Generalized discrete orbit function transforms of affine Weyl groups
GGENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS OF AFFINEWEYL GROUPS
TOMASZ CZY ˙ZYCKI AND JI ˇR´I HRIVN ´AK Abstract.
The affine Weyl groups with their corresponding four types of orbit functions areconsidered. Two independent admissible shifts, which preserve the symmetries of the weightand the dual weight lattices, are classified. Finite subsets of the shifted weight and the shifteddual weight lattices, which serve as a sampling grid and a set of labels of the orbit functions,respectively, are introduced. The complete sets of discretely orthogonal orbit functions overthe sampling grids are found and the corresponding discrete Fourier transforms are formulated.The eight standard one-dimensional discrete cosine and sine transforms form special cases of thepresented transforms. Institute of Mathematics, University of Bia(cid:32)lystok, Akademicka 2, 15–267 Bia(cid:32)lystok, Poland Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical Uni-versity in Prague, Bˇrehov´a 7, CZ–115 19 Prague, Czech Republic
E-mail: jiri.hrivnak@fjfi.cvut.cz, [email protected].
Introduction
This paper aims to extend the results of the discrete Fourier calculus of orbit functions [11,13]. Thesefour types of orbit functions are induced by the affine Weyl groups of the compact simple Lie groups.Firstly, notions of an admissible shift and an admissible dual shift are introduced and classified. Thentwo finite subsets of the shifted dual weight lattice and of the shifted weight lattice are introduced— the first set serves as a sampling grid and the second as a complete set of labels of the discretelyorthogonal orbit functions.The four types of orbit functions are a generalization of the standard symmetric and antisymmetricorbit sums [3]. They can also be viewed as multidimensional generalizations of one-dimensional cosineand sine functions with the symmetry and periodicity determined by the affine Weyl groups [15]. Indeed,for the case of rank one are the standard cosine and sine functions special cases of orbit functions. For adetailed review of the properties of the symmetric and antisymmetric orbit functions see [18,20] and thereferences therein. This paper focuses on discrete Fourier transforms of the orbit functions [11, 13, 23].For the case of rank one, the discrete Fourier transforms from [13] become one-dimensional discretecosine and sine transforms known as DCT–I and DST–I [5]. The two transforms DCT–I and DST–Iconstitute only a part of the collection of one-dimensional cosine and sine transforms. The other sixmost ubiquitous transforms DCT–II, DCT–III, DCT–IV and DST–II, DST–III, DST–IV are obtainedby imposing different boundary conditions on the corresponding difference equations [5]. The crucialfact is that the resulting grids and the resulting labels of the functions are shifted from their originalposition depending on the given boundary conditions while preserving their symmetry. This observationserves as a starting point for deriving and generalizing these transforms in the present paper. Besides thestandard multidimensional Cartesian product generalizations of DCT’s and DST’s, other approaches,which also develop multidimensional analogues of the four types of sine and cosine transforms, are basedon antisymmetric and symmetric trigonometric functions.The (anti)symmetric cosine and sine functions are introduced in [19] and are directly connectedto the four types of orbit functions