a short linear combination of basisfunctions from a limited number of evaluations. For multivariate functions, the case of the monomialbasis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshevpolynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systemsallows to define a variety of generalized multivariate Chebyshev polynomials that have connections totopics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithmto recover a function that is the linear combination of at most r such polynomials from the knowledgeof r and an explicitly bounded number of evaluations of this function. Keywords:
Chebyshev Polynomials, Hankel matrix, root systems, sparse interpolation, Weyl groups.
Mathematics Subject Classification: ∗ INRIA M´editerran´ee, 06902 Sophia Antipolis, France. [email protected] † North Carolina State University, Department of Mathematics, Box 8205, Raleigh, NC 27695-8205, [email protected] . Thesecond author was partially supported by a grant from the Simons Foundation ( a r X i v : . [ c s . S C ] J a n . Hubert & M.F. Singer Contents K [ x ± ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 Determining a basis of the quotient algebra . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.2 Eigenvalues and eigenvectors of the multiplication matrices . . . . . . . . . . . . . . . . 364.2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 The case of χ -invariant linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.1 Restriction to the invariant ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.2 Determining a basis of the quotient algebra . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.3 Multiplication maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.4 Algorithm and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
The goal of sparse interpolation is the exact recovery of a function as a short linear combination of elementsin a specific set of functions, usually of infinite cardinality, from a limited number of evaluations, or otherfunctional values. The function to recover is sometimes refered to as a blackbox : it can be evaluated, butits expression is unknown. We consider the case of a multivariate function f ( x , . . . , x n ) that is a sum ofgeneralized Chebyshev polynomials and present an algorithm to retrieve the summands. We assume we knowthe number of summands, or an upper bound for this number, and the values of the function at a finite setof well chosen points.Beside their strong impact in analysis, Chebyshev polynomials arise in the representation theory of simpleLie algebras. In particular, the Chebyshev polynomials of the first kind may be identified with orbit sumsof weights of the Lie algebra sl and the Chebyshev polynomials of the second kind may be identified withcharacters of this Lie algebra. Both types of polynomials are invariant under the action of the symmetricgroup { , − } , the associated associated Weyl group, on the exponents of the monomials. In presentationsof the theory of Lie algebras (c.f., [11, Ch.5, § generalized Chebyshevpolynomials associated to a root system, as similarly done in [27, 41, 43, 46]. Several authors have alreadyexploited the connection between Chebyshev polynomials and the theory of Lie algebras or root systems(e.g., [18], [47], [57]) and successfully used this in the context of quadrature problems [38, 42, 44, 46] ordifferential equations [53].A forebear of our algorithm is Prony’s method to retrieve a univariate function as a linear combination ofexponential functions from its values at equally spaced points [51]. The method was further developed in anumerical context [48]. In exact computation, mostly over finite fields, some of the algorithms for the sparseinterpolation of multivariate polynomial functions in terms of monomials bear similarities to Prony’s methodand have connections with linear codes [8, 3]. General frameworks for sparse interpolation were proposedin terms of sums of characters of Abelian groups and sums of eigenfunctions of linear operators [19, 25].The algorithm in [35] for the recovery of a linear combination of univariate Chebyshev polynomials doesnot fit in these frameworks though. Yet, as observed in [5], a simple change of variables turns Chebyshevpolynomials into Laurent polynomials with a simple symmetry in the exponents. This symmetry is mostnaturally explained in the context of root systems and Weyl groups and leads to a multivariate generalization.Previous algorithms [5, 22, 30, 35, 49] for sparse interpolation in terms of Chebyshev polynomials of onevariable depend heavily on the relations for the products, an identification property, and the commutationof composition. We show in this paper how analogous results hold for generalized Chebyshev polynomials ofseveral variables and stem from the underlying root system. As already known, expressing the multiplicationof generalized Chebyshev polynomials in terms of other generalized Chebyshev polynomials is presided overby the Weyl group. As a first original result we show how to select n points in Q n so that each n -variablegeneralized Chebyshev polynomial is determined by its values at these n points (Lemma 2.25, Theorem 2.27).A second original observation permits to generalize the commutation property in that we identify pointswhere commutation is available (Proposition 3.4).To provide a full algorithm, we revisit sparse interpolation in an intrinsically multivariate approach thatallows one to preserve and exploit symmetry. For the interpolation of sparse sums of Laurent monomials thealgorithm presented (Section 3.1) has strong ties with a multivariate Prony method [34, 45, 55]. It associatesto each sum of r monomials f ( x ) = ∑ α a α x α , where x α = x α . . . x α n n and a α in a field K , a linear formΩ ∶ K [ x, x − ] → K given by Ω ( p ) = ∑ α a α p ( ζ α ) where ζ α = ( ξ α , . . . , ξ α n ) for suitable ξ . This linear formallows us to define a Hankel operator from K [ x, x − ] to its dual (see Section 4.1) whose kernel is an ideal I having precisely the ζ α as its zeroes. The ζ α can be recovered as eigenvalues of multiplication maps on K [ x, x − ]/ I . The matrices of these multiplication maps can actually be calculated directly in terms of thematrices of a Hankel operator, without explicitly calculating I . One can then find the ζ α and the a α usingonly linear algebra and evaluation of the original polynomial f ( x ) at well-chosen points. The calculation of3. Hubert & M.F. Singerthe ( α , . . . , α n ) is then reduced to the calculation of logarithms.The usual Hankel or mixed Hankel-Toepliz matrices that appeared in the literature on sparse interpolation[8, 35] are actually the matrices of the Hankel operator mentioned above in the different univariate polynomialbases considered. The recovery of the support of a linear form with this type of technique also appears inoptimization, tensor decomposition and cubature [2, 9, 13, 15, 36, 37]. We present new developments to takeadvantage of the invariance or semi-invariance of the linear form. This allows us to reduce the size of thematrices involved by a factor equal to the order of the Weyl group (Section 4.3).For sparse interpolation in terms of Chebyshev polynomials (Section 3.2 and 3.3), one again recasts thisproblem in terms of a linear form on a Laurent polynomial ring. We define an action of the Weyl group onthis ring as well as on the underlying ambient space and note that the linear form is invariant or semi-invariantaccording to whether we consider generalized Chebyshev polynomials of the first or second kind. Evaluations,at specific points, of the function to interpolate provide the knowledge of the linear form on a linear basisof the invariant subring or semi-invariant module. In the case of interpolation of sparse sums of Laurentmonomials the seemingly trivial yet important fact that ( ξ β ) α = ( ξ α ) β is crucial to the algorithm. In themultivariate Chebyshev case we identify a family of evaluation points that provides a similar commutationproperty in the Chebyshev polynomials (Lemma 3.4).Since the linear form is invariant, or semi-invariant, the support consists of points grouped into orbits ofthe action of the Weyl group. Using tools developed in analogy to the Hankel formulation above, we showhow to recover the values of the fundamental invariants (Algorithm 4.15) on each of these orbits and, fromthese, the values of the Chebyshev polynomials that appear in the sparse sum. Furthermore, we show howto recover each Chebyshev polynomial from its values at n carefully selected points (Theorem 2.27).The relative cost of our algorithms depends on the linear algebra operations used in recovering the supportof the linear form and the number of evaluations needed. Recovering the support of a linear form on theLaurent polynomial ring is solved with linear algebra after introducing the appropriate Hankel operators.Symmetry reduces the size of matrices, as expected, by a factor the order of the group. Concerning evalu-ations of the function to recover, we need evaluations to determine certain sunbmatrices of maximum rankused in the linear algebra component of the algorithms. To bound the number of evaluations needed, werely on the interpolation property of sets of polynomials indexed by the hyperbolic cross (Proposition 4.5,Corollary 4.12), a result generalizing the case of monomials in [55]. The impact of this on the relative costsof the algorithms is discussed in Section 3.4.The paper is organized as follows. In Section 2, we begin by describing the connection between univariateChebyshev polynomials and the representation theory of traceless 2 × n -variable Cheby-shev polynomial, of the first or second kind, is determined by its values on n special points. In Section 3 weshow how multivariate sparse interpolation can be reduced to retrieving the support of certain linear formson a Laurent polynomial ring. For sparse interpolation in terms of multivariate Chebyshev polynomials ofthe first and second kind, we show how we can consider the restriction of the linear form to the ring of in-variants of the Weyl group or the module of semi-invariants. In addition, we discuss some of the costs of ouralgorithm as compared to treating generalized Chebyshev polynomials as sums of monomials. In Section 4we introduce Hankel operators and their use in determining algorithmically the support of a linear formthrough linear algebra operations. After reviewing the definitions of Hankel operators and multiplicationmatrices in the context of linear forms on a Laurent polynomial ring, we extend these tools to apply to linearforms invariant under a Weyl group and show how these developments allow one to scale down the size ofthe matrices by a factor equal to the order of this group. Throughout these sections we provide examples toillustrate the theory and the algorithms. In Section 5 we discuss the global algorithm and point out somedirections of further improvement. Acknowledgment:
The authors wish to thank the Fields institute and the organizers of the thematic4 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomialsprogram on computer algebra where this research was initiated. They also wish to thank Andrew Arnold fordiscussions on sparse interpolation and the timely pointer on the use of the hypercross in the multivariatecase. 5. Hubert & M.F. Singer
In this section we first discuss how the usual Chebyshev polynomials arise from considerations concerningroot systems and their Weyl group. This approach allows us to give higher dimensional generalizations ofthese polynomials [27, 46]. We review the results about root systems and representation theory allowing usto define the generalized Chebyshev polynomials of the first and second kind. This section concludes withthe first original result in this article necessary to our purpose: we show how one can determine the degreeof a Chebyshev polynomial from its values at few well chosen points.
The univariate Chebyshev polynomials of the first and second kind arise in many contexts; approximationtheory, polynomial interpolation, and quadrature formulas are examples. A direct and simple way to definethese polynomials is as follows.
Definition 2.1
1. The
Chebyshev polynomials of the first kind , { ˜ T n ( x ) ∣ n = , , , . . . } , are the uniquemonic polynomials satisfying ˜ T n ( cos ( θ )) = cos ( n θ ) or ˜ T n ( x + x − ) = x n + x − n .
2. The
Chebyshev polynomials of the second kind , { ˜ U n ( x ) ∣ n = , , , . . . } , are the unique monic poly-nomials satisfying ˜ U n ( cos ( θ )) = sin (( n + ) θ ) sin ( θ ) or ˜ U n ( x + x − ) = x ( n + ) − x −( n + ) x − x − = x n + x n − + ⋅ ⋅ ⋅ + x − n + x − n . The second set of equalities for ˜ T n and ˜ U n are familiar when written in terms of x = e iθ since cos nθ = ( e inθ + e − inθ ) and sin ( nθ ) = ( e inθ − e − inθ ) . We introduced these equalities in terms of x for a clearerconnection with the following sections.These polynomials also arise naturally when one studies the representation theory of the Lie algebra sl ( C ) of 2 × π ∶ sl ( C ) → gl n ( C ) is a direct sum ofirreducible representations. For each nonnegative integer n , there is a unique irreducible representation π n ∶ sl ( C ) → gl n + ( C ) of dimension n + { diag ( a, − a ) ∣ a ∈ C } , this map is given by π n ( diag ( a, − a )) = diag ( na, ( n − ) a, . . . , ( − n ) a, − n a ) . Each of the maps diag ( a, − a ) ↦ m a , for m = n, n − , . . . , − n, − n is called a weight of this representation. The set of weights appearing in the representations of sl ( C ) maytherefore be identified with the lattice of integers in the one-dimensional vector space R . The group ofautomorphisms of this vector space that preserves this lattice is precisely the two element group { id, σ } where id ( m ) = m and σ ( m ) = − m . This group is called the Weyl group W .We now make the connection between Lie theory and Chebyshev polynomials. Identify the weight corre-sponding to the integer m with the weight monomial x m in the Laurent polynomial ring Z [ x, x − ] and letthe generator σ of the group W act on this ring via the map σ ⋅ x m = x σ ( m ) . For each weight monomial x m , m ≥
0, we can define the orbit polynomial Θ m ( x ) = x m + x − m and the character polynomial Ξ m ( x ) = x m + x m − + . . . + x − m + x − m . th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsNote that for each m , both of these polynomials lie in the ring of invariants Z [ x, x − ] W = Z [ x + x − ] ofthe Weyl group. Therefore there exist polynomials T n ( X ) and U n ( X ) such that Θ n ( x ) = T n ( x + x − ) andΞ n ( x ) = U n ( x + x − ) . The Chebyshev polynomials of the first and second kind can be recovered using theformulas ˜ T n ( X ) = T n ( X ) and ˜ U n ( X ) = U n ( X ) . The previous discussion shows how the classical Chebyshev polynomials arise from representation of asemisimple Lie algebra and the action of the Weyl group on a Laurent polynomial ring. As noted above, thisdiscussion could have started just with the associated root system and its Weyl group and weights. This isprecisely what we do in Section 2.3 and 2.4 where we define a generalization of these polynomials for any(reduced) root system.
We review the definition and results on root systems that are needed to define generalized Chebyshevpolynomials. These are taken from [11, Chapitre VI],[26, Chapter 8] or [56, Chapitre V] where completeexpositions can be found.
Definition 2.2
Let V be a finite dimensional real vector space with an inner product ⟨⋅ , ⋅⟩ and R a finitesubset of V . We say R is a root system in V if1. R spans V and does not contain .2. If ρ, ˜ ρ ∈ R , then s ρ ( ˜ ρ ) ∈ R , where s ρ is the reflection defined by s ρ ( γ ) = γ − ⟨ γ, ρ ⟩⟨ ρ, ρ ⟩ ρ, γ ∈ V.
3. For all ρ, ˜ ρ ∈ R , ⟨ ˜ ρ, ρ ⟩⟨ ρ, ρ ⟩ ∈ Z .
4. If ρ ∈ R , and c ∈ R , then cρ ∈ R if and only if c = ± . The definition of s ρ above implies that ⟨ s ρ ( µ ) , s ρ ( ν )⟩ = ⟨ µ, ν ⟩ for any µ, ν ∈ V .In many texts, a root system is defined only using the first three of the above conditions and the last conditionis used to define a reduced root system. All root systems in this paper are reduced so we include this lastcondition in our definition and dispense with the adjective “reduced”. Furthermore, some texts define a rootsystem without reference to an inner product (c.f. [11, Chapitre VI],[56, Chapitre V]) and only introduce aninner product later in their exposition. The inner product allows one to identify V with its dual V ∗ in acanonical way and this helps us with many computations. Definition 2.3
The
Weyl group W of a root system R in V is the subgroup of the orthogonal group, withrespect to the inner product ⟨⋅ , ⋅⟩ , generated by the reflections s ρ , ρ ∈ R . One can find a useful basis of the ambient vector space V sitting inside the set of roots : Definition 2.4
Let R be a root system.1. A subset B = { ρ , . . . , ρ n } of R is a base if(a) B is a basis of the vector space V .(b) Every root µ ∈ R can be written as µ = α ρ + . . . + α n ρ n or µ = − α ρ − . . . − α n ρ n for some α ∈ N n .
7. Hubert & M.F. Singer
2. If B is a base, the roots of the form µ = α ρ + . . . + α n ρ n for some α ∈ N n are called the positive roots and the set of positive roots is denoted by R + . A standard way to show bases exist (c.f. [26, Chapter 8.4],[56, Chapitre V, § H that does not contain any of the roots and letting v be an element perpendicular to H . Onedefines R + = { ρ ∈ R ∣ ⟨ v, ρ ⟩ > } and then shows that B = { ρ ∈ R + ∣ ρ ≠ ρ ′ + ρ ′′ for any pair ρ ′ , ρ ′′ ∈ R + } , the indecomposable positive roots, forms a base. For any two bases B and B ′ there exists a σ ∈ W such that σ ( B ) = B ′ . We fix once and for all a base B of R.The base can be used to define the following important cone in V . Definition 2.5
The closed fundamental Weyl chamber in V relative to the base B = { ρ , . . . , ρ n } is ΛΛ ={ v ∈ V ∣ ⟨ v, ρ i ⟩ ≥ } . The interior of ΛΛ is called the open fundamental Weyl chamber . Of course, different bases have different open fundamental Weyl chambers. If L i is the hyperplane perpendic-ular to an element ρ i in the base B, then the connected components of V − ⋃ ni = L i correspond to the possibleopen fundamental Weyl chambers. Furthermore, the Weyl group acts transitively on these components.The element ρ ∨ = ρ ⟨ ρ, ρ ⟩ that appears in the definition of s ρ is called the coroot of ρ . The set of all coroots is denoted by R ∨ andthis set is again a root system called the dual root system with the same Weyl group as R [11, Chapitre VI, § ∨ is a base of R ∨ .A root system defines the following lattice in V , called the lattice of weights. This lattice and related conceptsplay an important role in the representation theory of semisimple Lie algebras. Definition 2.6
Let B = { ρ , . . . , ρ n } the base of R and B ∨ = { ρ ∨ , . . . , ρ ∨ n } its dual.1. An element µ of V is called a weight if ⟨ µ, ρ ∨ i ⟩ = ⟨ µ, ρ i ⟩⟨ ρ i , ρ i ⟩ ∈ Z for i = , . . . , n . The set of weights forms a lattice called the weight lattice Λ .2. The fundamental weights are elements { ω , . . . , ω n } such that ⟨ ω i , ρ ∨ j ⟩ = δ i,j , i, j = , . . . , n .3. A weight µ is strongly dominant if ⟨ µ, ρ i ⟩ > for all ρ i ∈ B . A weight µ is dominant if ⟨ µ, ρ i ⟩ ≥ forall ρ i ∈ B , i.e., µ ∈ ΛΛ . Weights are occasionally referred to as integral elements, [26, Chapter 8.7]. In describing the properties oftheir lattice it is useful to first define the following partial order on elements of V [29, Chapter 10.1]. Definition 2.7
For v , v ∈ V , we define v ≻ v if v − v is a sum of positive roots or v = v , that is, v − v = ∑ ni = n i ρ i for some n i ∈ N . The following proposition states three key properties of weights and of dominant weights which we will uselater.
Proposition 2.8
1. The weight lattice Λ is invariant under the action of the Weyl group W .2. Let B = { ρ , . . . , ρ n } be a base. If µ is a dominant weight and σ ∈ W , then µ ≻ σ ( µ ) . If µ is a stronglydominant weight, then σ ( µ ) = µ if and only if σ is the identity. thth
1. The weight lattice Λ is invariant under the action of the Weyl group W .2. Let B = { ρ , . . . , ρ n } be a base. If µ is a dominant weight and σ ∈ W , then µ ≻ σ ( µ ) . If µ is a stronglydominant weight, then σ ( µ ) = µ if and only if σ is the identity. thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials δ = ∑ ρ ∈ R + ρ is a strongly dominant weight equal to n ∑ i = ω i .4. If µ and µ are dominant weights, then ⟨ µ , µ ⟩ ≥ . proof: The proofs of items 1., 2., and 3. may be found in [29, Section 13.2 and 13.3]. For item 4. itis enough to show this when µ and µ are fundamental weights since dominant weights are nonnegativeinteger combinations of these. The fact for fundamental weights follows from Lemma 10.1 and Exercise 7 ofSection 13 of [29] (see also [26, Proposition 8.13, Lemma 8.14]). ∎ Example 2.9
The (reduced) root systems have been classified and presentations of these can be found inmany texts. We give three examples, A , A , B , here. In most texts, these examples are given so that theinner product is the usual inner product on Euclidean space. We have chosen the following representationsbecause we want the associated weight lattices (defined below) to be the integer lattices in the ambientvector spaces. Nonetheless there is an isomorphism of the underlying inner product spaces identifying theserepresentations. A . This system has two elements [ ] , [− ] in V = R . The inner product given by ⟨ u, v ⟩ = uv . A base isgiven by ρ = [ ] . The Weyl group has two elements, given by the matrices [ ] and [− ] . A . This system has 6 elements ±[ − ] T , ±[− ] T , ±[ ] T ∈ R when the inner product is given by ⟨ u, v ⟩ = u T S v where S = [ ] . A base is given by ρ = [ − ] T and ρ = [− ] T . We have ⟨ ρ i , ρ i ⟩ = so that ρ ∨ i = ρ i for i = { , } .The Weyl group is of order and represented by the matrices ⎡⎢⎢⎢⎢⎣ − ⎤⎥⎥⎥⎥⎦·„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ A , ⎡⎢⎢⎢⎢⎣ − ⎤⎥⎥⎥⎥⎦·„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ A , ⎡⎢⎢⎢⎢⎣ − − ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − −
11 0 ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − − ⎤⎥⎥⎥⎥⎦ . where A and A are the reflections associated with ρ and ρ . We implicitly made choices so thatthe fundamental weights are ω = [ ] T and ω = [ ] T . The lattice of weights is thus the integerlattice in R and orbits of weights are represented in Figure 2.1. B . This system has 8 elements ±[ − ] T , ±[− ] T , ±[ ] T , ±[ ] T when the inner product is givenby ⟨ u, v ⟩ = u T S v where S = [ ] . A base is given by ρ = [ − ] T and ρ = [− ] T . We have ⟨ ρ , ρ ⟩ = and ⟨ ρ , ρ ⟩ = . Hence ρ ∨ = ρ and ρ ∨ = ρ . The Weyl group is of order and represented by the matrices ⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − − ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − −
12 1 ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − −
10 1 ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − − ⎤⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎣ − − ⎤⎥⎥⎥⎥⎦ . We implicitly made choices so that the fundamental weights are ω = [ ] T and ω = [ ] T . Thelattice of weights is thus the integer lattice in R and orbits of weights are represented in Figure 2.1.