of the series of the root systems B n and C n — see for instancethe three-dimensional relations in [9]. These functions arise as Cartesian products of one-dimensionaltrigonometric functions which are symmetrized with respect to the permutation group S n . Four typesof generalized cosine and sine transforms of both symmetric and antisymmetric types are formulated Date : October 23, 2018. a r X i v : . [ m a t h - ph ] N ov T. CZY ˙ZYCKI AND J. HRIVN ´AK in [19] and detailed for two-variable functions in [10,12]. This approach, however, relies on the validity ofthe one-dimensional DCT’s and DST’s and obtained multidimensional shifted discrete grids are subsetsof the Cartesian product grids, used in the standard multidimensional version of DCT’s and DST’s.This paper uses a different approach — generalized DCT’s and DST’s are derived independently ontheir one-dimensional versions and the resulting grids are not subsets of Cartesian product grids.The physical motivation of this work stems from widespread use of various types of one-dimensionaland multidimensional DCT’s and DST’s. A textbook case of using one-dimensional DCT’s and DST’s isdescription of modes of a beaded string where the type of the transform is determined by the positionsof the beads and their boundary conditions. Similar straightforward applications might be expectedin two-dimensional and three-dimensional settings. Other more involved applications include generalinterpolation methods, for instance in chemical physics [27], quantum algorithms [6] and quantumcommunication processes [2].From a mathematical point of view, the present approach uses the following two types of homomor-phisms [13, 14, 23], • the standard retraction homomorphisms ψ, (cid:98) ψ of the affine and the dual affine Weyl groups, • the four sign homomorphisms σ , which determine the type of the special function,and adds two types of homomorphisms related to two shifts (cid:37) ∨ and (cid:37) , • the shift and the dual shift homomorphisms (cid:98) θ (cid:37) ∨ , θ (cid:37) , which control the affine boundaries condi-tions, • the two γ − homomorphisms, which combine the previous three types of homomorphisms andcontrol the behavior of the orbit functions on the boundaries.In Section 2, the notation and pertinent properties concerning the affine Weyl groups and the corre-sponding lattices are reviewed. The admissible shifts and the dual admissible shifts are classified andthe four types of homomorphisms are presented. In Section 3, the four types of orbit functions arerecalled and their symmetries, depending on the shifts, are determined. The finite set of grid of pointsand the labels are introduced and the numbers of their points calculated. In Section 4, the discreteorthogonality of orbit functions is shown and the corresponding discrete Fourier transforms presented.2. Pertinent properties of affine Weyl groups
Roots and reflections.
The notation, established in [13], is used. Recall that, to the simple Lie algebra of rank n , correspondsthe set of simple roots ∆ = { α , . . . , α n } of the root system Π [1, 3, 15]. The set ∆ spans the Euclideanspace R n , with the scalar product denoted by (cid:104) , (cid:105) . The set of simple roots determines partial ordering ≤ on R n — for λ, ν ∈ R n it holds that ν ≤ λ if and only if λ − ν = k α + · · · + k n α n with k i ≥ i ∈ { , . . . , n } . The root system Π and its set of simple roots ∆ can be defined independently on Lietheory and such sets which correspond to compact