9. Hubert & M.F. Singer
Convention:
We will always assume that the root systems are presented in such a way that the associatedweight lattices are the integer lattice. This implies that the associated Weyl group lies in GL n ( Z ) .We may assume that there is a matrix S with rational entries such that < v, w >= v T Sw . This is not obviousfrom the definition of a root system but follows from the classification of irreducible root systems. Any rootsystem is the direct sum of orthogonal irreducible root systems ([29, Section 10.4]) and these are isomorphicto root systems given by vectors with rational coordinates where the inner product is the usual inner producton affine space [11, Ch.VI, Planches I-IX]. Taking the direct sum of these inner product spaces one gets aninner product on the ambient space with S having rational entries. For the examples we furthermore choose S so as to have the longest roots to be of norm 2.Figure 2.1: A -orbits and B -orbits of all α ∈ N with ∣ α ∣ ≤
9. Elements of an orbit have the same shapeand color. The orbits with 3, or 4, elements are represented by circles, the orbits with 6, or 8, elements bydiamonds or squares. Squares and solid disc symbols are on the sublattice generated by the roots.
As seen in Section 2.1, the usual Chebyshev polynomials can be defined by considering a Weyl group acting onthe exponents of monomials in a ring of Laurent polynomials. We shall use this approach to define Chebyshevpolynomials of several variables as in [27, 46]. This section defines the generalized Chebyshev polynomialsof the first kind. The next section presents how those of the second kind appear in the representations ofsimple Lie algebras.Let Λ and W be the weight lattice and the Weyl group associated to a root system. With ω , . . . , ω n thefundamental weights, we identify Λ with Z n through ω → α = [ α , . . . , α n ] T where ω = α ω + . . . + α n ω n .An arbitrary weight ω = α ω + . . . + α n ω n ∈ Λ is associated with the weight monomial x α = x α . . . x α n n .10 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsIn this way one sees that the group algebra Z [ Λ ] can be identified with the Laurent polynomial ring Z [ x , . . . , x n , x − , . . . , x − n ] = Z [ x, x − ] . The action of W on Λ makes us identify W with subgroup of GL n ( Z ) .Let K be a field of characteristic 0 and denote K ∖ { } by K ∗ . The linear action of W on K [ x ± ] = K [ x , . . . , x n , x − , . . . , x − n ] is defined by ⋅ ∶ W × K [ x ± ] → K [ x ± ]( A, x α ) ↦ A ⋅ x α = x A α . . (2.1)We have ( A ⋅ f )( x ) = f ( x A ) . One can see the above action on K [ x ± ] as induced by the (nonlinear) actionon ( K ∗ ) n defined by the monomial maps: W × ( K ∗ ) n → ( K ∗ ) n ( A, ζ ) ↦ A ⋆ ζ = [ ζ , . . . , ζ n ] A − = [ ζ A − ⋅ , . . . , ζ A − ⋅ n ] (2.2)where A − ⋅ i is the i -th column vector of A − . Such actions are sometimes called multiplicative actions [39,Section 3].For a group morphism χ ∶ W → C ∗ , α, β ∈ Z n we defineΨ χα = ∑ B ∈W χ ( B − ) x Bα . (2.3)One sees that A ⋅ Ψ χα = Ψ χAα = χ ( A ) Ψ χα . Two morphisms are of particular interest: χ ( A ) = χ ( A ) = det ( A ) . In either case ( χ ( A )) = A ∈ W . In the former case we define the orbit polynomial Θ α . Inthe latter case we use the notation Υ α .Θ α = ∑ B ∈W x Bα , and Υ α = ∑ B ∈W det ( B ) x Bα , (2.4)where we used the simplificaion det ( B − ) = det ( B ) . Proposition 2.10
We have Θ α Θ β = ∑ B ∈W Θ α + Bβ , Υ α Θ β = ∑ B ∈W Υ α + Bβ , Υ α Υ β = ∑ B ∈W det ( B ) Θ α + Bβ . proof: This follows in a straightforward manner from the definitions. ∎ Note that Θ α is invariant under the Weyl group action: Θ α = A ⋅ Θ α = Θ Aα , for all A ∈ W . The ring of allinvariant Laurent polynomials is denoted Z [ x, x − ] W . This ring is isomorphic to a polynomial ring for whichgenerators are known [11, Chapitre VI, § Proposition 2.11
Let { ω , . . . , ω n } be the fundamental weights.1. { Θ ω , . . . , Θ ω n } is an algebraically independent set of invariant Laurent polynomials.2. Z [ x, x − ] W = Z [ Θ ω , . . . , Θ ω n ] We can now define the multivariate generalization of the Chebyshev polynomials of the first kind (cf. [27],[41], [43], [46])
Definition 2.12
Let α ∈ N n be a dominant weight. The Chebyshev polynomial of the first kind associatedto α is the polynomial T α in K [ X ] = K [ X , . . . X n ] such that Θ α = T α ( Θ ω , . . . , Θ ω n ) .
11. Hubert & M.F. SingerWe shall usually drop the phrase “associated to α ” and just refer to Chebyshev polynomials of the first kindwith the understanding that we have fixed a root systems and each of these polynomials is associated to adominant weight of this root system. Example 2.13
Following up on Example 2.9. A : As we have seen in Section 2.1, these are not the classical Chebyshev polynomials strictly speaking,but become these after a scaling. A : We can deduce from Proposition 2.10 the following recurrence formulas that allow us to write themultivariate Chebyshev polynomials associated to A in the monomial basis of K [ X, Y ] . We have T , = T , = X, T , = Y ; 4 T , = XY − and for a, b > T a + , = X T a + , − T a, , T ,b + = Y T b + − T ,b ;2 T a + ,b = X T a,b − T a,b + − T a,b − , T a,b + = Y T a,b − T a + ,b − − T a − ,b − . For instance T , = X − Y, T , = Y X − , T , = Y − X ; T , = X − Y X + , T , = X Y − Y − X, T , = XY − X − Y, T , = Y − Y X + T , = X − X Y + Y + X, T , = Y − XY + X + Y,T , = X Y − XY − X + Y, T , = Y X − X Y − Y + X,T , = X Y − X − Y + Y X − . B : Similarly we determine T , = T , = X, T , = Y ; T , = X − Y + X + , T , = Y X − Y, T , = Y − X − T , = X − XY + X + X, T , = Y − XY − Y,T , = X Y + XY − Y + Y, T , = XY − X − X ; T , = X − X Y + X + X − XY + Y − Y + X + T , = Y − XY − Y + X + X + ,T , = X Y + X Y − XY + XY − Y + Y , T , = Y X − X Y − XY + Y,T , = XY + X Y − Y + Y − X − X − X − . We now describe the role that root systems play in the representation theory of semisimple Lie algebrasand how the Chebyshev polynomials of the second kind arise in this context [12, Chapitre VIII, § Definition 2.14
Let g ⊂ gl n ( C ) be a semisimple Lie algebra and let h be a Cartan subalgebra, that is, amaximal diagonalizable subalgebra of g . Let π ∶ g → gl ( W ) be a representation of g .1. An element ν ∈ h ∗ is called a weight of π if W ν = { w ∈ W ∣ π ( h ) w = ν ( h ) w for all h ∈ h } is differentfrom { } .
12 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
2. The subspace W ν of W is a weight space and the dimension of W ν is called the multiplicity of ν in π .3. ν ∈ h ∗ is called a weight if it appears as the weight of some representation. An important representation of g is the adjoint representation ad ∶ g → gl ( g ) given by ad ( g )( h ) = [ g, h ] = gh − hg . For the adjoint representation, h is the weight space of 0. The nonzero weights of this representationare called roots and the set of roots is denoted by R. Let V be the real vector space spanned by R in h ∗ . Onecan show that there is a unique (up to constant multiple) inner product on V such that R is a root systemfor V in the sense of Section 2.2 The weights of this root system are the weights defined above coming fromrepresentations of g so there should be no confusion in using the same term for both concepts. In particular,the weights coming from representations form a lattice. The following is an important result concerningweights and representation. Proposition 2.15 [56, § VII-5,Th´eor`eme 1; § VII-12, Remarques] Let g ⊂ gl n ( C ) be a semisimple Lie algebraand π ∶ g → gl ( W ) be a representation of g . Let E = { µ , . . . , µ r } be the weights of π and let n i be themultiplicity of µ i .1. The sum r ∑ i = n i µ i ∈ Λ is invariant under the action of the Weyl group.2. If π is an irreducible representation then there is a unique µ ∈ E such that µ ≻ µ i for i = , . . . , r . Thisweight is called the highest weight of π and is a dominant weight for R . Two irreducible representationsare isomorphic if and only if they have the same highest weight.3. Any dominant weight µ for R appears as the highest weight of an irreducible representation of g . Note that property 1. implies that all weights in the same Weyl group orbit appear with the same multiplicityand so this sum is an integer combination of Weyl group orbits.In the usual expositions one denotes a basis of the group ring Z [ Λ ] by { e µ ∣ µ ∈ Λ } [12, Chapitre VIII, § { e ( µ ) ∣ µ ∈ Λ } ([29, § e µ ⋅ e λ = e µ + λ or e ( µ ) ⋅ e ( λ ) = e ( µ + λ ) . With the conventions introducedin the previous section, we define the character polynomial and state Weyl’s character formula. Definition 2.16
Let ω be a dominant weight. The character polynomial associated to ω is the polynomialin Z [ x, x − ] Ξ ω = ∑ λ ∈ Λ ω n λ x λ where Λ ω is the set of weights for the irreducible representation associated with ω and n λ is the multiplicityof λ in this representation. From Proposition 2.15 and the comment following it, one sees that Ξ α = ∑ β ≺ α n β Θ β . Here we abuse notationand include all Θ β with β ≺ α even if β ∉ Λ α in which case we let n β = Theorem 2.17 (Weyl character formula) δ = ∑ ρ ∈ R + ρ is a strongly dominant weight and Υ δ Ξ ω = Υ ω + δ where Υ α = ∑ B ∈W det ( B ) x Bα The earlier cited [11, Chapitre VI, § Definition 2.18
Let ω be a dominant weight. The Chebyshev polynomial of the second kind associated to ω is the polynomial U ω in K [ X ] = K [ X , . . . X n ] such that Ξ ω = U ω ( Θ ω , . . . , Θ ω n ) .
13. Hubert & M.F. SingerThis is the definition proposed in [46]. In [41], the Chebyshev polynomial of the second kind are defined asthe polynomial ˜ U ω such that Ξ ω = ˜ U ω ( Ξ ω , . . . , Ξ ω n ) . This is made possible thanks to [11, Chapitre VI, § Proposition 2.19
Let { ω , . . . , ω n } be the fundamental weights.1. { Ξ ω , . . . , Ξ ω n } is an algebraically independent set of invariant Laurent polynomials.2. Z [ x, x − ] W = Z [ Ξ ω , . . . , Ξ ω n ] One sees from [11, Chapitre VI, § { Θ ω , . . . , Θ ω n } to the basis { Ξ ω , . . . , Ξ ω n } so results using one definition can easily be applied to situations using the otherdefinition. The sparse interpolation algorithms to be presented in this article can also be directly modifiedto work for this latter definition as well. The only change is in Algorithm 3.8 where the evaluation pointsshould be ( Ξ ω ( ξ α T S ) , . . . , Ξ ω n ( ξ α T S )) instead of ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) . As with Chebyshev polynomials of the first kind, we shall usually drop the phrase “associated to ω ” andjust refer to Chebyshev polynomials of the second kind with the understanding that we have fixed a rootsystems and each of these polynomials is associated to a dominant weight of this root system. Example 2.20
Following up on Example 2.9 A : As we have seen in Section 2.1, the Chebyshev polynomials of the second kind associated to A arethe classical Chebyshev polynomials of the second kind after a scaling. A : We can deduce from Proposition 2.10 (as done in the proof of Proposition 2.23) the following recur-rence formulas that allow us to write the multivariate Chebyshev polynomials associated to A in themonomial basis of K [ X, Y ] . We have U , = and, for a, b ≥ , for a, b ≥ , U a + , = XU a, − U a − , , U ,b + = Y U ,b − U a + ,b − U a + ,b = XU a,b − U a − ,b + − U a,b − , U a,b + = Y U a,b − U a + ,b − − U a − ,b For instance U , = X, U , = Y ; U , = X − Y, U , = XY − , U , = Y − X ; U , = X − XY + , U , = Y − XY + ,U , = X Y − Y − X, U , = XY − X − Y ; U , = X − X Y + Y + X, U , = Y − XY + X + Y,U , = X Y − XY − X + Y, U , = XY − X Y − Y + X,U , = X Y − X − Y .
14 Monday 27 thth
14 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials B : Similarly we determine U , = U , = X − , U , = Y ; U , = X − X − Y , U , = XY − Y, U , = Y − X ; U , = X , U , = Y − XY + Y,U , = X Y − Y − Y, U , = XY − Y − X + X + U , = X − X − X Y + XY − X + Y + + X, U , = Y − XY + Y + X − ,U , = X Y − XY − XY + Y + X Y − Y, U , = XY − Y − X Y + XY + Y,U , = X Y − Y − Y − X + X + XY + X − . We note that the elements Υ α appearing in Theorem 2.17 are not invariant polynomials but are skew-symmetric polynomials, that is, polynomials p such that A ⋅ p = det ( A ) p . The K -span of all such polynomialsform a module over K [ x ± ] W which has a nice description. Theorem 2.21 [11, Ch. VI, § δ = ∑ ρ ∈ R + ρ , the map K [ x ± ] W → K [ x ± ] p ↦ Υ δ p is a K [ x ± ] W -module isomorphism between K [ x ± ] W and the K [ x ± ] W -module of skew-symmetric polynomials. This theorem allows us to denote the module of skew-symmetric polynomials by Υ δ K [ x ± ] W . In this section we gather properties about generalized Chebyshev polynomials that relate to orders on N n .They are needed in some of the proofs that underlie the sparse interpolation algorithms developed in thisarticle. Proposition 2.22
For any α, β ∈ N n there exist some a ν ∈ N with a α + β ≠ such that Θ α Θ β = ∑ ν ∈ N n ν ≺ α + β a ν Θ ν , Υ α Θ β = ∑ ν ∈ N n ν ≺ α + β a ν Υ ν . and the cardinality of the supports { ν ∈ N n ∣ ν ≺ α + β, and a ν ≠ } is at most ∣W∣ . proof: From Proposition 2.10 we have Θ α Θ β = ∑ B ∈W Θ α + Bβ , Υ α Θ β = ∑ B ∈W Υ α + Bβ . If µ ∈ N n is theunique dominant weight in the orbit of α + Bβ then Θ α + Bβ = Θ µ and Υ α + Bβ = Υ µ . We next prove that µ ≺ α + β .Let A ∈ W be such that A ( α + Bβ ) = µ . Since A, AB ∈ W we have Aα ≺ α and ABβ ≺ β (Proposition 2.8.2).Therefore Aα = α − ∑ m i ρ i an ABβ = β − ∑ n i ρ i for some m i , n i ∈ N . This implies A ( α + Bβ ) = Aα + ABβ = α − ∑ m i ρ i + β − ∑ n i ρ i = α + β − ∑( m i + n i ) ρ i so µ = A ( α + Bβ ) ≺ α + β . ∎ Proposition 2.23
For all α ∈ N n , T α = ∑ β ≺ α t β X β and U α = ∑ β ≺ α u β X β where t α ≠ and u α ≠ .
15. Hubert & M.F. Singer proof:
Note that the ⟨ ω i , ω j ⟩ are nonnegative rational numbers Proposition 2.8. Therefore the set of non-negative integer combinations of these rational numbers forms a well ordered subset of the rational numbers.This allows us to proceed by induction on ⟨ δ, α ⟩ to prove the first statement of the above proposition.Consider δ = ∑ ρ ∈ R + ρ = ∑ ri = ω i (Proposition 2.8). As a strongly dominant weight δ satisfies ⟨ δ, ρ ⟩ > ρ ∈ R + . Furthermore, for any dominant weight ω ≠ ⟨ δ, ω ⟩ > ⟨ ρ, ω ⟩ ≥ ρ ∈ R + , with at leastone inequality being a strict inequality. Hence ⟨ δ, α − ω i ⟩ < ⟨ δ, α ⟩ .The property is true for T and U . Assume it is true for all β ∈ N n such that ⟨ δ, β ⟩ < ⟨ δ, α ⟩ , α ∈ N n .There exists 1 ≤ i ≤ n such that α i ≥
1. By Lemma 2.10, Θ ω i Θ α − ω i = ∑ ν ≺ α a ν Θ ν with a α ≠
0. Hence a α T α = X i T α − ω i − ∑ ν ≺ αν ≠ α a ν T ν . Since ν ≺ α , ν ≠ α , implies that ⟨ δ, ν ⟩ < ⟨ δ, α ⟩ , the property thus holds byrecurrence for { T α } α ∈ N .By Proposition 2.15, Ξ α is invariant under the action of the Weyl group. Furthermore, any orbit of the Weylgroup will contain a unique highest weight. Therefore Ξ α = ∑ β ≺ α n β Θ β with n α ≠
0. Hence U α = ∑ β ≺ α n β T α and so the result follows from the above. The property holds for { U α } α ∈ N as it holds for { T α } α ∈ N . ∎ The following result shows that the partial order ≺ can be extended to an admissible order on N n . Admissibleorder on N n define term orders on the polynomial ring K [ X , . . . , X n ] upon which Gr¨obner bases can bedefined [7, 16]. In the proofs of Sections 3 and 4 some arguments stem from there. Proposition 2.24
Let B = { ρ , . . . , ρ n } be the base for R and consider δ = ∑ ρ ∈ R + ρ . Define the relation ≤ on N n by α ≤ β ⇔ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ⟨ δ, α ⟩ < ⟨ δ, β ⟩ or ⟨ δ, α ⟩ = ⟨ δ, β ⟩ and ⟨ ρ , α ⟩ < ⟨ ρ , β ⟩ or ⋮⟨ δ, α ⟩ = ⟨ δ, β ⟩ and ⟨ ρ , α ⟩ = ⟨ ρ , β ⟩ , . . . , ⟨ ρ n − , α ⟩ = ⟨ ρ n − , β ⟩ , ⟨ ρ n , α ⟩ < ⟨ ρ n , β ⟩ or α = β Then ≤ is an admissible order on N n , that is, for any α, β, γ ∈ N n [ . . . ] T ≤ γ, and α ≤ β ⇒ α + γ ≤ β + γ. Furthermore α ≺ β ⇒ α ≤ β. proof: We have that ⟨ ρ, α ⟩ ≥ ρ ∈ B and α ∈ N n . Hence, since δ = ∑ ρ ∈ R + ρ , ⟨ δ, α ⟩ > α . Furthermore since { ρ , ρ . . . , ρ n } is a basis for V = K n , so is { δ, ρ , . . . , ρ n } . Hence ≤ isan admissible order.We have already seen that δ is a strongly dominant weight (Proposition 2.8). As such ⟨ δ, ρ ⟩ > ρ ∈ R + .Hence, if α ≺ β , with α ≠ β , then β = α − m ρ − . . . − m n ρ n with m i ∈ N , at least one positive, so that ⟨ δ, α ⟩ < ⟨ δ, β ⟩ and thus α ≤ β . ∎ The algorithms for recovering the support of a linear combination of generalized Chebyshev polynomialswill first determine the values Θ ω ( ξ α T S ) for certain α but for unknown ω . To complete the determination,we will need to determine ω . We will show below that if { µ , . . . , µ n } are strongly dominant weights thatform a basis of the ambient vector space V , then one can choose an integer ξ that allows one to effectivelydetermine ω from the values ( Θ ω ( ξ µ i T S ) ∣ ≤ i ≤ n )
16 Monday 27 thth
16 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomialsor from the values ( Ξ ω ( ξ µ i T S ) ∣ ≤ i ≤ n ) We begin with two facts concerning strongly dominant weights which are crucial in what follows. • If µ and µ are dominant weights, then ⟨ µ , µ ⟩ ≥ • If B is a base of the roots, ρ ∈ B and µ is a strongly dominant weight, then ⟨ µ, ρ ⟩ >
0. This followsfrom the facts that ⟨ µ, ρ ∗ ⟩ > ρ ∗ is a positive multiple of ρ .Also recall our convention (stated at the end of Section 2.2) that the entries of S are in Q . We shall denoteby D their least common denominator. Note that with this notation we have that D ⟨ µ, ν ⟩ is an integer forany weights µ, ν . Lemma 2.25
Let µ be a strongly dominant weight and let ξ = ξ D where ξ ∈ N satisfies ξ > ( ∣W∣) .