simple Lie groups are called crystallographic [15].There are two types of sets of simple roots — the first type with roots of only one length, denotedstandardly as A n , n ≥ D n , n ≥ E , E , E and the second type with two different lengths of roots,denoted B n , n ≥ C n , n ≥ G and F . For the second type systems, the set of simple roots consistsof short simple roots ∆ s and long simple roots ∆ l , i.e. the following disjoint decomposition is given,∆ = ∆ s ∪ ∆ l . (1)The standard objects, related to the set ∆ ⊂ Π, are the following [1, 15]: • the highest root ξ ∈ Π with respect to the partial ordering ≤ restricted on Π • the marks m , . . . , m n of the highest root ξ ≡ − α = m α + · · · + m n α n , • the Coxeter number m = 1 + m + · · · + m n , • the root lattice Q = Z α + · · · + Z α n , • the Z -dual lattice to Q , P ∨ = (cid:8) ω ∨ ∈ R n | (cid:104) ω ∨ , α (cid:105) ∈ Z , ∀ α ∈ ∆ (cid:9) = Z ω ∨ + · · · + Z ω ∨ n , with (cid:104) α i , ω ∨ j (cid:105) = δ ij , (2) • the dual root lattice Q ∨ = Z α ∨ + · · · + Z α ∨ n , where α ∨ i = 2 α i / (cid:104) α i , α i (cid:105) , i ∈ { , . . . , n } , ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 3 • the dual marks m ∨ , . . . , m ∨ n of the highest dual root η ≡ − α ∨ = m ∨ α ∨ + · · · + m ∨ n α ∨ n ; the marksand the dual marks are summarized in Table 1 in [13], • the Z -dual lattice to Q ∨ P = (cid:8) ω ∈ R n | (cid:104) ω, α ∨ (cid:105) ∈ Z , ∀ α ∨ ∈ Q ∨ (cid:9) = Z ω + · · · + Z ω n , • the Cartan matrix C with elements C ij = (cid:104) α i , α ∨ j (cid:105) and with the property α ∨ j = C kj ω ∨ k , (3) • the index of connection c of Π equal to the determinant of the Cartan matrix C , c = det C, (4)and determining the orders of the isomorphic quotient groups P/Q and P ∨ /Q ∨ , c = | P/Q | = | P ∨ /Q ∨ | . The n reflections r α , α ∈ ∆ in ( n − a ∈ R n by r α a = a − (cid:104) α, a (cid:105) α ∨ . (5)and the affine reflection r with respect to the highest root ξ is given by r a = r ξ a + 2 ξ (cid:104) ξ, ξ (cid:105) , r ξ a = a − (cid:104) a, ξ (cid:105)(cid:104) ξ, ξ (cid:105) ξ , a ∈ R n . (6)The set of reflections r ≡ r α , . . . , r n ≡ r α n , together with the affine reflection r , is denoted by R , R = { r , r , . . . , r n } . (7)The dual affine reflection r ∨ , with respect to the dual highest root η , is given by r ∨ a = r η a + 2 η (cid:104) η, η (cid:105) , r η a = a − (cid:104) a, η (cid:105)(cid:104) η, η (cid:105) η, a ∈ R n . (8)The set of reflections r ∨ ≡ r α , . . . , r ∨ n ≡ r α n , together with the dual affine reflection r ∨ is denoted by R ∨ , R ∨ = { r ∨ , r ∨ , . . . , r ∨ n } . Weyl group and affine Weyl group.
The Weyl group W is generated by n reflections r α , α ∈ ∆. The set R of n + 1 generators (7)generates the affine Weyl group W aff . Except for the case A , the Coxeter presentation of W aff is ofthe form W aff = (cid:104) R | ( r i r j ) m ij = 1 (cid:105) , (9)where the numbers m ij , i, j ∈ { , , . . . , n } are given by the extended Coxeter-Dynkin diagrams — seee. g. [13]. It holds that m ii = 1 and if the i th and j th nodes in the diagram are not connected then m ij = 2; otherwise the single, double or triple vertices between the nodes indicate m ij equal to 3 , W aff is the semidirect product of the Abelian group of translations T ( Q ∨ ) by shifts from Q ∨ and of the Weyl group W , W aff = T ( Q ∨ ) (cid:111) W. Thus, for any w aff ∈ W aff , there exist a unique w ∈ W and a unique shift T ( q ∨ ) such that w aff = T ( q ∨ ) w .Taking any w aff = T ( q ∨ ) w ∈ W aff , the retraction homomorphism ψ : W aff → W and the mapping τ : W aff → Q ∨ are given by ψ ( w aff ) = w, (10) τ ( w aff ) = q ∨ . (11)The fundamental domain F of W aff , which consists of precisely one point of each W aff -orbit, is theconvex hull of the points (cid:110) , ω ∨ m , . . . , ω ∨ n m n (cid:111) . Considering n + 1 real parameters y , . . . , y n ≥
0, we have F = (cid:8) y ω ∨ + · · · + y n ω ∨ n | y + y m + · · · + y n m n = 1 (cid:9) . (12) T. CZY ˙ZYCKI AND J. HRIVN ´AK
Let us denote the isotropy subgroup of a point a ∈ R n and its order byStab W aff ( a ) = (cid:110) w aff ∈ W aff | w aff a = a (cid:111) , h ( a ) = | Stab W aff ( a ) | , and define a function ε : R n → N by the relation ε ( a ) = | W | h ( a ) . (13)Since the stabilizers Stab W aff ( a ) and Stab W aff ( w aff a ) are for any w aff ∈ W aff conjugated, one obtainsthat ε ( a ) = ε ( w aff a ) , w aff ∈ W aff . (14)Recall that the stabilizer Stab W aff ( a ) of a point a = y ω ∨ + · · · + y n ω ∨ n ∈ F is trivial, Stab W aff ( a ) = 1if the point a is in the interior of F , a ∈ int( F ). Otherwise the group Stab W aff ( a ) is generated by such r i for which y i = 0, i = 0 , . . . , n .Considering the standard action of W on the torus R n /Q ∨ , we denote for x ∈ R n /Q ∨ the isotropygroup by Stab( x ) and the orbit and its order by W x = (cid:8) wx ∈ R n /Q ∨ | w ∈ W (cid:9) , (cid:101) ε ( x ) ≡ | W x | . Recall the following three properties from Proposition 2.2 in [13] of the action of W on the torus R n /Q ∨ :(1) For any x ∈ R n /Q ∨ , there exists x (cid:48) ∈ F ∩ R n /Q ∨ and w ∈ W such that x = wx (cid:48) . (15)(2) If x, x (cid:48) ∈ F ∩ R n /Q ∨ and x (cid:48) = wx , w ∈ W , then x (cid:48) = x = wx. (16)(3) If x ∈ F ∩ R n /Q ∨ , i.e. x = a + Q ∨ , a ∈ F , then ψ (Stab W aff ( a )) = Stab( x ) andStab( x ) ∼ = Stab W aff ( a ) . (17)From (17) we obtain that for x = a + Q ∨ , a ∈ F it holds that ε ( a ) = (cid:101) ε ( x ) . (18)Note that instead of (cid:101) ε ( x ), the symbol ε ( x ) is used for | W x | , x ∈ F ∩ R n /Q ∨ in [11, 13]. The calculationprocedure of the coefficients ε ( x ) is detailed in § Admissible shifts.
For the development of the discrete Fourier calculus is crucial the notion of certain lattices invariantunder the action of the Weyl group W . In [11, 13] is formulated the discrete Fourier calculus on thefragment of the refined W − invariant lattice P ∨ . Considering any vector (cid:37) ∨ ∈ R n , we call (cid:37) ∨ anadmissible shift if the shifted lattice (cid:37) ∨ + P ∨ is still W − invariant, i.e. W ( (cid:37) ∨ + P ∨ ) = (cid:37) ∨ + P ∨ . (19)If (cid:37) ∨ ∈ P ∨ then the resulting lattice (cid:37) ∨ + P ∨ = P ∨ does not change — such admissible shifts are calledtrivial. Also any two admissible shifts (cid:37) ∨ , (cid:37) ∨ which would differ by a trivial shift, i.e. (cid:37) ∨ − (cid:37) ∨ ∈ P ∨ lead to the same resulting shifted lattice and are in this sense equivalent. Thus, in the following, weclassify admissible shifts up to this equivalence. The following proposition significantly simplifies theclassification of admissible shifts. Proposition 2.1.
Let (cid:37) ∨ ∈ R n . Then the following statements are equivalent.(1) (cid:37) ∨ is an admissible shift.(2) (cid:37) ∨ − W (cid:37) ∨ ⊂ P ∨ .(3) For all i ∈ { , . . . , n } it holds that (cid:37) ∨ − r i (cid:37) ∨ ∈ P ∨ . (20) ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 5 (cid:37) ∨ (cid:37)A ω ∨ ω C ω ∨ ω B n , n ≥ ω ∨ n − C n , n ≥ − ω n Table 1.