1. If ω be is a dominant weight then D ⋅ ⟨ µ, ω ⟩ = ⌊ log ξ ( Θ ω ( ξ µ T S ))⌋ where ⌊⋅⌋ is the usual floor function.2. If ω is a strongly dominant weight then D ⋅ ⟨ µ, ω ⟩ = nint [ log ξ ( Υ ω ( ξ µ T S )] where nint denotes the nearest integer proof:
1. Let s be the size of the stabilizer of ω in W . We have the followingΘ ω ( ξ µ T S ) = ∑ σ ∈W ξ µ T Sσ ( ω ) = ∑ σ ∈W ξ ⟨ µ, σ ( ω )⟩ = s ∑ σ ∈ C ξ D ⟨ µ, σ ( ω )⟩ where C is a set of coset representatives of W/ S tab ( ω ) . = sξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈ C ξ D ⟨ µ, σ ( ω )− ω ⟩ ) We now use the fact that for σ ∈ W , σ ( ω ) − ω = − ∑ ρ ∈ B n σρ ρ for some nonnegative integers n σρ . If σ ∈ C, σ ≠ n σρ are zero. Therefore we haveΘ ω ( ξ µ T S ) = sξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈ C ξ D ⟨ µ, − ∑ ρ ∈ B n σρ ρ ⟩ )= sξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈ C ξ − m σ ) , where each m σ is a positive integer. This follows from the fact that D ⟨ µ, ρ ⟩ is always a positive integer for µ a strongly dominant weight and ρ ∈ B. It is now immediate that sξ D ⟨ µ, ω ⟩ ≤ Θ ω ( ξ µ T S ) . (2.5) in the proof we show that the distance to the nearest integer is less than so this is well defined.
17. Hubert & M.F. SingerSince ξ > ( ∣W∣) > ∣W∣ we have 1 + ∑ σ ≠ ,σ ∈ C ξ − m σ ≤ + ∣W∣ ξ − < sξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈ C ξ − m σ ) < sξ D ⟨ µ, ω ⟩ (2.6)To prove the final claim, apply log ξ to (2.5) and (2.6) to yield D ⟨ µ, ω ⟩ + log ξ s ≤ log ξ ( Θ ω ( ξ µ T S )) ≤ D ⟨ µ, ω ⟩ + log ξ ( s ) Using the hypothesis on the lower bound for ξ , we have s ≤ ∣W∣ < ( ξ ) / (cid:212)⇒ log ξ ( s ) < . Therefore D ⟨ µ, ω ⟩ ≤ log ξ ( Θ ω ( ξ µ T S )) < D ⟨ µ, ω ⟩ + ω is a stongly dominant weight, we have that for any σ ∈ W , ω ≺ σ ( ω ) and σ ( ω ) = ω if and onlyif σ is the identity (c.f. Proposition 2.8). In particular, the stabilizer of ω is trivial. The proof begins in asimilar manner as above.We have Υ ω ( ξ µ T S ) = ∑ σ ∈W det ( σ ) ξ µ T Sσ ( ω ) = ∑ σ ∈W det ( σ ) ξ ⟨ µ, σ ( ω )⟩ = ξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈W det ( σ ) ξ D ⟨ µ, σ ( ω )− ω ⟩ ) We now use the fact that for σ ∈ W , σ ( ω ) − ω = − ∑ ρ ∈ B n σρ ρ for some nonnegative integers n σρ . If σ ∈ W , σ ≠ n σρ are zero. Therefore we haveΥ ω ( ξ µ T S ) = ξ ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈W det ( σ ) ξ ⟨ µ, − ∑ ρ ∈ B n σρ ρ ⟩ )= ξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈∣W∣ det ( σ ) ξ − m σ ) , where each m σ is a positive integer. This again follows from the fact that D ⟨ µ, ρ ⟩ is always a positive integerfor µ a strongly dominant weight and ρ ∈ B. At this point the proof diverges from the proof of 1. Since forany σ ∈ W , det ( σ ) = ± − ∣W∣ ξ − ≤ − ∑ σ ≠ ,σ ∈W ξ − m σ ≤ + ∑ σ ≠ ,σ ∈W det ( σ ) ξ − m σ ≤ + ∑ σ ≠ ,σ ∈W ξ − m σ ≤ + ∣W∣ ξ − . Therefore ξ D ⟨ µ, ω ⟩ ( − ∣W∣ ξ − ) ≤ Υ ω ( ξ µ T S ) ≤ ξ D ⟨ µ, ω ⟩ ( + ∣W∣ ξ − ) . (2.7)We will now show that 1 − ∣W∣ ξ − > ξ − and 1 + ∣W∣ ξ − < ξ .18 Monday 27 thth
17. Hubert & M.F. SingerSince ξ > ( ∣W∣) > ∣W∣ we have 1 + ∑ σ ≠ ,σ ∈ C ξ − m σ ≤ + ∣W∣ ξ − < sξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈ C ξ − m σ ) < sξ D ⟨ µ, ω ⟩ (2.6)To prove the final claim, apply log ξ to (2.5) and (2.6) to yield D ⟨ µ, ω ⟩ + log ξ s ≤ log ξ ( Θ ω ( ξ µ T S )) ≤ D ⟨ µ, ω ⟩ + log ξ ( s ) Using the hypothesis on the lower bound for ξ , we have s ≤ ∣W∣ < ( ξ ) / (cid:212)⇒ log ξ ( s ) < . Therefore D ⟨ µ, ω ⟩ ≤ log ξ ( Θ ω ( ξ µ T S )) < D ⟨ µ, ω ⟩ + ω is a stongly dominant weight, we have that for any σ ∈ W , ω ≺ σ ( ω ) and σ ( ω ) = ω if and onlyif σ is the identity (c.f. Proposition 2.8). In particular, the stabilizer of ω is trivial. The proof begins in asimilar manner as above.We have Υ ω ( ξ µ T S ) = ∑ σ ∈W det ( σ ) ξ µ T Sσ ( ω ) = ∑ σ ∈W det ( σ ) ξ ⟨ µ, σ ( ω )⟩ = ξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈W det ( σ ) ξ D ⟨ µ, σ ( ω )− ω ⟩ ) We now use the fact that for σ ∈ W , σ ( ω ) − ω = − ∑ ρ ∈ B n σρ ρ for some nonnegative integers n σρ . If σ ∈ W , σ ≠ n σρ are zero. Therefore we haveΥ ω ( ξ µ T S ) = ξ ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈W det ( σ ) ξ ⟨ µ, − ∑ ρ ∈ B n σρ ρ ⟩ )= ξ D ⟨ µ, ω ⟩ ( + ∑ σ ≠ ,σ ∈∣W∣ det ( σ ) ξ − m σ ) , where each m σ is a positive integer. This again follows from the fact that D ⟨ µ, ρ ⟩ is always a positive integerfor µ a strongly dominant weight and ρ ∈ B. At this point the proof diverges from the proof of 1. Since forany σ ∈ W , det ( σ ) = ± − ∣W∣ ξ − ≤ − ∑ σ ≠ ,σ ∈W ξ − m σ ≤ + ∑ σ ≠ ,σ ∈W det ( σ ) ξ − m σ ≤ + ∑ σ ≠ ,σ ∈W ξ − m σ ≤ + ∣W∣ ξ − . Therefore ξ D ⟨ µ, ω ⟩ ( − ∣W∣ ξ − ) ≤ Υ ω ( ξ µ T S ) ≤ ξ D ⟨ µ, ω ⟩ ( + ∣W∣ ξ − ) . (2.7)We will now show that 1 − ∣W∣ ξ − > ξ − and 1 + ∣W∣ ξ − < ξ .18 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials1 − ∣W∣ ξ − > ξ − : This is equivalent to ξ − ξ = ξ ( − ξ − ) > ∣W∣ . Since ξ > ( ∣W∣) , it is enough toshow that 1 − ξ − > ∣W∣ − . To achieve this it suffices to show 1 − ( ∣W∣) − > ∣W∣ − or equivalently, that f ( x ) = x − ( x ) − > x ≥
2. Observing that f ( ) > f ′ ( x ) = − ( x ) − > x ≥ + ∣W∣ ξ − < ξ : This is equivalent to ξ − ξ = ξ ( ξ − ) > ∣W∣ . In a similar manner as before, it sufficesto show ∣W∣ ( ξ − ) > ∣W∣ or ξ − > ∣W∣ − . To achieve this it suffices to show ∣W∣ − > ∣W∣ − orequivalently, f ( x ) = x − x − > x ≥
2. Observing that f ( ) > f ′ ( x ) > x ≥ ξ D ⟨ µ, ω ⟩ ξ − / < ξ D ⟨ µ, ω ⟩ ( − ∣W∣ ξ − ) ≤ Υ ω ( ξ µ T S ) ≤ ξ D ⟨ µ, ω ⟩ ( + ∣W∣ ξ − ) < ξ D ⟨ µ, ω ⟩ ξ / . Taking logarithms base ξ , we have D ⟨ µ, ω ⟩ − < log ξ ( Υ ω ( ξ µ T S )) < D ⟨ µ, ω ⟩ + ∎ The restriction in 2. that ω be a strongly dominant weight is necessary as Υ ω = ω belongs to thewalls of the Weyl chamber [11, Ch. VI, § Corollary 2.26 If β is a dominant weight and ξ = ξ D with ξ ∈ N and ξ > ∣W∣ , then Υ δ ( ξ ( δ + β ) T S ) ≠ . proof: Note that both δ and δ + β are strongly dominant weights. As in the proof of Lemma 2.25.2, wehave Υ δ ( ξ ( δ + β ) T S ) = ∑ σ ∈W det ( σ ) ξ ( δ + β ) T Sσ ( δ ) = ∑ σ ∈W det ( σ ) ξ ⟨ δ + β, σ ( δ )⟩ = ξ D ⟨ δ + β, δ ⟩ ( + ∑ σ ≠ ,σ ∈W det ( σ ) ξ D ⟨ δ + β, σ ( δ )− δ ⟩ ) We now use the fact that for σ ∈ W , σ ( δ ) − δ = − ∑ ρ ∈ B n σρ ρ for some nonnegative integers n σρ . If σ ∈ W , σ ≠ n σρ are zero. Therefore we haveΥ δ ( ξ ( δ + β ) T S ) = ξ ⟨ δ + β, δ ⟩ ( + ∑ σ ≠ ,σ ∈W det ( σ ) ξ ⟨ δ + β, − ∑ ρ ∈ B n σρ ρ ⟩ )= ξ D ⟨ δ + β, δ ⟩ ( + ∑ σ ≠ ,σ ∈∣W∣ det ( σ ) ξ − m σ ) , where each m σ is a positive integer. This follows from the fact that D ⟨ δ + β, ρ ⟩ is always a positive integer ρ ∈ B since δ + β is a strongly dominant weight. Therefore we haveΥ δ ( ξ ( δ + β ) T S ) = ξ D ⟨ δ + β, δ ⟩ ( + ∑ σ ≠ ,σ ∈∣W∣ det ( σ ) ξ − m σ ) ≥ ξ D ⟨ δ + β, δ ⟩ ( − ∣W∣ ξ − ) > . ∎ Theorem 2.27
Let { µ , . . . , µ n } be a basis of strongly dominant weights and let ξ = ( ξ ) D with ξ > ( ∣W∣) .One can effectively determine the dominant weight ω from either of the sets of the numbers { Θ ω ( ξ µ i T S ) ∣ i = , . . . , n } or { Ξ ω ( ξ µ i T S ) ∣ i = , . . . , n } (2.8)19. Hubert & M.F. Singer proof: Lemma 2.25.1 allows us to determine the rational numbers {⟨ µ i , ω ⟩ ∣ i = , . . . , n } from { Θ ω ( ξ µ i T S ) ∣ i = , . . . , n. } . Since the µ i are linearly independent, this allows us to determine ω .To determine the rational numbers {⟨ µ i , ω ⟩ ∣ i = , . . . , n } from { Ξ ω ( ξ µ i T S ) ∣ i = , . . . , n } we proceed asfollows. Since we know δ and the µ i we can evaluate the elements of the set { Υ δ ( ξ µ i T S ) ∣ i = , . . . , n } . TheWeyl Character Formula (Theorem 2.17) then allows us to evaluate Υ δ + ω ( ξ µ i T S ) = Υ δ ( ξ µ i T S ) Ξ ω ( ξ µ i T S ) for i = , . . . , n . Since δ + ω is a dominant weight, Lemma 2.25.2 allows us to determine {⟨ µ i , δ + ω ⟩ ∣ i = , . . . , n } from { Υ δ + ω ( ξ µ i T S ) ∣ i = , . . . , n } . Proceeding as above we can determine ω . ∎ Example 2.28
Following up on Example 2.9. A : We consider the strongly dominant weight µ = . Then for β ∈ N we have Θ β ( ξ ) = ξ β + ξ − β = ξ β ( + ξ − β ) from which we can deduce how to retrieve β for ξ sufficiently large. A : We can choose µ = [ , ] T and µ = [ , ] T as the elements of our basis of strongly dominant weights.To illustrate Theorem 2.27 and the proof of Lemma 2.25, for β = [ β β ] T : Θ β ( ξ µ T S ) = Θ µ ( ξ β T S ) = ξ β + β ( + ξ − β − β + ξ − β − β + ξ − β + ξ − β + ξ − β − β ) , Θ β ( ξ µ T S ) = Θ µ ( ξ β T S ) = ξ ( β + β ) ( + ξ − β + ξ − β + ξ − β − β + ξ − β − β + ξ − β − β ) . For ξ = ξ sufficiently large, the integer part of log ξ ( Θ β ( ξ µ T S )) is β + β and the integer part of log ξ ( Θ β ( ξ µ T S )) is β + β . From these we can determine β and β . B : We choose again { µ = [ , ] T , µ = [ , ] T } . Θ µ ( ξ β T S ) = ξ ( β + β ) ( + ξ − β + ξ − β + ξ − β − β + ξ − β − β + ξ − β − β + ξ − β − β + ξ − β − β ) Θ µ ( ξ β T S ) = ξ ( β + β ) ( + ξ − β + ξ − β + ξ − β − β + ξ − β − β + ξ − β − β + ξ − β − β + ξ − β − β ) .
20 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
We turn to the problem of sparse multivariate interpolation, that is, finding the support (with respect to agiven basis) and the coefficients of a multivariate polynomial from its values at chosen points. In Section 3.1,we consider the case of Laurent polynomials written with respect to the monomial basis. In Sections 3.2and 3.3 we consider the interpolation of a sparse sum of generalized Chebyshev polynomials, of the firstand second kind respectively. In Section 3.4, we discuss an important measure of the complexity of thealgorithms: the number of evaluations to be made.The goal in this section is to recast sparse interpolation into the problem of finding the suport of a (semi-invariant) linear form on the ring of Laurent polynomials. Evaluation of the function to interpolate, atspecific points, gives the values of the linear form on certain polynomials.Multivariate sparse interpolation has been often addressed by reduction to the univariate case [6, 8, 23, 31, 33].The essentially univariate sparse interpolation method initiated in [8] is known to be reminiscent of Prony’smethod [51]. The function f is evaluated at ( p k , . . . , p kn ) , for k = , , , . . . , where the p i are chosen as distinctprime numbers [8], or roots of unity [4, 23].Our approach builds on a multivariate generalization of Prony’s interpolation of sums of exponentials [34,45, 55]. It is designed to take the group invariance into account. This latter is destroyed when reducing to aunivariate problem. The evaluation points to be used for sparsity in the monomial basis are ( ξ α , . . . , ξ α n ) for a chosen ξ ∈ Q , ξ >
1, and for α ranging in an appropriately chosen finite subset of N n related to thepositive orthant of the hypercross C nr = { α ∈ N n ∣ r ∏ i = ( α i + ) ≤ r } . The hypercross and related relevant sets that will appear below are illustrated for n = n = C , C + C and C + C + C .The evaluation points to be used for sparsity in the generalized Chebyshev basis are ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) for ξ = ξ D , ξ ∈ N , ξ > ∣W∣ as described in Theorem 2.27. This can be recognized to generalize sparseinterpolation in terms of univariate Chebyshev polynomials [5, 22, 35, 49].In Section 4 we then show how to recover the support of a linear form. More precisely, we provide thealgorithms to solve the following two problems. Given r ∈ N :1. Consider the unknowns ζ , . . . , ζ r ∈ K n and a , . . . , a r ∈ K . They define the linear formΩ ∶ K [ x ± ] → K p ↦ ∑ ri = a i p ( ζ i )
21. Hubert & M.F. Singerthat we write as Ω = ∑ ri = a i e ζ i , where e ζ i ( p ) = p ( ζ i ) . From the values of Ω on { x α + β + γ ∣ α ∈ C nr , ∣ γ ∣ ≤ } ,Algorithm 4.8 retrieves the set of pairs {( a , ζ ) , . . . , ( a r , ζ r )} .2. Consider the Weyl group W acting on ( K ∗ ) n as in (2.2) , the unknowns ζ , . . . , ζ r ∈ K n and a , . . . , a r ∈ K ∗ . They define the χ -invariant linear formΩ ∶ K [ x ± ] → K p ↦ r ∑ i = a i ∑ A ∈W χ ( A ) p ( A ⋆ ζ i ) that we write as Ω = ∑ ri = a i ∑ A ∈W χ ( A ) e A ⋆ ζ i . From the values of Ω on { Θ α Θ β Θ γ ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } if χ ( A ) =
1, or { Υ δ + α Θ β Θ γ ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } if χ ( A ) = det ( A ) , Algorithm 4.15 retrieves the set of pairs {( ˜ a , ϑ ) , . . . , ( ˜ a r , ϑ r )} , where • ˜ a i = a i Θ ( ζ i ) = a i ∣W∣ or ˜ a i = a i Υ δ ( ζ i ) ≠ χ = • ϑ i = [ Θ ω ( ξ β T i S ) , . . . , Θ ω n ( ξ β T i S )] .The second problem appears as a special case of the first one, yet the special treatment allows one to reducethe size of the matrices by a factor ∣W∣ . These algorithms rely solely on linear algebra operations andevaluations of polynomial functions: • Determine a nonsingular principal submatrix of size r in a matrix of size ∣C nr ∣ ; • Compute the generalized eigenvectors of a pair of matrices of size r × r ; • Solve a nonsingular square linear system of size r .Figure 3.2: C , C + C and C + C + C . Consider a Laurent polynomial in n variables that is r -sparse in the monomial basis. This means that f = r ∑ i = a i x β i , for some a i ∈ K ∗ and β i ∈ Z n . The function it defines is a black box : we can evaluate it at chosen pointsbut know neither its coefficients { a , . . . , a r } ⊂ K ∗ nor its support { β , . . . , β r } ⊂ Z n ; only the size r of itssupport. The problem we address is to find the pairs ( a i , β i ) ∈ K ∗ × Z n from a small set of evaluations of f
22 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsTo f = r ∑ i = a i x β i and ξ ∈ Q , ξ >
1, we associate the linear formΩ ∶ K [ x ± ] → K p ↦ r ∑ i = a i p ( ζ i ) where ζ i = ξ β T i = [ ξ β i, . . . ξ β i,n ] ∈ ( K ∗ ) n . By denoting e ζ the linear form that is the evaluation at ζ ∈ K n we can write Ω = ∑ ri = a i e ζ i . We observe thatΩ ( x α ) = r ∑ i = a i ( ξ β T i ) α = r ∑ i = a i ( ξ α T ) β i = f ( ξ α T ) since β T α = α T β . In other words, the value of Ω on the monomial basis { x α ∣ α ∈ N n } is known from theevaluation of f at the set of points { ξ α T ∣ α ∈ N n } ⊂ ( K ∗ ) n . Though trite in the present case, a commutationproperty such as β T α = α T β is at the heart of sparse interpolation algorithms. Algorithm 3.1
LaurentInterpolation
Input: r ∈ N > , ξ ∈ Q , ξ > , and a function f that can be evaluated at arbitrary points and is known tobe a sum of r monomials. Output:
The pairs ( a , β ) , . . . , ( a r , β r ) ∈ K ∗ × Z n such that f = r ∑ i = a i x β i Perform the evaluations of f on {( ξ ( γ + α + β ) T ) ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } ⊂ Q n .Apply Algorithm 4.8(Support & Coefficients) to determine the pairs ( a , ζ ) , . . . , ( a r , ζ r ) ∈ K ∗ ×( K ∗ ) n suchthat the linear form Ω = ∑ ri = a i e ζ i satisfies Ω ( x α ) = f ( ξ α ) . For ≤ i ≤ r , determine β i from ζ i by taking logarithms. Indeed ζ i = ξ β T i . Hence for ≤ i ≤ r and ≤ j ≤ nζ i,j = ξ β i,j so that β i,j = ln ( ζ i,j ) ln ( ξ ) Example 3.2 In K [ x, y, x − , y − ] , let us consider a 2-sparse polynomial in the monomial basis. Thus f ( x, y ) = a x α y α + b x β y β . We have C = {[ ] T , [ ] T , [ ] T } . Hence { α + β + γ ∣ α, β ∈ C , ∣ γ ∣ ≤ } = {[ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ]} . To retrieve the pairs ( a, α ) and ( b, β ) in K ∗ × N one thus need to evaluate f at the points {[ , ] , [ , ξ ] , [ , ξ ] , [ , ξ ] , [ ξ, ] , [ ξ, ξ ] , [ ξ, ξ ] , [ ξ , ] , [ ξ , ξ ] , [ ξ , ]} ⊂ Q From these values, Algorithm 4.8 will recover the pairs ( a, [ ξ α , ξ α ]) , ( b, [ ξ β , ξ β ]) . Taking some logarithms on this output we get ( a, α ) , ( b, β ) .