Non-trivial admissible shifts (cid:37) ∨ and admissible dual shifts (cid:37) of affine Weylgroups with the standard realization of the root systems [4]. Proof. (1) ⇒ (2): If (cid:37) ∨ is admissible then for every w ∈ W and every p ∨ ∈ P ∨ there exists p ∨ ∈ P ∨ such that w ( (cid:37) ∨ + p ∨ ) = (cid:37) ∨ + p ∨ . Then (cid:37) ∨ − w(cid:37) ∨ = wp ∨ − p ∨ ∈ P ∨ .(2) ⇒ (3): If for every w ∈ W it holds that (cid:37) ∨ − w(cid:37) ∨ ∈ P ∨ , thus this equality is also valid for all r i ∈ W .(3) ⇒ (1): Any w ∈ W can be expressed as a product of generators, i.e. there exist indices i , i , . . . , i s ∈ { , . . . , n } such that w = r i r i . . . r i s . Thus, we have from the assumption that thereexist vectors p ∨ i , . . . , p ∨ i s ∈ P ∨ such that (cid:37) ∨ − r i (cid:37) ∨ = p ∨ i , (cid:37) ∨ − r i (cid:37) ∨ = p ∨ i , . . . , (cid:37) ∨ − r i s (cid:37) ∨ = p ∨ i s . Thenwe derive (cid:37) ∨ − w(cid:37) ∨ = (cid:37) ∨ − r i r i . . . r i s (cid:37) ∨ = (cid:37) ∨ − r i r i . . . r i s − ( (cid:37) ∨ − p ∨ i s )= (cid:37) ∨ − r i r i . . . r i s − ( (cid:37) ∨ − p ∨ i s − ) + r i r i . . . r i s − p ∨ i s = p ∨ i + r i p ∨ i + r i r i p ∨ i + · · · + r i r i . . . r i s − p ∨ i s . Denoting (cid:101) p ∨ = p ∨ i + r i p ∨ i + r i r i p ∨ i + · · · + r i r i . . . r i s − p ∨ i s we have that (cid:101) p ∨ ∈ P ∨ since P ∨ is W − invariant. Thus for all w ∈ W there exists (cid:101) p ∨ ∈ P ∨ such that (cid:37) ∨ − w(cid:37) ∨ = (cid:101) p ∨ and for all p ∨ ∈ P ∨ it holds that w ( (cid:37) ∨ + p ∨ ) = − (cid:101) p ∨ + (cid:37) ∨ + wp ∨ , i.e. there exists (cid:101) p (cid:48) ∨ = wp ∨ − (cid:101) p ∨ ∈ P ∨ such that w ( (cid:37) ∨ + p ∨ ) = (cid:37) ∨ + (cid:101) p (cid:48) ∨ and (cid:37) ∨ is admissible. (cid:3) Analyzing the condition (20), we note that it is advantageous to consider a shift (cid:37) ∨ – up to equivalence– in ω ∨ − basis, i.e. (cid:37) ∨ = y j ω ∨ j , y j ∈ (cid:104) , . (21)Using the relations (2), (3) and (5), we calculate that (cid:37) ∨ − r i (cid:37) ∨ = y i C ki ω ∨ k . Thus from (20), in order to (cid:37) ∨ be admissible, it has to hold y i C ki ∈ Z for all i ∈ { , . . . , n } . Let usdefine the numbers d i = gcd( C i , C i , . . . , C ni ) (22)i.e. the integer d i is the greatest common divisor of the i − th column of the Cartan matrix. Each d i can be expressed as the integer combination of C i , C i , . . . , C ni in the form d i = k i C i + k i C i + · · · + k i C i , k ki ∈ Z and we obtain from y i C ki ∈ Z that y i d i ∈ Z . Conversely from y i d i ∈ Z and C ki /d i ∈ Z we have that y i C ki ∈ Z and thus we conclude: Corollary 2.2.
A shift of the form (21) is admissible if and only if for all i ∈ { , . . . , n } it holds that d i y i ∈ Z where the numbers d i are defined by (22).The Cartan matrices taken from e.g. [4] are examined and cases for which a non-trivial admissibleshift exists, i.e. those with some d i = 2 and yielding y i = 1 /
2, are singled out. It appears that non-trivialadmissible shifts exist for the cases A , C and B n , n ≥
3. These are summarized in Table 1.
T. CZY ˙ZYCKI AND J. HRIVN ´AK
Dual affine Weyl group.