23. Hubert & M.F. Singer
We consider now the polynomial ring K [ X ] = K [ X , . . . , X n ] and a black box function F that is a r -sparsepolynomial in the basis of Chebyshev polynomials { T β } β ∈ N n of the first kind associated to the Weyl group W : F ( X , . . . , X n ) = r ∑ i = a i T β i ( X , . . . , X n ) ∈ K [ X , . . . , X n ] . By Definition 2.12, T β ( Θ ω ( x ) , . . . , Θ ω n ( x )) = Θ β ( x ) where Θ β ( x ) = ∑ A ∈W x Aβ . Upon introducing f ( x ) = F ( Θ ω ( x ) , . . . , Θ ω n ( x )) = r ∑ i = a i ∑ A ∈W x Aβ i we could apply Algorithm 3.1 to recover the pairs ( a i , Aβ i ) . Instead we examine how to recover the pairs ( a i , β i ) only. For that we associate to F and ξ ∈ N , ξ >
0, the linear formΩ ∶ K [ x ± ] → K p ↦ r ∑ i = a i ∑ A ∈W p ( A ⋆ ζ i ) where ζ i = ξ β T i S ∈ ( K ∗ ) n . The linear form Ω is W -invariant, that is Ω ( A ⋅ p ) = Ω ( p ) . The property relevant to sparse interpolation isthat the value of Ω on { Θ α } α ∈ N n is obtained by evaluating F . Proposition 3.3 Ω ( Θ α ) = F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) . The proof of this proposition is a consequence of the following commutation property . Lemma 3.4
Consider χ ∶ W → K ∗ a group morphism such that χ ( A ) = for all A ∈ W , and Ψ χα =∑ A ∈W χ ( A ) − x Aα . If S is a positive definite symmetric matrix such that A T SA = S for all A ∈ W , then forany ξ ∈ K ∗ Ψ χα ( ξ β T S ) = Ψ χβ ( ξ α T S ) , where Ψ χα = ∑ B ∈W χ ( B ) − x Bα as defined in (2.3) . proof: We have Ψ χα ( ξ β T S ) = ∑ A ∈W χ ( A ) − ( ξ β T S ) Aα = ∑ A ∈W χ ( A ) − ξ β T SAα . Since A T SA = S , we have SA = A − T S so thatΨ χα ( ξ β T S ) = ∑ A ∈W χ ( A ) − ξ β T A − T Sα = ∑ A ∈W χ ( A ) − ξ ( A − β ) T Sα . Since, trivially, β T Sα = α T Sβ for all α, β ∈ Z n , we haveΨ χα ( ξ β T S ) = ∑ A ∈W χ ( A ) − ξ α T S ( A − β ) = ∑ A ∈W χ ( A ) − ( ξ α T S ) A − β . The conclusion comes from the fact that χ ( A ) = χ ( A ) = χ ( A ) − . ∎
24 Monday 27 thth
24 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials proof of proposition 3.3:
When χ ( A ) = A ∈ W we have Ψ α = Θ α . Therefore Lemma 3.4 implies F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) = r ∑ i = a i T β i ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S ))= r ∑ i = a i Θ β i ( ξ α T S ) = r ∑ i = a i Θ α ( ξ β T i S ) = Ω ( Θ α ) . ∎ In the following algorithm to recover the support of F we need to have the value of Ω on the polynomialsΘ α Θ β Θ γ for α, β ∈ C nr and ∣ γ ∣ ≤
1. We have access to the values of Ω on Θ µ , for any µ ∈ N n , by evaluating F at ( Θ ω ( ξ µ T S ) , . . . , Θ ω n ( ξ µ T S )) . To get the values of Ω on Θ α Θ β Θ γ we consider the relationships stemmingfrom Proposition 2.22 Θ γ Θ α Θ β = ∑ ν ∈ S ( α,β,γ ) a ν Θ ν where S ( α, β, γ ) is a finite subset of { µ ∈ N n ∣ µ ≺ α + β + γ } . Then the set X W r = ⋃ α,β ∈C nr ∣ γ ∣≤ S ( α, β, γ ) (3.1)indexes the evaluations needed to determine the support of a r -sparse sum of Chebyshev polynomials asso-ciated to the Weyl group W .As we noted in the paragraph preceding Lemma 2.25, the entries of S are in Q and we shall denote by D the least common denominator of these entries. Algorithm 3.5
FirstKindInterpolation
Input: r ∈ N > , ξ ∈ N > , where ξ > ( ∣W∣) and ξ = ξ D , and a function F that can be evaluated atarbitrary points and is known to be the sum of r generalized Chebyshev polynomials of the first kind. Output:
The pairs ( a , β ) , . . . , ( a r , β r ) ∈ K ∗ × Z n such that F ( X , . . . , X n ) = r ∑ i = a i T β i ( X , . . . , X n ) . From the evaluations { F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) ∣ α ∈ X W r } determine { Ω ( Θ α Θ β Θ γ ) ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } % The hypothesis on ξ = ξ D guarantees that ξ α T S is a row vector of integers. Apply Algorithm 4.15 (Invariant Support & Coefficients) to calculate the vectors ϑ i = [ Θ ω ( ξ β T i S ) , . . . , Θ ω n ( ξ β T i S )] , ≤ i ≤ r and the vector [ ˜ a , . . . , ˜ a r ] = [∣W∣ a , . . . , ∣W∣ a r ] . Deduce [ a , . . . , a r ] . Calculate [ Θ β i ( ξ µ T S ) , . . . , Θ β i ( ξ µ T n S )] = [ T µ ( ϑ i ) , . . . , T µ n ( ϑ i )] using the precomputed Chebyshev polynomials { T µ , . . . , T µ n } , where µ , . . . , µ n are linearly independentstrongly dominant weights.
25. Hubert & M.F. Singer
Using Theorem 2.27, recover each β i from [ Θ β i ( ξ µ T S ) , . . . , Θ β i ( ξ µ T n S )] . As will be remarked after its description, Algorithm 4.15 may, in some favorable cases, return directly thevector [ Θ µ j ( ξ β T S ) , . . . , Θ µ j ( ξ β T r S )] = [ Θ β ( ξ µ T j S ) , . . . , Θ β r ( ξ µ T j S )] . for some or all 1 ≤ j ≤ n . This then saves on evaluating T µ j at the points ϑ , . . . , ϑ r . Example 3.6
We consider the Chebyshev polynomials of the first kind { T α } α ∈ N associated to the Weylgroup A and a 2-sparse polynomial F ( X, Y ) = a T α ( X, Y ) + b T β ( X, Y ) in this basis of K [ X, Y ] .We need to consider C = {[ ] T , [ ] T , [ ] T } . The following relations Θ , = , , Θ , Θ , = , , Θ , Θ , = , , Θ , = , + , , Θ , Θ , = , + , , Θ , = , + , and Θ , Θ , = , , Θ , Θ , = , + , , Θ , Θ , = , + , , Θ , Θ , = , , Θ , Θ , = , + , + , , Θ , Θ , = , + , + , , Θ , Θ , = , , Θ , Θ , = , + , , Θ , Θ , = , + , , allow one to express any product Θ α Θ β Θ γ , α, β ∈ C n , ∣ γ ∣ ≤ as a linear combination of elements from { Θ α ∣ α ∈ X A } where X A = {[ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ]} . For example Θ , Θ , = , + , + , .We consider f ( x, y ) = F ( Θ ω ( x, y ) , Θ ω ( x, y )) where Θ ω ( x, y ) = x + yx − + y − , and Θ ω ( x, y ) = y + xy − + x − . We introduce the invariant linear form Ω on K [ x, y, x − , y − ] determined by Ω ( Θ γ ) = f ( ξ γ + γ , ξ γ + γ ) The first step of the algorithm requires us to determine { Ω ( Θ α Θ β Θ γ ) ∣ α, β ∈ C , ∣ γ ∣ ≤ } . Expanding thesetriple products as linear combinations of orbit polynomials, we see from Proposition 3.3 that to determinethese values it is enough to evaluate f ( x, y ) at the 10 points { ξ α T S ∣ α ∈ X A } , that is, at the points {[ , ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ, ξ ] , [ ξ , ξ ] , [ ξ, ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ]} Note that D = so ξ = ( ξ ) for some ξ ∈ N > . Therefore the above vectors have integer entries.From these values, Algorithm 4.15 will recover the pairs ( a, ϑ α ) and ( b, ϑ β ) where ϑ α = [ Θ ω ( ξ α T S ) , Θ ω ( ξ α T S )] and ϑ β = [ Θ ω ( ξ β T S ) , Θ ω ( ξ β T S )] .
26 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
One can then form [ T µ ( ϑ α ) , T µ ( ϑ α )] = [ Θ α ( ξ µ T S ) , Θ α ( ξ µ T S )] and [ T µ ( ϑ β ) , T µ ( ϑ β )] = [ Θ β ( ξ µ T S ) , Θ β ( ξ µ T S )] using the polynomials calculated in Example 2.13 and find α and β as illustrated in Example 2.28.Note that the function f is a 12-sparse polynomial in the monomial basis f ( x, y ) = a Θ α ( x ) + b Θ β ( x )= a ( x α y α + x − α y α + α + x α + α y − α + x α y − α − α + x − α − α y α + x − α y − α )+ b ( x β y β + x − β y β + β + x β + β y − β + x β y − β − β + x − β − β y β + x − β y − β ) , Yet to retrieve its support we only need to evaluate f at points indexed by X A , which is equal to C + C + C and has cardinality .Note though that is actually an upper bound on the sparsity of f in the monomial basis. If α or β hasa component that is zero then the actual sparsity can be , , or . We shall comment on dealing withupper bounds on the sparsity rather than the exact sparsity in Section 5. We consider now the polynomial ring K [ X ] = K [ X , . . . , X n ] and a black box function F that is an r -sparsepolynomial in the basis of Chebyshev polynomials { U β } β ∈ N n of the second kind associated to the Weyl group W . Hence F ( X , . . . , X n ) = r ∑ i = a i U β i ( X , . . . , X n ) ∈ K [ X , . . . , X n ] . By Definition 2.16 and thanks to Theorem 2.17 U β ( Θ ω ( x ) , . . . , Θ ω n ( x )) = Ξ β ( x ) = Υ δ + β ( x ) Υ δ ( x ) . Hence uponintroducing f ( x ) = Υ δ ( x ) F ( Θ ω ( x ) , . . . , Θ ω n ( x )) = r ∑ i = a i Υ δ + β i ( x ) = r ∑ i = a i ∑ A ∈W det ( A ) − x A ( δ + β i ) we could apply Algorithm 3.1 to recover the pairs ( a i , A ( δ + β i )) . We examine how to recover only the pairs ( a i , δ + β i ) . For that we defineΩ ∶ K [ x ± ] → K p ↦ r ∑ i = a i ∑ A ∈W det ( A ) p ( ζ Ai ) where ζ i = ξ ( δ + β i ) T S ∈ ( K ∗ ) n . The linear form Ω is skew invariant, i.e. Ω ( A ⋅ p ) = det ( A ) − Ω ( p ) . The property relevant to sparse interpo-lation is that the value of Ω on { Υ α } α ∈ N n is obtained by evaluating F . Proposition 3.7 Ω ( Υ α ) = Υ δ ( ξ α T S ) F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) proof: Note that F ( Θ ω ( x ) , . . . , Θ ω n ( x )) = r ∑ i = a i Ξ β i ( x ) so that Υ δ ( x ) F ( Θ ω ( x ) , . . . , Θ ω n ( x )) = r ∑ i = a i Υ δ + β i ( x ) .27. Hubert & M.F. SingerLemma 3.4 impliesΥ δ ( ξ α T S ) F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) = r ∑ i = a i Υ δ + β i ( ξ α T S )= r ∑ i = a i Υ α ( ξ ( δ + β i ) T S ) = Ω ( Υ α ) . ∎ We are now in a position to describe the algorithm to recover the support of F from its evaluations at aset of points {( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) ∣ α ∈ ˇ X W r } . The set ˇ X W r is defined similarly to the set X W r in theprevious section (Equation (3.1)). ˇ X W r = ⋃ α,β ∈C nr ∣ γ ∣≤ ˇ S ( α, β, γ ) (3.2)where the subsets ˇ S ( α, β, γ ) of { µ ∈ δ + N n ∣ µ ≺ δ + α + β + γ } are defined by the fact thatΥ δ + α Θ β Θ γ = ∑ ν ∈ ˇ S ( α,β,γ ) a ν Υ ν . Algorithm 3.8
SecondKindInterpolation
Input: r ∈ N > , ξ ∈ N > , where ξ > ( ∣W∣) and ξ = ξ D , and a function F that can be evaluated atarbitrary points and is known to be the sum of r generalized Chebyshev polynomials of the second kind. Output:
The pairs ( a , β ) , . . . , ( a r , β r ) ∈ K ∗ × Z n such that F ( X , . . . , X n ) = r ∑ i = a i U β i ( X , . . . , X n ) . From { Υ δ ( ξ α T S ) F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) ∣ α ∈ ˇ X W r } determine { Ω ( Υ δ + α Θ β Θ γ ) ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } Apply Algorithm 4.15 (Invariant Support & Coefficients) to calculate the vectors ˇ ϑ i = [ Θ ω ( ξ ( δ + β i ) T S ) , . . . , Θ ω n ( ξ ( δ + β i ) T S )] , ≤ i ≤ r and the vector [ ˜ a , . . . , ˜ a r ] = [ Υ δ ( ξ ( δ + β ) T S ) a , . . . , Υ δ ( ξ ( δ + β r ) T S ) a r ] Calculate [ Θ δ + β i ( ξ µ T S ) , . . . , Θ δ + β i ( ξ µ T n S )] = [ T µ ( ˇ ϑ i ) , . . . , T µ n ( ˇ ϑ i )] using the Chebyshev polynomials { T µ , . . . , T µ n } , where µ , . . . , µ n are linearly independent strongly dom-inant weights.Using Theorem 2.27, recover each δ + β i , and hence β i , from [ Θ δ + β i ( ξ µ T S ) , . . . , Θ δ + β i ( ξ µ T n S )] . Compute [ Υ δ ( ξ ( δ + β ) T S ) , . . . , Υ δ ( ξ ( δ + β r ) T S )] and deduce [ a , . . . , a r ] . Our hypothesis for ξ imply that ξ > ∣W∣ so Corollary 2.26 implies that none of the components are zero.
28 Monday 27 thth
28 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
Example 3.9
We consider the Chebyshev polynomials of the second kind { U γ } γ ∈ N associated to the Weylgroup A and a 2-sparse polynomial F ( X, Y ) = a U α ( X, Y ) + b U β ( X, Y ) in this basis of K [ X, Y ] .We need to consider C = {[ ] T , [ ] T , [ ] T } . The following relations Υ , Θ , = , , Υ , Θ , = , , Υ , Θ , = , , Υ , Θ , = , , Υ , Θ , = , + , , Υ , Θ , = , + , , Υ , Θ , = , , Υ , Θ , = , + , , Υ , Θ , = , + , and Υ , Θ , = , , Υ , Θ , = , + , , Υ , Θ , = , + , Υ , Θ , = , , Υ , Θ , = , + , + , , Υ , Θ , = , + , + , Υ , Θ , = , , Υ , Θ , = , + , , Υ , Θ , = , + , allow one to express any product Υ δ + α Θ β Θ γ , α, β ∈ C n , ∣ γ ∣ ≤ as a linear combination of elements from { Υ α ∣ α ∈ ˇ X A } where ˇ X A = {[ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ] , [ , ]} . We consider f ( x, y ) = Υ δ ( x, y ) F ( Θ ω ( x, y ) , Θ ω ( x, y )) where Θ ω ( x, y ) = x + yx − + y − , Θ ω ( x, y ) = y + xy − + x − , and Υ δ ( x, y ) = xy − x − y − x y − + xy − + yx − − x − y − . We introduce the χ -invariant linear form Ω on K [ x, y, x − , y − ] determined by Ω ( Υ γ ) = f ( ξ γ + γ , ξ γ + γ ) The first step of the algorithm requires us to determine { Ω ( Υ δ + α Θ β Θ γ ) ∣ α, β ∈ C , ∣ γ ∣ ≤ } . Expanding theseproducts as linear combinations of skew orbit polynomials, we see that it is enough to evaluate f at the 10points { ξ ( δ + α ) T S ∣ α ∈ ˇ X A } , that is, at the points {[ ξ, ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ] , [ ξ , ξ ]} Note that D = so ξ = ( ξ ) for some ξ ∈ N and therefore the above vectors have integer entries.From these values, Algorithm 4.15 will recover the pairs ( Υ δ ( ξ ( δ + α ) T S ) a, ˇ ϑ α ) and ( Υ δ ( ξ ( δ + β ) T S ) b, ˇ ϑ β ) where ˇ ϑ α = [ Θ ω ( ξ ( δ + α ) T S ) , Θ ω ( ξ ( δ + α ) T S ) and ˇ ϑ β = [ Θ ω ( ξ ( δ + β ) T S ) , Θ ω ( ξ ( δ + β ) T S )] . One then can form [ T µ ( ˇ ϑ α ) , T µ ( ˇ ϑ α )] = [ Θ δ + α ( ξ µ T S ) , Θ δ + α ( ξ µ T S )] and [ T µ ( ˇ ϑ β ) , T µ ( ˇ ϑ β )] = [ Θ δ + β ( ξ µ T S ) , Θ δ + β ( ξ µ T S )] using the polynomials calculated in Example 2.13 and find δ + α and δ + β as illustrated in Example 2.28.We can then compute Υ δ ( ξ ( δ + α ) T S ) and Υ δ ( ξ ( δ + β ) T S ) and hence a and b .Note that the function f is a 12-sparse polynomial in the monomial basis since f ( x, y ) = a Υ δ + α ( x )+ b Υ δ + β ( x ) . Yet to retrieve its support we only need to evaluate f at points indexed by ˇ X A that has cardinality .
29. Hubert & M.F. Singer
There are two factors that are the main contributions to the cost of the algorithms described above: thecost of the linear algebra operations in Algorithm 4.8 or Algorithm 4.15 and the needed number of functionevaluations.For Algorithm 3.1, one calls upon the linear algebra operations of Algorithm 4.8 to calculate the supportand coefficients of the sparse polynomial that is being interpolated. This involves one ∣C nr ∣ × ∣C nr ∣ matrix andseveral of its r × r submatrices. Algorithm 4.8 is fed with the evaluation at the points { ξ ( γ + α + β ) T ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } ⊂ Q n . Since ∣C nr ∣ ≤ r log n − ( r ) [40, Lemma 1.4], ∣C nr + C nr ∣ ≤ r log n − ( r ) and ∣C nr + C nr + C n ∣ ≤ ( n + ) r log n − ( r ) .This latter number is a crude upper bound on the number of evaluations of f in Algorithm 3.1. This boundwas given in [55] in the context of the multivariate generalization of Prony’s method.Turning to sums of Chebyshev polynomials of the first kind, we wish to compare the cost of the interpolationof the r -sparse polynomial F = ∑ ri = a i T β i , with Algorithm 3.5, to the cost of the the r ∣W∣ -sparse polynomial f ( x ) = ∑ ri = ∑ A ∈W a i x Aβ i , with Algorithm 3.1. The analysis for the sparse interpolation of F = ∑ ri = a i U β i with Algorithm 3.8 compared with the sparse interpolation of f ( x ) = ∑ ri = ∑ A ∈W a i det ( A ) x Aβ i with Algo-rithm 3.1 is the same.First note that Algorithm 4.15 will involve a matrix of the size ∣C nr ∣ and some of its submatrices of size r .This is to be constrasted with Algorithm 3.1 involving in theses cases a matrix of size ∣C n ∣W∣ r ∣ and some of itssubmatrices of size ∣W∣ r .The number of evaluations is the cardinality of X W r defined by Equation (3.1). X W r is a superset of C nr +C nr +C n .In the case where W is B or A , X W r is a proper superset and the discrepancy is illustrated in Figure 3.3and 3.4. On the other hand there is experimental evidence that X A r is equal to C nr + C nr + C n . The terms thatappear in the sets S ( α, β, ) , S ( α, β, ω ) , . . . , S ( α, β, ω n ) (see the definition of X W r given by Equation (3.1))and hence in X W r strongly depend on the group W . Specific analysis for each group would provide a refinedbound on the cardinal of X W r .Nonetheless, taking the group structure and action of W into account, one can make the following estimate.Proposition 2.22 implies that S ( α, β, ) is of cardinality at most ∣W∣ while S ( α, β, γ ) is bounded by ∣W∣ in general. Yet, the isotropy group W ω i of ω i is rather large: among the n generators of the group, n − ω i unchanged. Since W ω i contains the identity as well we have ∣W ω i ∣ ≥ n . Therefore ∣ S ( α, β, ω i )∣ ≤∣ S ( α, β, )∣∣W/W ω i ∣ ≤ n ∣W∣ . Hence ∣ X W r ∣ = RRRRRRRRRRRRRRRRR ⋃ α,β ∈C nr ∣ γ ∣≤ S ( α, β, γ )RRRRRRRRRRRRRRRRR ≤ RRRRRRRRRRRRRRRRR ⋃ α,β ∈C nr i = ,...,n S ( α, β, ω i )RRRRRRRRRRRRRRRRR + RRRRRRRRRRRR ⋃ α,β ∈C nr S ( α, β, )RRRRRRRRRRRR≤ ( n ( n ∣W∣ ) + ∣W∣) r log n − ( r ) ≤ (∣W∣ r ) log n − ( r ) . This is to be compared to interpolating a ∣W∣ r -sparse Laurent polynomial that would use at most ∣ C n ∣W∣ r + C n ∣W∣ r + C n ∣ ≤ ( n + ) (∣W∣ r ) log n − (∣ W ∣ r ) evaluations. Therefore, even with this crude estimate, the number of evaluations to be performed to applyAlgorithm 3.5 is less than with the approach using Algorithm 3.1 considering the given polynomial as beinga ∣W∣ r -sparse Laurent polynomial.30 Monday 27 thth
There are two factors that are the main contributions to the cost of the algorithms described above: thecost of the linear algebra operations in Algorithm 4.8 or Algorithm 4.15 and the needed number of functionevaluations.For Algorithm 3.1, one calls upon the linear algebra operations of Algorithm 4.8 to calculate the supportand coefficients of the sparse polynomial that is being interpolated. This involves one ∣C nr ∣ × ∣C nr ∣ matrix andseveral of its r × r submatrices. Algorithm 4.8 is fed with the evaluation at the points { ξ ( γ + α + β ) T ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } ⊂ Q n . Since ∣C nr ∣ ≤ r log n − ( r ) [40, Lemma 1.4], ∣C nr + C nr ∣ ≤ r log n − ( r ) and ∣C nr + C nr + C n ∣ ≤ ( n + ) r log n − ( r ) .This latter number is a crude upper bound on the number of evaluations of f in Algorithm 3.1. This boundwas given in [55] in the context of the multivariate generalization of Prony’s method.Turning to sums of Chebyshev polynomials of the first kind, we wish to compare the cost of the interpolationof the r -sparse polynomial F = ∑ ri = a i T β i , with Algorithm 3.5, to the cost of the the r ∣W∣ -sparse polynomial f ( x ) = ∑ ri = ∑ A ∈W a i x Aβ i , with Algorithm 3.1. The analysis for the sparse interpolation of F = ∑ ri = a i U β i with Algorithm 3.8 compared with the sparse interpolation of f ( x ) = ∑ ri = ∑ A ∈W a i det ( A ) x Aβ i with Algo-rithm 3.1 is the same.First note that Algorithm 4.15 will involve a matrix of the size ∣C nr ∣ and some of its submatrices of size r .This is to be constrasted with Algorithm 3.1 involving in theses cases a matrix of size ∣C n ∣W∣ r ∣ and some of itssubmatrices of size ∣W∣ r .The number of evaluations is the cardinality of X W r defined by Equation (3.1). X W r is a superset of C nr +C nr +C n .In the case where W is B or A , X W r is a proper superset and the discrepancy is illustrated in Figure 3.3and 3.4. On the other hand there is experimental evidence that X A r is equal to C nr + C nr + C n . The terms thatappear in the sets S ( α, β, ) , S ( α, β, ω ) , . . . , S ( α, β, ω n ) (see the definition of X W r given by Equation (3.1))and hence in X W r strongly depend on the group W . Specific analysis for each group would provide a refinedbound on the cardinal of X W r .Nonetheless, taking the group structure and action of W into account, one can make the following estimate.Proposition 2.22 implies that S ( α, β, ) is of cardinality at most ∣W∣ while S ( α, β, γ ) is bounded by ∣W∣ in general. Yet, the isotropy group W ω i of ω i is rather large: among the n generators of the group, n − ω i unchanged. Since W ω i contains the identity as well we have ∣W ω i ∣ ≥ n . Therefore ∣ S ( α, β, ω i )∣ ≤∣ S ( α, β, )∣∣W/W ω i ∣ ≤ n ∣W∣ . Hence ∣ X W r ∣ = RRRRRRRRRRRRRRRRR ⋃ α,β ∈C nr ∣ γ ∣≤ S ( α, β, γ )RRRRRRRRRRRRRRRRR ≤ RRRRRRRRRRRRRRRRR ⋃ α,β ∈C nr i = ,...,n S ( α, β, ω i )RRRRRRRRRRRRRRRRR + RRRRRRRRRRRR ⋃ α,β ∈C nr S ( α, β, )RRRRRRRRRRRR≤ ( n ( n ∣W∣ ) + ∣W∣) r log n − ( r ) ≤ (∣W∣ r ) log n − ( r ) . This is to be compared to interpolating a ∣W∣ r -sparse Laurent polynomial that would use at most ∣ C n ∣W∣ r + C n ∣W∣ r + C n ∣ ≤ ( n + ) (∣W∣ r ) log n − (∣ W ∣ r ) evaluations. Therefore, even with this crude estimate, the number of evaluations to be performed to applyAlgorithm 3.5 is less than with the approach using Algorithm 3.1 considering the given polynomial as beinga ∣W∣ r -sparse Laurent polynomial.30 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsFigure 3.3: X B r , for r ∈ { , , } : the elements that do not belong to C r + C r + C are represented by carminsquares. Figure 3.4: X A r , for r ∈ { , , } : the elements in purple do not belong to C r + C r + C . 31. Hubert & M.F. Singer In Section 3 we converted the recovery of the support of a polynomial in the monomial or Chebyshev basesto the recovery of the support of a linear form. For f ( x ) = r ∑ i = a i x β i , F ( X ) = r ∑ i = a i T β i ( X ) , or F ( X ) = r ∑ i = a i U β i ( X ) we respectively introduced the linear forms on K [ x ± ] Ω = r ∑ i = a i e ζ i , r ∑ i = a i ∑ A ∈W e A ⋆ ζ i , or r ∑ i = a i ∑ A ∈W det ( A ) e A ⋆ ζ i where, for some chosen ξ ∈ N > , ζ i = ξ β T i , ζ i = ξ β T i S , or ζ i = ξ ( δ + β i ) T S . The linear forms are known from theirvalues at some polynomials, respectively:Ω ( x α ) = f ( ξ α ) , Ω ( Θ α ) = F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) , or Ω ( Υ α ) = Υ δ ( ξ α T S ) F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) . This section provides the technology to recover the support of these linear forms. We shall either retrieve { ζ , . . . , ζ r } ⊂ Q n , or {( Θ ω ( ζ i ) , . . . , Θ ω n ( ζ i )) ∣ ≤ i ≤ r } ⊂ N n Identifying the support of a linear form on a polynomial ring K [ x ] already has applications in optimization,tensor decomposition and cubature [1, 2, 9, 13, 15, 36, 37]. How to take advantage of symmetry in some ofthese applications appears in [14, 21, 52]. To a linear form Ω ∶ K [ x ] → K one associates [13, 15, 45, 50] aHankel operator ̂H ∶ K [ x ] → K [ x ] ∗ whose kernel I Ω is the ideal of the support { ζ , . . . , ζ n } of Ω. We cancompute directly these points as eigenvalues of the multiplication maps on the quotient algebra K [ x ]/ I Ω .The present application to sparse interpolation is related to a multivariate version of Prony’s method, astackled in [34, 45, 55]. Contrary to the previously mentioned applications, where the symmetry is given bythe linear action of a finite group on the ambient space of the support, here the Weyl groups act linearlyon K [ x ± ] but nonlinearly on the ambient space of the support. Thanks to Theorem 2.27, we can satisfyourselves with recovering only the values of the freely generating invariant polynomials on the support, i.e. {( Θ ω ( ζ i ) , . . . , Θ ω n ( ζ i )) ∣ ≤ i ≤ r } .In Section 4.1 we review the definitions of Hankel operators associated to a linear form, multiplicationmaps and their relationship to each other in the context of K [ x ± ] rather than K [ x ] . In Section 4.2 wepresent an algorithm to calculate the matrix representation of certain multiplication maps and determinethe support of the original linear form as eigenvalues of the multiplication maps. In Section 4.3, to retrievethe orbits forming the support of an invariant or semi-invariant form, we introduce the Hankel operators ̂H ∶ K [ x ± ] W → ( K [ x ± ] W χ ) ∗ , where K [ x ± ] W is the ring of invariants for the action of the Weyl group W on K [ x ± ] ; K [ x ± ] W χ is the K [ x ± ] W -module of χ -invariant polynomials where χ ∶ W → { , − } is either given by χ ( A ) = χ ( A ) = det A , depending whether we consider Chebyshev polynomials of the first or second kind.In this latter section, in analogy to the previous sections, we also introduce the appropriate multiplicationmaps and give an algorithm to determine the support of the associated χ -invariant linear from in terms ofthe eignevalues of these multiplication maps.The construction could be extended to other group actions on the ring of (Laurent) polynomials. Yet weshall make use of the fact that, for a Weyl group W , K [ x ± ] W is isomorphic to a polynomial ring and K [ x ± ] W χ is a free K [ x ± ] W -module of rank one.32 Monday 27 thth
There are two factors that are the main contributions to the cost of the algorithms described above: thecost of the linear algebra operations in Algorithm 4.8 or Algorithm 4.15 and the needed number of functionevaluations.For Algorithm 3.1, one calls upon the linear algebra operations of Algorithm 4.8 to calculate the supportand coefficients of the sparse polynomial that is being interpolated. This involves one ∣C nr ∣ × ∣C nr ∣ matrix andseveral of its r × r submatrices. Algorithm 4.8 is fed with the evaluation at the points { ξ ( γ + α + β ) T ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } ⊂ Q n . Since ∣C nr ∣ ≤ r log n − ( r ) [40, Lemma 1.4], ∣C nr + C nr ∣ ≤ r log n − ( r ) and ∣C nr + C nr + C n ∣ ≤ ( n + ) r log n − ( r ) .This latter number is a crude upper bound on the number of evaluations of f in Algorithm 3.1. This boundwas given in [55] in the context of the multivariate generalization of Prony’s method.Turning to sums of Chebyshev polynomials of the first kind, we wish to compare the cost of the interpolationof the r -sparse polynomial F = ∑ ri = a i T β i , with Algorithm 3.5, to the cost of the the r ∣W∣ -sparse polynomial f ( x ) = ∑ ri = ∑ A ∈W a i x Aβ i , with Algorithm 3.1. The analysis for the sparse interpolation of F = ∑ ri = a i U β i with Algorithm 3.8 compared with the sparse interpolation of f ( x ) = ∑ ri = ∑ A ∈W a i det ( A ) x Aβ i with Algo-rithm 3.1 is the same.First note that Algorithm 4.15 will involve a matrix of the size ∣C nr ∣ and some of its submatrices of size r .This is to be constrasted with Algorithm 3.1 involving in theses cases a matrix of size ∣C n ∣W∣ r ∣ and some of itssubmatrices of size ∣W∣ r .The number of evaluations is the cardinality of X W r defined by Equation (3.1). X W r is a superset of C nr +C nr +C n .In the case where W is B or A , X W r is a proper superset and the discrepancy is illustrated in Figure 3.3and 3.4. On the other hand there is experimental evidence that X A r is equal to C nr + C nr + C n . The terms thatappear in the sets S ( α, β, ) , S ( α, β, ω ) , . . . , S ( α, β, ω n ) (see the definition of X W r given by Equation (3.1))and hence in X W r strongly depend on the group W . Specific analysis for each group would provide a refinedbound on the cardinal of X W r .Nonetheless, taking the group structure and action of W into account, one can make the following estimate.Proposition 2.22 implies that S ( α, β, ) is of cardinality at most ∣W∣ while S ( α, β, γ ) is bounded by ∣W∣ in general. Yet, the isotropy group W ω i of ω i is rather large: among the n generators of the group, n − ω i unchanged. Since W ω i contains the identity as well we have ∣W ω i ∣ ≥ n . Therefore ∣ S ( α, β, ω i )∣ ≤∣ S ( α, β, )∣∣W/W ω i ∣ ≤ n ∣W∣ . Hence ∣ X W r ∣ = RRRRRRRRRRRRRRRRR ⋃ α,β ∈C nr ∣ γ ∣≤ S ( α, β, γ )RRRRRRRRRRRRRRRRR ≤ RRRRRRRRRRRRRRRRR ⋃ α,β ∈C nr i = ,...,n S ( α, β, ω i )RRRRRRRRRRRRRRRRR + RRRRRRRRRRRR ⋃ α,β ∈C nr S ( α, β, )RRRRRRRRRRRR≤ ( n ( n ∣W∣ ) + ∣W∣) r log n − ( r ) ≤ (∣W∣ r ) log n − ( r ) . This is to be compared to interpolating a ∣W∣ r -sparse Laurent polynomial that would use at most ∣ C n ∣W∣ r + C n ∣W∣ r + C n ∣ ≤ ( n + ) (∣W∣ r ) log n − (∣ W ∣ r ) evaluations. Therefore, even with this crude estimate, the number of evaluations to be performed to applyAlgorithm 3.5 is less than with the approach using Algorithm 3.1 considering the given polynomial as beinga ∣W∣ r -sparse Laurent polynomial.30 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsFigure 3.3: X B r , for r ∈ { , , } : the elements that do not belong to C r + C r + C are represented by carminsquares. Figure 3.4: X A r , for r ∈ { , , } : the elements in purple do not belong to C r + C r + C . 31. Hubert & M.F. Singer In Section 3 we converted the recovery of the support of a polynomial in the monomial or Chebyshev basesto the recovery of the support of a linear form. For f ( x ) = r ∑ i = a i x β i , F ( X ) = r ∑ i = a i T β i ( X ) , or F ( X ) = r ∑ i = a i U β i ( X ) we respectively introduced the linear forms on K [ x ± ] Ω = r ∑ i = a i e ζ i , r ∑ i = a i ∑ A ∈W e A ⋆ ζ i , or r ∑ i = a i ∑ A ∈W det ( A ) e A ⋆ ζ i where, for some chosen ξ ∈ N > , ζ i = ξ β T i , ζ i = ξ β T i S , or ζ i = ξ ( δ + β i ) T S . The linear forms are known from theirvalues at some polynomials, respectively:Ω ( x α ) = f ( ξ α ) , Ω ( Θ α ) = F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) , or Ω ( Υ α ) = Υ δ ( ξ α T S ) F ( Θ ω ( ξ α T S ) , . . . , Θ ω n ( ξ α T S )) . This section provides the technology to recover the support of these linear forms. We shall either retrieve { ζ , . . . , ζ r } ⊂ Q n , or {( Θ ω ( ζ i ) , . . . , Θ ω n ( ζ i )) ∣ ≤ i ≤ r } ⊂ N n Identifying the support of a linear form on a polynomial ring K [ x ] already has applications in optimization,tensor decomposition and cubature [1, 2, 9, 13, 15, 36, 37]. How to take advantage of symmetry in some ofthese applications appears in [14, 21, 52]. To a linear form Ω ∶ K [ x ] → K one associates [13, 15, 45, 50] aHankel operator ̂H ∶ K [ x ] → K [ x ] ∗ whose kernel I Ω is the ideal of the support { ζ , . . . , ζ n } of Ω. We cancompute directly these points as eigenvalues of the multiplication maps on the quotient algebra K [ x ]/ I Ω .The present application to sparse interpolation is related to a multivariate version of Prony’s method, astackled in [34, 45, 55]. Contrary to the previously mentioned applications, where the symmetry is given bythe linear action of a finite group on the ambient space of the support, here the Weyl groups act linearlyon K [ x ± ] but nonlinearly on the ambient space of the support. Thanks to Theorem 2.27, we can satisfyourselves with recovering only the values of the freely generating invariant polynomials on the support, i.e. {( Θ ω ( ζ i ) , . . . , Θ ω n ( ζ i )) ∣ ≤ i ≤ r } .In Section 4.1 we review the definitions of Hankel operators associated to a linear form, multiplicationmaps and their relationship to each other in the context of K [ x ± ] rather than K [ x ] . In Section 4.2 wepresent an algorithm to calculate the matrix representation of certain multiplication maps and determinethe support of the original linear form as eigenvalues of the multiplication maps. In Section 4.3, to retrievethe orbits forming the support of an invariant or semi-invariant form, we introduce the Hankel operators ̂H ∶ K [ x ± ] W → ( K [ x ± ] W χ ) ∗ , where K [ x ± ] W is the ring of invariants for the action of the Weyl group W on K [ x ± ] ; K [ x ± ] W χ is the K [ x ± ] W -module of χ -invariant polynomials where χ ∶ W → { , − } is either given by χ ( A ) = χ ( A ) = det A , depending whether we consider Chebyshev polynomials of the first or second kind.In this latter section, in analogy to the previous sections, we also introduce the appropriate multiplicationmaps and give an algorithm to determine the support of the associated χ -invariant linear from in terms ofthe eignevalues of these multiplication maps.The construction could be extended to other group actions on the ring of (Laurent) polynomials. Yet weshall make use of the fact that, for a Weyl group W , K [ x ± ] W is isomorphic to a polynomial ring and K [ x ± ] W χ is a free K [ x ± ] W -module of rank one.32 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