The dual affine Weyl group (cid:99) W aff is generated by the set R ∨ and, except for the case A , the Coxeterpresentation of (cid:99) W aff is of the form (cid:99) W aff = (cid:104) R ∨ | ( r ∨ i r ∨ j ) m ∨ ij = 1 (cid:105) . The numbers m ∨ ij , i, j ∈ { , , . . . , n } are deduced, following the identical rules as for W aff , from thedual extended Coxeter-Dynkin diagrams [13].The dual affine Weyl group (cid:99) W aff is a semidirect product of the group of shifts T ( Q ) and the Weylgroup W (cid:99) W aff = T ( Q ) (cid:111) W. For any w aff ∈ (cid:99) W aff there exist a unique element w ∈ W and a unique shift T ( q ), q ∈ Q such that w aff = T ( q ) w . Taking any w aff = T ( q ) w ∈ (cid:99) W aff , the dual retraction homomorphism (cid:98) ψ : (cid:99) W aff → W andthe mapping (cid:98) τ : (cid:99) W aff → Q are given by (cid:98) ψ ( w aff ) = w, (23) (cid:98) τ ( w aff ) = q. (24)The dual fundamental domain F ∨ of (cid:99) W aff is the convex hull of vertices (cid:110) , ω m ∨ , . . . , ω n m ∨ n (cid:111) . Considering n + 1 real parameters z , . . . , z n ≥
0, we have F ∨ = (cid:8) z ω + · · · + z n ω n | z + z m ∨ + · · · + z n m ∨ n = 1 (cid:9) . (25)Let us denote the isotropy subgroup of a point b ∈ R n by Stab (cid:99) W aff ( b ) and define for any M ∈ N afunction h ∨ M : R n → N by the relation h ∨ M ( b ) = (cid:12)(cid:12)(cid:12)(cid:12) Stab (cid:99) W aff (cid:18) bM (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (26)Recall that for a point b = z ω + · · · + z n ω n ∈ F ∨ such that z + z m ∨ + · · · + z n m ∨ n = 1 is theisotropy group trivial, Stab (cid:99) W aff ( b ) = 1, if b ∈ int( F ∨ ), i.e. all z i > i = 0 , . . . , n . Otherwise the groupStab (cid:99) W aff ( b ) is generated by such r ∨ i for which z i = 0, i = 0 , . . . , n .From F ∨ being a fundamental region follows that(1) For any b ∈ R n there exists b (cid:48) ∈ F ∨ , w ∈ W and q ∈ Q such that b = wb (cid:48) + q. (27)(2) If b, b (cid:48) ∈ F ∨ and b (cid:48) = w aff b , w aff ∈ (cid:99) W aff then b = b (cid:48) , i.e. if there exist w ∈ W and q ∈ Q suchthat b (cid:48) = wb + q then b (cid:48) = b = wb + q. (28)Considering a natural action of W on the quotient group R n /M Q , we denote for λ ∈ R n /M Q theisotropy group and its order byStab ∨ ( λ ) = { w ∈ W | wλ = λ } , (cid:101) h ∨ M ( λ ) ≡ | Stab ∨ ( λ ) | . (29)Recall the following property from Proposition 3.6 in [13] of the action of W on the quotient group R n /M Q . If λ ∈ M F ∨ ∩ R n /M Q , i.e. λ = b + M Q , b ∈ M F ∨ , then (cid:98) ψ (Stab (cid:99) W aff ( b/M )) = Stab ∨ ( λ ) andStab ∨ ( λ ) ∼ = Stab (cid:99) W aff ( b/M ) . (30)From (30) we obtain that for λ = b + M Q , b ∈ F ∨ it holds that h ∨ M ( b ) = (cid:101) h ∨ M ( λ ) . (31)Note that instead of (cid:101) h ∨ M ( λ ), the symbol h ∨ λ is used for | Stab ∨ ( λ ) | , λ ∈ R n /M Q in [11,13]. The calculationprocedure of the coefficients h ∨ λ is detailed in § ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 7
Admissible dual shifts.