We consider a commutative K -algebra R and a R -module S . R will later be either K [ x ± ] or the invariantsubring K [ x ± ] W while S will be either K [ x ± ] , K [ x ± ] W or Υ δ K [ x ± ] W , the module of skew-invariant poly-nomials (Lemma 2.21). Hence S shall be a free R -modules of rank one: there is an element that we shalldenote Υ in S s.t. S = Υ R . In the cases of interest Υ is either 1 or Υ δ . We shall keep the explicit mentionof Υ though the R -module isomorphism between R and S would allow us to forego the use of S . R and S are also K -vector spaces. To a K -linear form Ω on S we associate a Hankel operator ̂H ∶ R → S ∗ ,where S ∗ is the dual of S , i.e. the K -vector space of K -linear forms on S . The kernel of this operator is anideal I Ω in R , considered as a ring. The matrices of the multiplication maps in R/ I Ω are given in terms ofthe matrix of ̂H . Hankel operator
For a linear form Ω ∈ S ∗ , the associated Hankel operator ̂H is the K -linear map ̂H ∶ R → S ∗ p ↦ Ω p , where Ω p ∶ S → K q ↦ Ω ( q p ) . If U and V are K -linear subspaces of R and S respectively we define ̂H ∣ U,V using the restrictions Ω p ∣ V of Ω p to V : ̂H ∣ U,V ∶ U → V ∗ p ↦ Ω p ∣ V , Assume U is the K -linear span ⟨ B ⟩ = ⟨ b , . . . , b r ⟩ of a linearly independent set B = { b , . . . , b r } in R and V isthe K -linear span ⟨ Υ c , . . . , Υ c s ⟩ in S , denoted ⟨ Υ C ⟩ , where C = { c , . . . , c s } is a linearly independent subsetof R . Then the matrix of ̂H ∣ U,V in B and the dual basis of Υ C is H C,B = ( Ω ( Υ c i b j )) ≤ i ≤ s ≤ j ≤ r . The kernel of ̂H I Ω = { p ∈ R ∣ Ω p = } = { p ∈ R ∣ Ω ( q p ) = , ∀ q ∈ S} is an ideal of R . We shall consider both the quotient spaces R/ I Ω and S/ Υ I Ω where Υ I Ω is the submodule I Ω S of S . Lemma 4.1
The image of ̂H lies in ( Υ I Ω ) ⊥ and ̂H induces an injective morphism H ∶ R/ I Ω → (S/ Υ I Ω ) ∗ that has the following diagram commute. R ( Υ I Ω ) ⊥ (S/ Υ I Ω ) ∗ R/ I Ω ̂H ≅ π H proof: A basis of R/ I Ω is the image by the natural projection π ∶ R → R/ I Ω of a linearly independent set C ⊂ R s.t. R = ⟨ C ⟩ ⊕ I Ω . Hence S = ⟨ Υ C ⟩ ⊕ Υ I Ω . Recall from linear algebra, see for instance [24, PropositionV, Section 2.30], that this latter equality implies: S ∗ = ( Υ I Ω ) ⊥ ⊕ ( Υ C ) ⊥ and ( Υ I Ω ) ⊥ → ⟨ Υ C ⟩ ∗ Φ ↦ Φ ∣⟨ Υ C ⟩ is an isomorphism, 33. Hubert & M.F. Singerwhere, for any set V ⊂ S , V ⊥ = { Φ ∈ S ∗ ∣ Φ ( v ) = , ∀ v ∈ V } is a K -linear subspace of S ∗ .Note that the image of ̂H lies in ( Υ I Ω ) ⊥ . With the natural identification of ⟨ Υ C ⟩ ∗ with (S/ Υ I Ω ) ∗ , thefactorisation of ̂H by the natural projection π ∶ R → R/ I Ω defines the injective morphism H ∶ R/ I Ω →(S/ Υ I Ω ) ∗ through the announced commuting diagram. ∎ If the rank of ̂H is finite and equal to r , then the dimension of R/ I Ω , S/ Υ I Ω and (S/ Υ I Ω ) ∗ , as K -vectorspaces, is r and the injective linear operator H is then an isomorphism. The point here is the followingcriterion for detecting bases of R/ I Ω . Theorem 4.2
Assume that rank ̂H = r < ∞ and consider B = { b , . . . , b r } and C = { c , . . . , c r } subsets of R . Then the image of B and C by π ∶ R → R/ I Ω are both bases of R/ I Ω if and only if the matrix H C,B isnon-singular. proof: Assume that B and C are both bases for R/ I Ω , we can identify R/ I Ω with ⟨ B ⟩ and S/ Υ I Ω with ⟨ Υ C ⟩ . Hence H C,B is the matrix of H in the basis B and the dual basis of Υ C . Since H is an isomorphism, H C,B is nonsingular.Assume H C,B is nonsingular. We need to show that B and C are linearly independent modulo I Ω , i.e. thatany linear combination of their elements that belongs to the ideal is trivial. Take a = ( a , . . . , a r ) ∈ K r suchthat a b + ⋅ ⋅ ⋅ + a r b r ∈ I Ω . Using the definition of I Ω , we get a Ω ( Υ c i b ) + ⋅ ⋅ ⋅ + a r Ω ( Υ c i b r ) = ∀ i = , . . . , r .These equalities amount to H C,B a = a =
0. Similarly a c + ⋅ ⋅ ⋅ + a r c r ≡ I Ω leads to a T H C,B = a = ∎ Multiplication maps
We now assume that the Hankel operator ̂H associated to Ω has finite rank r . Then R/ I Ω is of dimension r when considered as a linear space over K . For p ∈ R , consider the multiplication map ̂M p ∶ R → R q ↦ q p and M p ∶ R/ I Ω → R/ I Ω q ′ ↦ π ( q p ) where q ∈ R satisfies π ( q ) = q ′ . (4.1) M p is a well defined linear map respecting the following commuting diagram [17, Proposition 4.1] R RR/ I Ω R/ I Ω ̂M p ππ M p Let us temporarily introduce the Hankel operator ̂H p associated to Ω p . This is the map defined by ̂H p =̂H ○ ̂M p . Therefore the image of ̂H p is included in the image of ̂H and ker ̂H ⊂ ker ̂H p . We can thus constructthe maps H p that satisfy ̂H p = H p ○ π . Then H p = H ○ M p and we have the following commuting diagram.34 Monday 27 thth
We now assume that the Hankel operator ̂H associated to Ω has finite rank r . Then R/ I Ω is of dimension r when considered as a linear space over K . For p ∈ R , consider the multiplication map ̂M p ∶ R → R q ↦ q p and M p ∶ R/ I Ω → R/ I Ω q ′ ↦ π ( q p ) where q ∈ R satisfies π ( q ) = q ′ . (4.1) M p is a well defined linear map respecting the following commuting diagram [17, Proposition 4.1] R RR/ I Ω R/ I Ω ̂M p ππ M p Let us temporarily introduce the Hankel operator ̂H p associated to Ω p . This is the map defined by ̂H p =̂H ○ ̂M p . Therefore the image of ̂H p is included in the image of ̂H and ker ̂H ⊂ ker ̂H p . We can thus constructthe maps H p that satisfy ̂H p = H p ○ π . Then H p = H ○ M p and we have the following commuting diagram.34 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
R ( Υ I Ω ) ⊥ R (S/ Υ I Ω ) ∗ R/ I Ω R/ I Ω ̂H p ≅H p π ̂M p M p ̂HH π Theorem 4.3
Assume the Hankel operator ̂H associated to the linear form Ω has finite rank r . Let B ={ b , . . . , b r } and C = { c , . . . , c r } be bases of R/ I Ω . Then the matrix M Bp of the multiplication by an element p of R in R/ I Ω is given by M Bp = ( H C,B ) − H C,Bp where H C,B = ( Ω ( Υ c i b j )) ≤ i,j ≤ r and H C,Bp = ( Ω ( Υ c i b j p )) ≤ i,j ≤ r proof: The matrix of H p in B and the dual basis of Υ C is H C,Bp . Then H C,Bp = H C,B M Bp since H p = H○M p .From Theorem 4.2, H C,B is invertible. ∎ K [ x ± ] We now consider R and S to be the ring of Laurent polynomials K [ x ± ] . As before, the evaluations e ζ ∶ K [ x ± ] → K at a point ζ ∈ ( K ∗ ) n are defined as follow: For p ∈ K [ x ± ] , e ζ ( p ) = p ( ζ ) . For a , . . . , a r ∈ K ∗ anddistinct ζ , . . . , ζ r ∈ ( K ∗ ) n we write Ω = ∑ ri = a i e ζ i for the linear formΩ ∶ K [ x ± ] → K p ↦ r ∑ i = a i p ( ζ i ) . In this section we characterize such a linear form in terms of its associated Hankel operator. We show howto compute ζ , . . . , ζ r from the knowledge of the values of Ω on a finite dimensional subspace of K [ x ± ] . If Ω = r ∑ i = a i e ζ i , where a i ∈ K ∗ , and ζ , . . . , ζ r are distinct points in ( K ∗ ) n then the associatedHankel operator ̂H has finite rank r and its kernel I Ω is the annihilating ideal of { ζ , . . . , ζ r } . proof: It is easy to see that p ( ζ ) = . . . = p ( ζ r ) = p ∈ I Ω . For the converse inclusion, considersome interpolation polynomials p , . . . , p r at ζ , . . . , ζ r , i.e. p i ( ζ i ) = p j ( ζ i ) = i ≠ j [17, Lemma2.9]. For q ∈ I Ω we have Ω ( q p i ) = a i q ( ζ i ) =
0. Hence I Ω is the annihilating ideal of { ζ , . . . , ζ r } .It is thus a radical ideal with dim K K [ x ± ]/ I Ω = r . ∎ Theorem 4.2 gives necessary and sufficient condition for a set B = { b , . . . , b r } in K [ x ± ] to be a basis of K [ x ± ]/ I Ω when the dimension of this latter, as a K -vector space, is r . This condition is that the matrix H B = ( Ω ( b i b j )) ≤ i,j ≤ r is nonsingular. The problem of where to look for this basis was settled in [55] wherethe author introduces lower sets and the positive octant of the hypercross of order r . 35. Hubert & M.F. SingerA subset Γ of N n is a lower set if whenever α + β ∈ Γ, α, β ∈ N n , then α ∈ Γ. The positive octant of thehypercross of order r is C nr = { α ∈ N n ∣ r ∏ i = ( α i + ) ≤ r } . It is the union all the lower sets of cardinality r or less [55, Lemma 10]. We extend [55, Corollary 11] forfurther use in Section 4.3. Proposition 4.5
Let ⩽ be an order on N n such that ⩽ γ and α ⩽ β ⇒ α + γ ⩽ β + γ for all α, β, γ ∈ N n .Consider two families of polynomials { P α ∣ α ∈ N n } and { Q α ∣ α ∈ N n } in K [ X ] such that P α = ∑ β ⩽ α p β X β and Q α = ∑ β ⩽ α q β X β with p α , q α ≠ .If J is an ideal in K [ X ] = K [ X , . . . , X n ] such that dim K K [ X ]/ J = r then there exists a lower set Γ ofcardinal r such that both { P α ∣ α ∈ Γ } and { Q α ∣ α ∈ Γ } are bases of K [ X ]/ J . proof: For the chosen term order ⩽ , a Gr¨obner basis of J defines a lower set Γ that has r monomials andis a basis of K [ x ]/ J [17, Chapter 2].Consider a polynomial P = ∑ β ∈ Γ a β P β , for some a β ∈ K not all zero. Take α to be highest element of Γ forwhich a α ≠
0. Then X α is the leading term of P . As X α does not belong to the initial ideal, P ∉ J [16,Chapter 2]. It follows that { P α ∣ α ∈ Γ } is linearly independent modulo J and hence is a basis of K [ X ]/ J .The same is applies to { Q α ∣ α ∈ Γ } . ∎ Corollary 4.6 If I is an ideal in K [ x ± ] such that dim K K [ x ± ]/ I = r then K [ x ± ]/ I admits a basis in { x α ∣ α ∈ C nr } . proof: A monomial basis of K [ x ]/ J , where J = I ∩ K [ x ] , is a basis for K [ x ± ]/ I . ∎ The eigenvalues of the multiplication map M p , introduced in Equation (4.1), are the values of p on thevariety of I Ω ; as I Ω is a radical ideal, this is part of the following result, which is a simple extension of [17,Chapter 2, Proposition 4.7 ] to the Laurent polynomial ring. The proof appears as a special case of the laterProposition 4.14. Theorem 4.7
Let I be a radical ideal in K [ x ± ] whose variety consists of r distinct points ζ , . . . , ζ r in ( ¯ K ∗ ) n then: • A set B = { b , . . . , b r } is a basis of K [ x ± ]/ I if and only if the matrix W Bζ = ( b j ( ζ i )) ≤ i,j ≤ r is non singular; • The matrix M Bp of the multiplication M p by p in a basis B of K [ x ± ]/ I satisfies W Bζ M Bp = D pζ W Bζ where D pζ is the diagonal matrix diag ( p ( ζ ) , . . . , p ( ζ r )) . This theorem gives us a basis of left eigenvectors for M Bp : The i -th row of W Bζ , [ b ( ζ i ) . . . b r ( ζ i )] , is aleft eigenvector associated to the eigenvalue p ( ζ i ) . One can furthermore observe that H C,B = ( W Cζ ) T A W Bζ where A = diag ( a , . . . , a r ) . (4.2)36 Monday 27 thth
Let I be a radical ideal in K [ x ± ] whose variety consists of r distinct points ζ , . . . , ζ r in ( ¯ K ∗ ) n then: • A set B = { b , . . . , b r } is a basis of K [ x ± ]/ I if and only if the matrix W Bζ = ( b j ( ζ i )) ≤ i,j ≤ r is non singular; • The matrix M Bp of the multiplication M p by p in a basis B of K [ x ± ]/ I satisfies W Bζ M Bp = D pζ W Bζ where D pζ is the diagonal matrix diag ( p ( ζ ) , . . . , p ( ζ r )) . This theorem gives us a basis of left eigenvectors for M Bp : The i -th row of W Bζ , [ b ( ζ i ) . . . b r ( ζ i )] , is aleft eigenvector associated to the eigenvalue p ( ζ i ) . One can furthermore observe that H C,B = ( W Cζ ) T A W Bζ where A = diag ( a , . . . , a r ) . (4.2)36 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
Assuming that a linear form Ω on K [ x ± ] is a weighted sum of evaluations at some points of ( K ∗ ) n , wewish to determine its support and its coefficients. We assume we know the cardinal r of this supportand that we can evaluate Ω at the monomials { x α } α ∈ N n . In other words, we assume that Ω = ∑ ri = a i e ζ i where ζ , . . . , ζ r ∈ ( K ∗ ) n and then a , . . . , a r ∈ K ∗ are the unknowns. For that we have access as input to { Ω ( x α + β + γ ) ∣ α, β ∈ C nr ; ∣ γ ∣ ≤ } .The ideal I Ω of these points is the kernel of the Hankel operator associated to Ω. One strategy would consistin determining a set of generators, or even a Gr¨obner basis, of this ideal and then find its roots with amethod to be chosen. In the present case there is nonetheless the possibility to directly form the matricesof the multiplication maps in K [ x ± ]/ I Ω (applying Theorem 4.3) once a basis for K [ x ± ]/ I Ω is determined(applying Theorem 4.2 and Corollary 4.6). The key fact that is used is that the set of j th coordinates ofthe ζ i , { ζ ,j , . . . , ζ r,j } , are the left eigenvalues of the multiplication map M x j ∶ K [ x ± ]/ I Ω → K [ x ± ]/ I Ω , where M x j ( p ) = x j p . The matrices of these maps commute and are simultaneously diagonalizable (Theorem 4.7or [16, Chapter 2, §
4, Exercise 12]). One could calculate the eigenspaces of the first matrix and proceedby induction to give such a common diagonalization since these eigenspaces are left invariant by the othermatrices. A more efficient approach given in the algorithm is to take a generic linear combination of thesematrices that ensures that this new matrix has distinct eigenvalues and calculate a basis of eigenvectors forit. In this basis each of the original matrices is diagonal.
Algorithm 4.8
Support & Coefficients
Input: r ∈ N > and { Ω ( x γ + α + β ) ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } Output: • The points ζ i = [ ζ i, , . . . , ζ i,n ] ∈ K n , for ≤ i ≤ r , • The vector [ a , . . . , a r ] ∈ ( K ∗ ) n of coefficients,such that Ω = r ∑ i = a i e ζ i .Form the matrix H C nr = [ Ω ( x α + β )] α,β ∈C nr Determine a lower set Γ within C nr of cardinal r such that the principal submatrix H Γ0 indexed by Γ isnonsingular. % Γ = { , γ , . . . , γ r } and { x γ ∣ γ ∈ Γ } is a basis of K [ x ± ]/ I Ω (Theorem 4.2). Form the matrices H Γ j = [ Ω ( x j x α + β )] α,β ∈ Γ and the matrices M Γ j = ( H Γ1 ) − H Γ j , for ≤ j ≤ n . % M Γ j is the matrix of multiplication by x j in K [ x ± ]/ I Ω (Theorem 4.3)% The matrices M Γ1 , . . . , M Γ n are simultaneously diagonalisable (Theorem 4.7). Consider L = (cid:96) M + . . . + (cid:96) n M n a generic linear combination of M , . . . , M n % The eigenvalues of L are λ i = ∑ nj = (cid:96) j ζ i,j , for 1 ≤ i ≤ r . For most [ (cid:96) , . . . , (cid:96) n ] ∈ K n they are distinct . Compute W a matrix whose rows are r linearly independent left eigenvectors of L appropriately nomalized % A left eigenvector associated to λ i is a nonzero multiple of the row vector [ , ( ζ i ) γ , . . . , ( ζ i ) γ r ] (Theorem 4.7)% The normalization of the first component to 1 allows us to assume the rows of W are exactly these vectors. We note that in forming L we desire that the eigenvalues of L are distinct. This is violated only when the characteristicpolynomial P L ( x ) of L has repeated roots. This latter condition is given by the vanishing of the resultant Res x ( P L , dP L dx ) which yields a polynomial condition on the (cid:96) i that fail to meet the required condition.
37. Hubert & M.F. Singer
For ≤ j ≤ n , determine the matrix D j = diag ( ζ ,j , . . . , ζ r,j ) such that W M Γ x j = D j W .For ≤ i ≤ r , form the points ζ i = [ ζ i, , . . . , ζ i,n ] from the diagonal entries of the matrices D j , ≤ j ≤ n .Determine the matrix diag ( a , . . . , a r ) such that H Γ0 = W T AW . % Only the first row of the left and right handside matrices need to be considered,% resulting in the linear system [ a . . . a r ] W = [ Ω ( ) Ω ( x γ ) . . . Ω ( x γ r )] Depending on the elements of Γ it might be possible to retrieve the coordinates of the points ζ j directly from W . The easiest case is when [ , . . . , ] T , . . . , [ . . . , , ] T ∈ Γ : the coordinates of ζ j can be read directlyfrom the normalized left eigenvectors of L , i.e. the rows of W .The determination of a lower set Γ of cardinality r whose associated principal submatrix is not singular isactually an algorithmic subject on its own. It is strongly tied to determining the Gr¨obner bases of I Ω and isthe focus of, for instance, [10, 54]. In a complexity meticulous approach to the problem, one would not formthe matrix H C nr at once, but construct Γ and the associated submatrix degree by degree or following someterm order. The number of evaluations of the function to interpolate is then reduced. This actual numberof evaluation heavily depends on the shape of Γ. Example 4.9
In Example 3.2 we called on Algorithm 4.8 with r = and Ω ( x γ y γ ) = f ( ξ γ , ξ γ ) = a ξ α γ + α γ + b ξ β γ + β γ . Hence H C = ⎡⎢⎢⎢⎢⎢⎣ f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ )⎤⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ a + b aξ α + bξ β aξ α + bξ β aξ α + bξ β aξ α + bξ β aξ α + α + bξ β + β aξ α + bξ β aξ α + α + bξ β + β aξ α + bξ β ⎤⎥⎥⎥⎥⎥⎦ The lower sets of cardinality 2 are Γ = {[ ] T , [ ] T } and Γ = {[ ] T , [ ] T } . One can check thatthe determinant of H C is zero while the determinants of the principal submatrices indexed by Γ and Γ are respectively ab ( ξ α − ξ β ) and ab ( ξ α − ξ β ) . Hence, whenever α ≠ β , Γ is a valid choice, i.e. H Γ isnon singular. Similarly for Γ when α ≠ β .Let us assume we can take Γ = Γ . We form: H Γ0 = [ f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ )] = [ a + b aξ α + bξ β aξ α + bξ β aξ α + bξ β ] ,H Γ1 = [ f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ )] = [ aξ α + bξ β aξ α + bξ β aξ α + bξ β aξ α + bξ β ] , and H Γ2 = [ f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ ) f ( ξ , ξ )] = [ aξ α + bξ β aξ α + α + bξ β + β aξ α + α + bξ β + β aξ α + β + bξ β + β ] . It follows that the multiplication matrices are: M = ( H Γ0 ) − H Γ1 = [ − ξ α + β ξ α + ξ β ] and M = ( H Γ0 ) − H Γ2 = ⎡⎢⎢⎢⎢⎢⎢⎢⎣ ξ α + β − ξ α + β ξ α − ξ β − ξ α + β ξ α − ξ β ξ α − ξ β ξ α − ξ β ξ α − ξ β ξ α + α − ξ β + β ξ α − ξ β ⎤⎥⎥⎥⎥⎥⎥⎥⎦ . The matrix of common left eigenvectors of M and M , with only in the first column, is W = [ ξ α ξ β ] . The diagonal matrices of eigenvalues are D = diag ( ξ α , ξ β ) and D = diag ( ξ α , ξ β ) . We shall thus outputthe points [ ξ α , ξ α ] T and [ ξ β , ξ β ] T of K .
38 Monday 27 thth
38 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
The first row of H Γ0 is [ a b ] W so that the vector of coefficients [ a b ] can be retrieved by solving therelated linear system. χ -invariant linear forms We now consider
R = K [ x ± ] W and S = K [ x ± ] W χ where W is a Weyl group acting on K [ x ± ] according to (2.1).The group morphism χ ∶ W → { , − } is given by either χ ( A ) = χ ( A ) = det ( A ) . K [ x ± ] W χ is a free K [ x ± ] W -module of rank one. When χ ( A ) = det ( A ) a basis for it is Υ δ (Proposition 2.21). We may write K [ x ± ] W χ = Υ K [ x ± ] W where Υ can be 1 or Υ δ . The starting point is a linear form Ω on K [ x ± ] that is χ -invariant, i.e. Ω ( A ⋅ p ) = χ ( A ) Ω ( p ) for all A ∈ W and p ∈ K [ x ± ] . We show how the restricted Hankel operator ̂H ∶ K [ x ± ] W → K [ x ± ] W χ allows one to recover the orbits in the support of the χ -invariant formΩ = r ∑ i = a i ∑ A ∈W χ ( A ) e A ⋆ ζ i where the ζ i ∈ ( K ∗ ) n have distinct orbits. By that we mean that we shall retrieve the values of the invariantpolynomials Θ ω , . . . , Θ ω n on { ζ , . . . , ζ n } .The linear map p χ ∶ K [ x ± ] → K [ x ± ] W χ q ↦ ∣W∣ ∑ A ∈W χ ( A ) − A ⋅ q is a projection that satisfies • p χ ( p q ) = p p χ ( q ) for all p ∈ K [ x ± ] W , q ∈ K [ x ± ] , and • p χ ( A ⋅ q ) = χ ( A ) p χ ( q ) for all q ∈ K [ x ± ] .Then, for any χ -invariant form Ω we have Ω ( p ) = Ω ( p χ ( p )) . Hence Ω is fully determined by its restrictionto K [ x ± ] W χ . We shall write Ω W when we consider the restriction of Ω to K [ x ± ] W χ . Similarly, we denote ̂H W and I W Ω the Hankel operator associated to Ω W and its kernel. Hence ̂H W ∶ K [ x ± ] W → ( K [ x ± ] W χ ) ∗ and I W Ω is an ideal of K [ x ± ] W . Lemma 4.10 If Ω = r ∑ i = a i ∑ A ∈W χ ( A ) e A ⋆ ζ i then I W Ω = I Ω ∩ K [ x ± ] W and the dimension of K [ x ± ] W / I W Ω is r . proof: Take p ∈ I W Ω ⊂ K [ x ± ] W . One wishes to show that for any q ∈ K [ x ± ] we have Ω ( p q ) =
0. This is truebecause Ω ( p q ) = Ω ( p χ ( p q )) = Ω ( p p χ ( q )) and Ω ( p q ′ ) = q ′ ∈ K [ x ± ] W . Hence I W Ω ⊂ I Ω ∩ K [ x ± ] W .The other inclusion is obvious.The proof that dim K [ x ± ] W / I W Ω = r follows the structure of [17, Ch.2,Proposition 2.10]. Let Z = { A ⋆ ζ i ∣ A ∈W , ≤ i ≤ r } be the union of the orbits of the ζ i . According to [17, Lemma 2.9], for each i there exists apolynomial ˜ p i such that for z ∈ Z ˜ p i ( z ) = { z = ζ i W i be the stabilizer of ζ i . Note that for A ∈ W , ˜ p i ( A ⋆ ζ j ) = j ≠ i and ˜ p i ( A ⋆ ζ i ) = A ∈ W i . Define p i = ∣W∣∣W i ∣ p ( ˜ p i ) . We have p i ∈ K [ x ± ] W and p i ( ζ j ) = δ i,j . Hence the linear map φ ∶ K [ x ± ] W → K r q ↦ [ q ( ζ ) . . . q ( ζ r )] is onto. We proceed to determine its kernel.One easily sees that I Ω ∩ K [ x ± ] W ⊂ ker φ . Consider q ∈ ker φ . Since q is invariant q ( A ⋆ ζ i ) = q ( ζ i ) = A ∈ W . By Theorem 4.4, I Ω is the annihilating ideal of { A ∗ ζ i ∣ ≤ i ≤ r, A ∈ W/W ζ i } . Hence q ∈ I Ω ∩ K [ x ± ] W .Since I W Ω = I Ω ∩ K [ x ± ] W , we have proved that ker φ = I W Ω . Hence K [ x ± ] W / I W Ω is isomorphic to K r . ∎ Theorem 4.11 If Ω = r ∑ i = a i ∑ A ∈W χ ( A ) e A ⋆ ζ i , where a i ∈ K ∗ and ζ , . . . , ζ r have distinct orbits in ( K ∗ ) n ,then the Hankel operator ̂H W associated to Ω W is of rank r . The variety of the extension of I W Ω to K [ x ± ] is { A ⋆ ζ i ∣ A ∈ W , ≤ i ≤ r } . proof: Follows from Lemma 4.10 and Theorem 4.4. ∎ K [ x ± ] W is isomorphic to a polynomial ring K [ X ] = K [ X , . . . , X n ] (Proposition 2.11). Then Proposition 4.5implies the following. Corollary 4.12
Let I W be an ideal of K [ x ± ] W such that the dimension of K [ x ± ] W / I W is of dimension r as a K -linear space. There exists a lower set Γ of cardinal r such that { Θ α ∣ α ∈ Γ } and { Ξ α ∣ α ∈ Γ } are bothbases of K [ x ± ] W / I W . proof: The Chebyshev polynomials of the first and second kind, { T α } α and { U α } α , were defined inDefinition 2.3 and 2.4 as the (only) polynomials in K [ X ] = K [ X , . . . , X n ] such that T α ( Θ ω , . . . , Θ ω ) = Θ α and U α ( Θ ω , . . . , Θ ω ) = Ξ α .Consider J the ideal in K [ X ] that corresponds to I W through the isomorphism between K [ x ± ] W and K [ X ] .Then dim K K [ X ]/ J = r . With the order ≤ on N n defined in Proposition 2.24, and by Proposition 2.23, { T α } α and { U α } α satisfy the hypothesis of Proposition 4.5. Hence there is a lower set Γ of cardinality r s.t. { T α ∣ α ∈ Γ } and { U α ∣ α ∈ Γ } are both bases of K [ X ]/ J . This particular Γ provides the announced conclusionthrough the isomorphism between K [ x ± ] W and K [ X ] . ∎ Proposition 4.13
Assume Ω is a χ -invariant linear form on K [ x ± ] whose restricted Hankel operator ̂H W ∶ K [ x ± ] W → K [ x ± ] W χ is of rank r . Then there is a non singular principal submatrix of size r in H = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ [ Ω ( Θ α Θ β )] α,β ∈ C nr if χ = , [ Ω ( Υ δ + α Θ β )] α,β ∈ C nr if χ = det . Let Γ be the index set of such a submatrix. Then { Θ α ∣ α ∈ Γ } is a basis of K [ x ± ] W / I W Ω considered as a K -linear space. Furthermore one can always find such a Γ that is a lower set. proof: When ̂H W is of rank r , r is the dimension of K [ x ± ] W / I W Ω . By Corollary 4.12, applied to I W Ω , thereis a lower set Γ of size r such that { Θ α ∣ α ∈ Γ } and { Ξ α ∣ α ∈ Γ } are both bases of K [ x ± ] W / I W Ω .40 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsAs Γ ⊂ C nr [55, Lemma 10], by Theorem 4.2, both [ Ω ( Υ Θ α Θ β )] α,β ∈ Γ and [ Ω ( Υ Ξ α Θ β )] α,β ∈ Γ . are non-singular. When χ = Υ = H . When χ = det then Υ = Υ δ and Υ δ Ξ α = Υ δ + α by Theorem 2.17. Hence the right handside matrix above is asubmatrix of H . ∎ Introducing the matrices A = diag ( a , . . . , a r ) , W Θ ζ = [ Θ α ( ζ i )] ≤ i ≤ r, α ∈ Γ and W Υ ζ = [ Υ δ + α ( ζ i )] ≤ i ≤ r, α ∈ Γ , one observes that [ Ω ( Θ α Θ β )] α,β ∈ Γ = ( W Θ ζ ) T A W Θ ζ and [ Ω ( Υ δ + α Θ β )] α,β ∈ Γ = ( W Υ ζ ) T A W Θ ζ (4.3)according to whether χ = Assume that the ideal I W of K [ x ± ] W is radical with K [ x ± ] W / I W of dimension r . Con-sider ζ , . . . , ζ r in ( ¯ K ∗ ) n whose distinct orbits form the variety of I W ⋅ K [ x ± ] . Then • A set B = { b , . . . , b r } is a basis of K [ x ± ] W / I W if and only if the matrix W Bζ = ( b j ( ζ i )) ≤ i,j ≤ r is nonsingular; • The matrix M Bp of the multiplication M p by p ∈ K [ x ± ] W in a basis B of K [ x ± ] W / I W satisfies W Bζ M Bp = D pζ W Bζ where D pζ is the diagonal matrix diag ( p ( ζ ) , . . . , p ( ζ r )) . proof: Clearly if B is linearly dependent modulo I W then det W Bζ = B = { b , . . . , b r } is a basis of K [ x ± ] W / I W . For any q ∈ K [ x ± ] W there thus exist unique ( q , . . . , q r ) ∈ K r such that q ≡ q b + . . . + q r b r mod I W . Observe that W Bζ ⎡⎢⎢⎢⎢⎢⎣ q ⋮ q r ⎤⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ q ( ζ )⋮ q ( ζ r )⎤⎥⎥⎥⎥⎥⎦ Thus W Bζ M Bp ⎡⎢⎢⎢⎢⎢⎣ q ⋮ q r ⎤⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ p ( ζ ) q ( ζ )⋮ p ( ζ r ) q ( ζ r )⎤⎥⎥⎥⎥⎥⎦ = D pζ ⎡⎢⎢⎢⎢⎢⎣ q ( ζ )⋮ q ( ζ r )⎤⎥⎥⎥⎥⎥⎦ = D pζ W Bζ ⎡⎢⎢⎢⎢⎢⎣ q ⋮ q r ⎤⎥⎥⎥⎥⎥⎦ . This thus shows the equality W Bζ M Bp = D pζ W Bζ , for all p ∈ K [ x ± ] W . This latter equality means that the i th row of W Bζ is a left eigenvector of M Bp associated to the eigenvalue p ( ζ i ) . If we choose q ∈ K [ x ± ] W so that itseparates the orbits of zeros of I W , then the left eigenvectors associated to the r distinct eigenvalues q ( ζ i ) are linearly independent. Those are nonzero multiples of the rows of W Bζ . Therefore det W Bζ ≠ ∎ Let Ω = ∑ ri = a i ∑ A ∈W χ ( A ) e A ⋆ ζ i for ζ , . . . , ζ r with distinct orbits. The underlying ideas of the followingalgorithm are similar to those of Algorithm 4.8. 41. Hubert & M.F. Singer Algorithm 4.15
Invariant Support & Coefficients
Input: r ∈ N > and • { Ω ( Θ α Θ β Θ γ ) ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } if χ = • { Ω ( Υ δ + α Θ β Θ γ ) ∣ α, β ∈ C nr , ∣ γ ∣ ≤ } if χ = det . Output: • The vectors [ Θ ω ( ζ i ) , . . . , Θ ω n ( ζ i )] for ≤ i ≤ r , where ω i = ( , . . . , , , , . . . , ) is the i th fundamentalweight. • The row vector ˜a = [ ˜ a . . . ˜ a r ] such that – ˜a = [∣W∣ a . . . ∣W∣ a r ] when χ = – ˜a = [ Υ δ ( ζ ) a . . . Υ δ ( ζ r ) a r ] when χ = det such that Ω = r ∑ i = a i ∑ A ∈W χ ( A ) e A ⋆ ζ i Form the matrix H C nr = [ Ω ( Θ α Θ β )] α,β ∈ C nr or [ Ω ( Υ δ + α Θ β )] α,β ∈ C nr according to whether χ is or det .Determine a lower set Γ of cardinal r such that the principal submatrix H Γ0 indexed by Γ is nonsingular. % Γ = { , γ , . . . , γ r } and the subset { Θ γ ∣ γ ∈ Γ } is a basis of K [ x ± ] W / I W Ω . For ≤ j ≤ n , form the matrices- H Γ j = [ Ω ( Θ α Θ β Θ ω j )] α,β ∈ Γ or [ Ω ( Υ δ + α Θ β Θ ω j )] α,β ∈ Γ according to whether χ is or det ;- the matrices M Γ j = ( H Γ0 ) − H Γ j ; % M Γ j is the matrix of multiplication by Θ ω i in K [ x ± ] W / I W Ω (Theorem 4.3)% The matrices M Γ1 , . . . , M Γ n are simultaneously diagonalisable (Theorem 4.14). Consider L = (cid:96) M + . . . + (cid:96) n M n a generic linear combination of M , . . . , M n % The eigenvalues of L are λ i = ∑ nj = (cid:96) j Θ ω j ( ζ i ) , for 1 ≤ i ≤ r .% For most [ (cid:96) , . . . , (cid:96) n ] ∈ K n these eigenvalues are distinct. Compute W a matrix whose rows are the r linearly independent normalized left eigenvectors of L . % A left eigenvector associated to λ i is a scalar multiple of [ Θ γ ( ζ i ) ∣ γ ∈ Γ ] .% Since Θ ( ζ i ) = ∣W∣ the eigenvectors can be rescaled so that they are exactly [ Θ γ ( ζ i ) ∣ γ ∈ Γ ] . For ≤ j ≤ n , determine the matrix D ( j ) = diag ( Θ ω j ( ζ ) , . . . , Θ ω j ( ζ r )) s.t. W M Γ j = D ( j ) W .From the diagonal entries of the matrices D ( j ) form the vectors [ Θ ω ( ζ i ) . . . Θ ω r ( ζ i )] for ≤ i ≤ r % If { ω , . . . , ω n } ⊂ Γ, we can form the vectors [ Θ ω ( ζ i ) . . . Θ ω r ( ζ i )] directly from the entries of W . Take h to be the first row of H Γ0 and solve the linear system ˜a W = h for the row vector ˜a = [ ˜ a . . . ˜ a r ] . % From Equation (4.3) H Γ0 = W T diag ( a , . . . , a r ) W if χ = H Γ0 = ( W Υ ζ ) T diag ( a , . . . , a r ) W if χ = det.% The first row of this equality is ˜a W = h where% ˜a = [ Θ ( ζ ) a . . . Θ ( ζ r ) a r ] , when χ =
1, and ˜a = [ Υ δ ( ζ ) a . . . Υ δ ( ζ r ) a r ] , when χ = det.
42 Monday 27 thth
42 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev PolynomialsAlgorithm 4.15 is called within Algorithm 3.5 and 3.8. At the next step of these algorithms one computes T µ ( Θ ω ( ζ i ) , . . . , Θ ω r ( ζ i )) for µ runing through a set of n linearly independent strongly dominant weights.We have that T µ ( Θ ω ( ζ i ) , . . . , Θ ω r ( ζ i )) = Θ µ ( ζ i ) . Hence if Γ includes some strongly dominant weights the entries of the related row of W could be output tosave on these evaluations. Example 4.16
In Example 3.6 we called on Algorithm 4.15 with r = and Ω ( Θ γ ,γ ) = f ( ξ γ + γ , ξ γ + γ ) where f ( x, y ) = F ( Θ ω ( x, y ) , Θ ω ( x, y )) = a Θ α ( x, y ) + b Θ β ( x, y ) . The underlying ideas of Algorithm 4.15 follow these of Algorithm 4.8 which was fully illustrated in Ex-ample 4.9. The same level of details would be very cumbersome in the present case and probably notenlightening. We shall limit ourselves to illustrate the formation of the matrices H C , H Γ0 , H Γ1 and H Γ2 interms of evaluation of the function to interpolate and make explicit the matrix W to be computed.We first need to consider the matrix H indexed by C = {[ ] T , [ ] T , [ ] T } H C = ⎡⎢⎢⎢⎢⎢⎣ Ω ( Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , ) ⎤⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ ( Θ , ) ( Θ , ) ( Θ , ) ( Θ , ) ( Θ , ) + ( Θ , ) ( Θ , ) + ( Θ , ) ( Θ , ) ( Θ , ) + ( Θ , ) ( Θ , ) + ( Θ , )⎤⎥⎥⎥⎥⎥⎦= ⎡⎢⎢⎢⎢⎢⎣ f ( , ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) f ( ξ, ξ ) + f ( , ) f ( ξ / , ξ / ) f ( ξ, ξ ) + f ( , ) f ( ξ / , ξ / ) + f ( ξ / , ξ / )⎤⎥⎥⎥⎥⎥⎦ One can check that this matrix has determinant zero whatever α and β . The possible lower sets Γ of cardinal-ity are {[ ] T , [ ] T } or {[ ] T , [ ] T } . One can actually check that the respective determinantsof the associated principal submatrices are f ( , ) ( f ( ξ / , ξ / ) + f ( ξ / , ξ / )) − ( f ( ξ / , ξ / )) = ab ( Θ α ( ξ / , ξ / ) − Θ β ( ξ / , ξ / )) and f ( , ) ( f ( ξ / , ξ / ) + f ( ξ / , ξ / )) − ( f ( ξ / , ξ / )) = ab ( Θ α ( ξ / , ξ / ) − Θ β ( ξ / , ξ / )) . At least one of these is non zero if α and β have distinct orbits. Assume it is the former, so that we choose Γ = {[ ] T , [ ] T } . Then H Γ0 = [ f ( , ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / )] ,H Γ1 = [ Ω ( Θ , Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , Θ , ) Ω ( Θ , ) ] = [
36 Ω ( Θ , )
12 Ω ( Θ , ) +
24 Ω ( Θ , )
12 Ω ( Θ , ) +
24 Ω ( Θ , )
24 Ω ( Θ , ) + ( Θ , ) + ( Θ , )]= [ f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) f ( , ) + f ( ξ, ξ ) + f ( ξ , ξ )] ,H Γ2 = [ Ω ( Θ , Θ , ) Ω ( Θ , Θ , Θ , ) Ω ( Θ , Θ , Θ , ) Ω ( Θ , Θ , ) ] = [
36 Ω ( Θ , )
24 Ω ( Θ , ) +
12 Ω ( Θ , )
24 Ω ( Θ , ) +
12 Ω ( Θ , ) ( Θ , ) +
20 Ω ( Θ , ) + ( Θ , )]= [ f ( ξ / , ξ / ) f ( , ) + f ( ξ, ξ ) f ( , ) + f ( ξ, ξ ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) + f ( ξ / , ξ / )] .
43. Hubert & M.F. Singer
The matrix of left eigenvectors common to M = ( H Γ0 ) − H Γ1 and M = ( H Γ0 ) − H Γ2 to be computed is W = [ Θ , ( ξ α T S ) Θ , ( ξ α T S ) Θ , ( ξ β T S ) Θ , ( ξ β T S )] = [ , ( ξ α T S ) , ( ξ β T S )] = [ α ( ξ / , ξ / ) β ( ξ / , ξ / )] . We have
W M = diag ( Θ , ( ξ α T S ) , Θ , ( ξ β T S )) W and W M = diag ( Θ , ( ξ α T S ) , Θ , ( ξ β T S )) W so thatthe points ϑ α = [ Θ , ( ξ α T S ) Θ , ( ξ α T S )] T and ϑ β = [ Θ , ( ξ β T S ) Θ , ( ξ β T S )] T can be output. We know that H Γ0 = W T diag ( a, b ) W . Extracting the first rows of this equality provides thelinear system [ a b ] W = [ f ( , ) f ( ξ / , ξ / )] to be solved in order to provide the second component of the output. Example 4.17
In Example 3.9 we called on Algorithm 4.15 with r = and Ω ( Υ γ ,γ ) = f ( ξ γ + γ , ξ γ + γ ) where f ( x, y ) = Υ δ ( x, y ) F ( Θ ω ( x, y ) , Θ ω ( x, y )) = a Υ δ + α ( x, y ) + b Υ δ + β ( x, y ) . As in previous example, we illustrate the formation of the matrices H C , H Γ0 , H Γ1 and H Γ2 in terms ofevaluation of the function to interpolate and make explicit the matrix W to be computed.We first need to consider the matrix H indexed by C = {[ ] T , [ ] T , [ ] T } H C = ⎡⎢⎢⎢⎢⎢⎣ Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , )⎤⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎢⎣ , , , , , + , , + , , , + , , + , ⎤⎥⎥⎥⎥⎥⎥⎦= ⎡⎢⎢⎢⎢⎢⎣ f ( ξ, ξ ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) f ( ξ , ξ ) + f ( ξ, ξ ) f ( ξ / , ξ / ) f ( ξ , ξ ) + f ( ξ, ξ ) f ( ξ / , ξ / ) + f ( ξ / , ξ / )⎤⎥⎥⎥⎥⎥⎦ One can check that this matrix has determinant zero whatever α and β . The possible lower sets Γ of cardinal-ity are {[ ] T , [ ] T } or {[ ] T , [ ] T } . One can actually check that the respective determinantsof the associated principal submatrices are f ( ξ, ξ ) ( f ( ξ / , ξ / ) + f ( ξ / , ξ / )) − f ( ξ / , ξ / ) = ab ( Θ δ + β ( ξ / , ξ / ) − Θ δ + α ( ξ / , ξ / )) ( Υ δ + α ( ξ, ξ ) Υ δ + β ( ξ / , ξ / ) − Υ δ + β ( ξ, ξ ) Υ δ + α ( ξ / , ξ / )) and f ( ξ, ξ ) ( f ( ξ / , ξ / ) + f ( ξ / , ξ / )) − f ( ξ / , ξ / ) = ab ( Θ δ + β ( ξ / , ξ / ) − Θ δ + α ( ξ / , ξ / )) ( Υ δ + α ( ξ, ξ ) Υ δ + β ( ξ / , ξ / ) − Υ δ + β ( ξ, ξ ) Υ δ + α ( ξ / , ξ / )) . At least one of these is non zero. Assume the former is and choose Γ = {[ ] T , [ ] T } . Then H Γ0 = [ f ( ξ, ξ ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / )] ,H Γ1 = [ Ω ( Υ , Θ , Θ , ) Ω ( Υ , Θ , ) Ω ( Υ , Θ , Θ , ) Ω ( Υ , Θ , )] = [
12 Υ , , + ,
12 Υ , +
12 Υ , , + , + , ]= ⎡⎢⎢⎢⎢⎣ f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) f ( ξ , ξ ) + f ( ξ, ξ ) + f ( ξ , ξ )⎤⎥⎥⎥⎥⎦ ,
44 Monday 27 th January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials H Γ2 = [ Ω ( Υ , Θ , Θ , ) Ω ( Υ , Θ , Θ , ) Ω ( Υ , Θ , Θ , ) Ω ( Υ , Θ , Θ , )] = [
12 Υ , , + ,
12 Υ , +
12 Υ , , + , + , ]= ⎡⎢⎢⎢⎢⎣ f ( ξ / , ξ / ) f ( ξ , ξ ) + f ( ξ, ξ ) f ( ξ , ξ ) + f ( ξ, ξ ) f ( ξ / , ξ / ) + f ( ξ / , ξ / ) + f ( ξ / , ξ / )⎤⎥⎥⎥⎥⎦ . The matrix of left eigenvectors common to M = ( H Γ0 ) − H Γ1 and M = ( H Γ0 ) − H Γ2 to be computed is W = [ Θ , ( ξ ( δ + α ) T S ) Θ , ( ξ ( δ + α ) T S ) Θ , ( ξ ( δ + β ) T S ) Θ , ( ξ ( δ + β ) T S )] = [ , ( ξ ( δ + α ) T S ) , ( ξ ( δ + β ) T S )] = [ δ + α ( ξ / , ξ / ) δ + β ( ξ / , ξ / )] . We have
W M = diag ( Θ , ( ξ ( δ + α ) T S ) , Θ , ( ξ ( δ + β ) T S )) W and W M = diag ( Θ , ( ξ ( δ + α ) T S ) , Θ , ( ξ ( δ + β ) T S )) W so that the points ϑ α = [ Θ , ( ξ ( δ + α ) T S ) Θ , ( ξ ( δ + α ) T S )] T and ϑ β = [ Θ , ( ξ ( δ + β ) T S ) Θ , ( ξ ( δ + β ) T S )] T can be output. We know that H Γ0 = ̂ W T diag ( a, b ) W where ̂ W = [ Υ , ( ξ ( δ + α ) T S ) Υ , ( ξ ( δ + α ) T S ) Υ , ( ξ ( δ + β ) T S ) Υ , ( ξ ( δ + β ) T S )] = [ Υ δ + α ( ξ, ξ ) Υ δ + α ( ξ / , ξ / ) Υ δ + β ( ξ, ξ ) Υ δ + β ( ξ / , ξ / )] . Extracting the first rows of this equality provides the linear system [ Υ δ + α ( ξ, ξ ) a Υ δ + β ( ξ, ξ ) b ] W = [ f ( ξ, ξ ) f ( ξ / , ξ / )] to be solved in order to provide the second component of the output, namely [ Υ δ + α ( ξ, ξ ) a Υ δ + β ( ξ, ξ ) b ] .
45. Hubert & M.F. Singer
For the benefit of clarity we have decribed the algorithms for sparse interpolation, be it in terms of Laurentmonomials or generalized Chebyshev polynomials, in two separate phases : in Section 3 we basically massagedthe sparse interpolation problem into the recovery of the support of the linear form and offered to performthere all the evaluations of the functions that may be needed to cover all the possible cases. Once we examinethe algorithms to recover the support of the linear forms, in Section 4, it becomes apparent that not all theseevaluations are used. First, as commented upon after Algorithm 4.8 determining the lower set Γ of theappropriate cardinality r can be approached iteratively and should not require forming the whole matrix H C nr . Then only the evaluations indexed by Γ + Γ + C n (rather than C nr + C nr + C n ) are required to form thesubsequent matrices. It is thus clear that going further with our intrinsically mutivariate approach to sparseinterpolation needs a holistic approach.All along the article we have mostly worked under the assumption that we know the number r of summandsexactly. Much of the litterature on sparse interpolation considers an upper bound R to the number ofsummands. It is not a theoretical difficulty. The algorithms work similarly with R instead of r as input.The exact number of summands can then be retrieved as the rank of the matrix H C nR . This would indicatethat, in this case where we only know an upper bound, we actually need to form the whole matrix H C nR first. But the practical approach to sparse interpolation is to design early termination strategies that provideprobabilistic certificate on the actual number of summands [32, 33, 28]. Such strategies would deserve anextension to the generalized Chebyshev polynomials considered here.As noted in Section 3, one can consider an r -sparse sum of generalized Chebyshev polynomials as a ˜ r -sparsesum of monomials where ˜ r is bounded by r ∣W∣ . Yet the approach we presented for r -sparse sum of generalizedChebyshev polynomials allows to restrict the size of matrices to ∣C nr ∣ instead of ∣C n ∣W∣ r ∣ . Our initial hope was tohave an analogous benefit, by a factor ∣W∣ , on the number of evaluations. The number of evaluations neededfor the sparse interpolation of a sum of r ∣W∣ -monomials, is bounded by the cardinality of C n ∣W∣ r + C n ∣W∣ r + C n .We nonetheless bounded the number of evaluations to be made by the cardinality of X W r , which is only asuperset of C nr + C nr + C n . Our initial estimate of the cardinality of X W r still shows a benefit of our approachalso in terms of the number of evaluations. Yet we feel that a more refined analysis, taking into account thespecific properties of the different Weyl groups, would testify to a stronger benefit.In our generalized approach to sparse interpolation the emphasis is on the associated Hankel operator ratherthan the matrices that arose when laying down the problem as a set of linear equations. In [8, 35] the structureof these matrices is exploited to work out the best complexity of the linear algebra used in the algorithmfor the univariate cases. One has to recognize that it is the multiplication rules on the polynomial basis(monomial or Chebyshev respectively) that gives the specific structure to the matrix of the Hankel operator.A deeper understanding of how the action of the Weyl group can be used to express these multiplicationrules in the most economical form should lead to a better control of the complexity of our approach.46 Monday 27 thth
For the benefit of clarity we have decribed the algorithms for sparse interpolation, be it in terms of Laurentmonomials or generalized Chebyshev polynomials, in two separate phases : in Section 3 we basically massagedthe sparse interpolation problem into the recovery of the support of the linear form and offered to performthere all the evaluations of the functions that may be needed to cover all the possible cases. Once we examinethe algorithms to recover the support of the linear forms, in Section 4, it becomes apparent that not all theseevaluations are used. First, as commented upon after Algorithm 4.8 determining the lower set Γ of theappropriate cardinality r can be approached iteratively and should not require forming the whole matrix H C nr . Then only the evaluations indexed by Γ + Γ + C n (rather than C nr + C nr + C n ) are required to form thesubsequent matrices. It is thus clear that going further with our intrinsically mutivariate approach to sparseinterpolation needs a holistic approach.All along the article we have mostly worked under the assumption that we know the number r of summandsexactly. Much of the litterature on sparse interpolation considers an upper bound R to the number ofsummands. It is not a theoretical difficulty. The algorithms work similarly with R instead of r as input.The exact number of summands can then be retrieved as the rank of the matrix H C nR . This would indicatethat, in this case where we only know an upper bound, we actually need to form the whole matrix H C nR first. But the practical approach to sparse interpolation is to design early termination strategies that provideprobabilistic certificate on the actual number of summands [32, 33, 28]. Such strategies would deserve anextension to the generalized Chebyshev polynomials considered here.As noted in Section 3, one can consider an r -sparse sum of generalized Chebyshev polynomials as a ˜ r -sparsesum of monomials where ˜ r is bounded by r ∣W∣ . Yet the approach we presented for r -sparse sum of generalizedChebyshev polynomials allows to restrict the size of matrices to ∣C nr ∣ instead of ∣C n ∣W∣ r ∣ . Our initial hope was tohave an analogous benefit, by a factor ∣W∣ , on the number of evaluations. The number of evaluations neededfor the sparse interpolation of a sum of r ∣W∣ -monomials, is bounded by the cardinality of C n ∣W∣ r + C n ∣W∣ r + C n .We nonetheless bounded the number of evaluations to be made by the cardinality of X W r , which is only asuperset of C nr + C nr + C n . Our initial estimate of the cardinality of X W r still shows a benefit of our approachalso in terms of the number of evaluations. Yet we feel that a more refined analysis, taking into account thespecific properties of the different Weyl groups, would testify to a stronger benefit.In our generalized approach to sparse interpolation the emphasis is on the associated Hankel operator ratherthan the matrices that arose when laying down the problem as a set of linear equations. In [8, 35] the structureof these matrices is exploited to work out the best complexity of the linear algebra used in the algorithmfor the univariate cases. One has to recognize that it is the multiplication rules on the polynomial basis(monomial or Chebyshev respectively) that gives the specific structure to the matrix of the Hankel operator.A deeper understanding of how the action of the Weyl group can be used to express these multiplicationrules in the most economical form should lead to a better control of the complexity of our approach.46 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials
References [1] M. Abril Bucero, C. Bajaj, and B. Mourrain. On the construction of general cubature formula by flatextensions.
Linear Algebra and its Applications , 502:104 – 125, 2016. Structured Matrices: Theory andApplications.[2] M. Abril Bucero and B. Mourrain. Border basis relaxation for polynomial optimization.
Journal ofSymbolic Computation , 74:378 – 399, 2016.[3] A. Arnold.
Sparse Polynomial Interpolation and Testing . PhD thesis, University of Waterloo, 3 2016.[4] A. Arnold, M. Giesbrecht, and D. Roche. Sparse interpolation over finite fields via low-order roots ofunity. In
ISSAC 2014—Proceedings of the 39th International Symposium on Symbolic and AlgebraicComputation , pages 27–34. ACM, New York, 2014.[5] A. Arnold and E. Kaltofen. Error-correcting sparse interpolation in the Chebyshev basis. In
Proceedingsof the 2015 ACM on International Symposium on Symbolic and Algebraic Computation , ISSAC ’15,pages 21–28, New York, NY, USA, 2015. ACM.[6] A. Arnold and D. Roche. Multivariate sparse interpolation using randomized Kronecker substitutions. In
ISSAC 2014—Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation ,pages 35–42. ACM, New York, 2014.[7] T. Becker and V. Weispfenning.
Gr¨obner Bases - A Computational Approach to Commutative Algebra .Springer-Verlag, New York, 1993.[8] M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation.In
Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing , STOC ’88, pages301–309, New York, NY, USA, 1988. ACM.[9] A. Bernardi and D. Taufer. Waring, tangential and cactus decompositions. arXiv:1812.02612 , Dec 2018.[10] J. Berthomieu, B. Boyer, and J.-C. Faug`ere. Linear algebra for computing Gr¨obner bases of linearrecursive multidimensional sequences.
Journal of Symbolic Computation , 83:36 – 67, 2017. Special issueon the conference ISSAC 2015: Symbolic computation and computer algebra.[11] N. Bourbaki. ´El´ements de math´ematique. Fasc. XXXIV. Groupes et alg`ebres de Lie. Chapitre IV:Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr´es par des r´eflexions. ChapitreVI: syst`emes de racines . Actualit´es Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968.[12] N. Bourbaki. ´El´ements de math´ematique. Fasc. XXXVIII: Groupes et alg`ebres de Lie. Chapitre VII:Sous-alg`ebres de Cartan, ´el´ements r´eguliers. Chapitre VIII: Alg`ebres de Lie semi-simples d´eploy´ees .Actualit´es Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975.[13] J. Brachat, P. Comon, B. Mourrain, and E. Tsigaridas. Symmetric tensor decomposition.
Linear AlgebraAppl. , 433(11-12):1851–1872, 2010.[14] M. Collowald and E. Hubert. A moment matrix approach to computing symmetric cubatures. https://hal.inria.fr/hal-01188290 , August 2015.[15] M. Collowald and E. Hubert. Algorithms for computing cubatures based on moment theory.
Studies inApplied Mathematics , 141(4):501–546, 2018.[16] D. Cox, J. Little, and D. O’Shea.
Ideals, varieties, and algorithms . Undergraduate Texts in Mathemat-ics. Springer, Cham, fourth edition, 2015. An introduction to computational algebraic geometry andcommutative algebra. 47. Hubert & M.F. Singer[17] D. A. Cox, J. Little, and D. O’Shea.
Using algebraic geometry , volume 185 of
Graduate Texts inMathematics . Springer, New York, second edition, 2005.[18] J. Dieudonn´e.
Special functions and linear representations of Lie groups , volume 42 of
CBMS RegionalConference Series in Mathematics . American Mathematical Society, Providence, R.I., 1980. Exposi-tory lectures from the CBMS Regional Conference held at East Carolina University, Greenville, NorthCarolina, March 5–9, 1979.[19] A. Dress and J. Grabmeier. The interpolation problem for k -sparse polynomials and character sums. Adv. in Appl. Math. , 12(1):57–75, 1991.[20] W. Fulton and J. Harris.
Representation theory , volume 129 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.[21] K. Gatermann and P. A. Parrilo. Symmetry groups, semidefinite programs, and sums of squares.
J.Pure Appl. Algebra , 192(1-3):95–128, 2004.[22] M. Giesbrecht, G. Labahn, and W. Lee. Symbolic-numeric sparse polynomial interpolation in Chebyshevbasis and trigonometric interpolation. In
CASC 2004 , 2004.[23] M. Giesbrecht, G. Labahn, and W.-S. Lee. Symbolic-numeric sparse interpolation of multivariate poly-nomials.
J. Symbolic Comput. , 44(8):943–959, 2009.[24] W. H. Greub.
Linear algebra . Third edition. Die Grundlehren der Mathematischen Wissenschaften,Band 97. Springer-Verlag New York, Inc., New York, 1967.[25] D. Grigoriev, M. Karpinski, and M. Singer. The interpolation problem for k -sparse sums of eigenfunc-tions of operators. Adv. in Appl. Math. , 12(1):76–81, 1991.[26] B. Hall.
Lie groups, Lie algebras, and representations , volume 222 of
Graduate Texts in Mathematics .Springer, Cham, second edition, 2015. An elementary introduction.[27] M. Hoffman and W. Withers. Generalized Chebyshev polynomials associated with affine Weyl groups.
Trans. Amer. Math. Soc. , 308(1):91–104, 1988.[28] Q. Huang. An improved early termination sparse interpolation algorithm for multivariate polynomials.
J. Syst. Sci. Complex. , 31(2):539–551, 2018.[29] J. Humphreys.
Introduction to Lie algebras and representation theory . Springer-Verlag, New York-Berlin,1972. Graduate Texts in Mathematics, Vol. 9.[30] E. Imamogli and E. Kaltofen. On computing the degree of a Chebyshev polynomial from its value.Manuscript, November 2018.[31] E. Kaltofen and Y. Lakshman. Improved sparse multivariate polynomial interpolation algorithms. In
Symbolic and algebraic computation (Rome, 1988) , volume 358 of
Lecture Notes in Comput. Sci. , pages467–474. Springer, Berlin, 1989.[32] E. Kaltofen and W.-S. Lee. Early termination in sparse interpolation algorithms.
J. Symbolic Comput. ,36(3-4):365–400, 2003. International Symposium on Symbolic and Algebraic Computation (ISSAC’2002)(Lille).[33] E. Kaltofen, W.-S. Lee, and A. Lobo. Early termination in Ben-Or/Tiwari sparse interpolation anda hybrid of Zippel’s algorithm. In
Proceedings of the 2000 International Symposium on Symbolic andAlgebraic Computation (St. Andrews) , pages 192–201, New York, 2000. ACM.[34] S. Kunis, T. Peter, T. R¨omer, and U. von der Ohe. A multivariate generalization of Prony’s method.
Linear Algebra Appl. , 490:31–47, 2016.48 Monday 27 thth
Linear Algebra Appl. , 490:31–47, 2016.48 Monday 27 thth January, 2020 [01:40]parse Interpolation in Terms of Multivariate Chebyshev Polynomials[35] Y. Lakshman and D. Saunders. Sparse polynomial interpolation in nonstandard bases.
SIAM J. Com-put. , 24(2):387–397, 1995.[36] J. B. Lasserre.
Moments, positive polynomials and their applications , volume 1 of
Imperial College PressOptimization Series . Imperial College Press, London, 2010.[37] M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In
Emergingapplications of algebraic geometry , volume 149 of
IMA Vol. Math. Appl. , pages 157–270. Springer, NewYork, 2009.[38] H. Li and Y. Xu. Discrete Fourier analysis on fundamental domain and simplex of A d lattice in d -variables. J. Fourier Anal. Appl. , 16(3):383–433, 2010.[39] M. Lorenz.
Multiplicative invariant theory , volume 135 of
Encyclopaedia of Mathematical Sciences .Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, VI.[40] C. Lubich.
From quantum to classical molecular dynamics: reduced models and numerical analysis .Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Z¨urich, 2008.[41] V. D. Lyakhovsky and Ph. V. Uvarov. Multivariate Chebyshev polynomials.
J. Phys. A , 46(12):125201,22, 2013.[42] R. Moody, L. Motlochov´a, and J. Patera. Gaussian cubature arising from hybrid characters of simpleLie groups.
J. Fourier Anal. Appl. , 20(6):1257–1290, 2014.[43] R. Moody and J. Patera. Computation of character decompositions of class functions on compactsemisimple Lie groups.
Math. Comp. , 48(178):799–827, 1987.[44] R. Moody and J. Patera. Cubature formulae for orthogonal polynomials in terms of elements of finiteorder of compact simple Lie groups.
Adv. in Appl. Math. , 47(3):509–535, 2011.[45] B. Mourrain. Polynomial–exponential decomposition from moments.
Foundations of ComputationalMathematics , 18(6):1435–1492, Dec 2018.[46] H. Munthe-Kaas, M. Nome, and B. Ryland. Through the kaleidoscope: symmetries, groups andChebyshev-approximations from a computational point of view. In
Foundations of computational math-ematics, Budapest 2011 , volume 403 of
London Math. Soc. Lecture Note Ser. , pages 188–229. CambridgeUniv. Press, Cambridge, 2013.[47] M. Nesterenko, J. Patera, and A. Tereszkiewicz. Orthogonal polynomials of compact simple Lie groups.
Int. J. Math. Math. Sci. , 2011.[48] V. Pereyra and G. Shcerer, editors.
Exponential Data Fitting and its Applications . Bentham e-books, , 2010.[49] D. Potts and M. Tasche. Sparse polynomial interpolation in Chebyshev bases.
Linear Algebra Appl. ,441:61–87, 2014.[50] S. Power. Finite rank multivariable Hankel forms.
Linear Algebra Appl. , 48:237–244, 1982.[51] C. (Baron de Prony) Riche. Essai exp´erimental et analytique sur les lois de la dilatabilit´e des fluides´elastique et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, `a diff´erentestemp´eratures.
J. de l’ ´Ecole Polytechnique , 1:24–76, 1795.[52] C. Riener, T. Theobald, L. J. Andr´en, and J. B. Lasserre. Exploiting symmetries in SDP-relaxationsfor polynomial optimization.
Math. Oper. Res. , 38(1):122–141, 2013. 49. Hubert & M.F. Singer[53] B. Ryland and H. Munthe-Kaas. On multivariate Chebyshev polynomials and spectral approximationson triangles. In
Spectral and high order methods for partial differential equations , volume 76 of
Lect.Notes Comput. Sci. Eng. , pages 19–41. Springer, Heidelberg, 2011.[54] S. Sakata. The BMS algorithm. In M. Sala, S. Sakata, T. Mora, C. Traverso, and L. Perret, ed-itors,
Gr¨obner Bases, Coding, and Cryptography , pages 143–163. Springer Berlin Heidelberg, Berlin,Heidelberg, 2009.[55] T. Sauer. Prony’s method in several variables: symbolic solutions by universal interpolation.
J. SymbolicComput. , 84:95–112, 2018.[56] J.-P. Serre.
Alg`ebres de Lie semi-simples complexes . W. A. Benjamin, inc., New York-Amsterdam, 1966.[57] N. Vilenkin.
Special functions and the theory of group representations . Translated from the Russianby V. N. Singh. Translations of Mathematical Monographs, Vol. 22. American Mathematical Society,Providence, R. I., 1968.50 Monday 27 thth