The second key ingredient of the discrete Fourier calculus is a lattice, invariant under the action ofthe Weyl group W , which will label the sets of orthogonal functions. In [11,13] is formulated the discreteFourier calculus with special functions labeled by the labels from the W − invariant weight lattice P .Considering any vector (cid:37) ∈ R n , we call (cid:37) an admissible dual shift if the shifted lattice (cid:37) + P is still W − invariant, i.e. W ( (cid:37) + P ) = (cid:37) + P. Similarly to the shifts, if (cid:37) ∈ P then the resulting lattice (cid:37) + P = P does not change — such admissibledual shifts are called trivial. Also any two dual admissible shifts (cid:37) , (cid:37) which would differ by a trivialshift, i.e. (cid:37) − (cid:37) ∈ P lead to the same resulting shifted lattice and are equivalent. Rephrasing ofProposition 2.1 leads to the classification of the admissible dual shifts. Proposition 2.3.
Let (cid:37) ∈ R n . Then the following statements are equivalent.(1) (cid:37) is an admissible dual shift.(2) (cid:37) − W (cid:37) ⊂ P .(3) For all i ∈ { , . . . , n } it holds that (cid:37) − r i (cid:37) ∈ P. (32)Analyzing similarly the condition (32) and considering a dual shift (cid:37) up to equivalence in ω − basis,i.e. (cid:37) = y j ω j , y j ∈ (cid:104) , . (33)we calculate that (cid:37) − r i (cid:37) = y i C ik ω k . Defining the numbers d (cid:48) i = gcd( C i , C i , . . . , C in ) (34)i.e. the integer d (cid:48) i is the greatest common divisor of the i − th row of the Cartan matrix, we againconclude: Corollary 2.4.
A dual shift of the form (33) is admissible if and only if for all i ∈ { , . . . , n } it holdsthat d (cid:48) i y i ∈ Z where the numbers d (cid:48) i are defined by (34).The Cartan matrices are repeatedly examined and cases for which a non-trivial dual admissible shiftexists, i.e. those with some d (cid:48) i = 2 and yielding y i = 1 /
2, are singled out. It appears that non-trivialdual admissible shifts exist for the cases A , C and C n , n ≥ Shift homomorphisms.
An admissible shift (cid:37) ∨ induces a homomorphism from the dual affine Weyl group to the multiplicativetwo-element group {± } . This ’shift’ homomorphism (cid:98) θ (cid:37) ∨ : (cid:99) W aff → {± } is defined for w aff ∈ (cid:99) W aff ,using the mapping (24), via the equation (cid:98) θ (cid:37) ∨ ( w aff ) = e πi (cid:104) (cid:98) τ ( w aff ) , (cid:37) ∨ (cid:105) . (35)For trivial admissible shifts we obtain a trivial homomorphism (cid:98) θ (cid:37) ∨ ( w aff ) = 1. Since all non-trivialadmissible shifts (cid:37) ∨ are from P ∨ , the map (35) indeed maps to {± } . The key point to see that (35)defines a homomorphism is that the equality e πi (cid:104) wq, (cid:37) ∨ (cid:105) = e πi (cid:104) q, (cid:37) ∨ (cid:105) , q ∈ Q, w ∈ W, (36)which is equivalent to (cid:104) q, (cid:37) ∨ − w − (cid:37) ∨ (cid:105) ∈ Z , is valid because (cid:37) ∨ − w − (cid:37) ∨ ∈ P ∨ is the second statement ofProposition 2.1 and P ∨ is Z − dual to Q . Considering two dual affine Weyl group elements w aff1 , w aff2 ∈ (cid:99) W aff of the form w aff1 a = w a + q , w aff2 a = w a + q with w , w ∈ W and q , q ∈ Q we calculate that (cid:98) τ ( w aff1 w aff2 ) = w q + q and thus (cid:98) θ (cid:37) ∨ ( w aff1 w aff2 ) = e πi (cid:104) (cid:98) τ ( w aff1 w aff2 ) , (cid:37) ∨ (cid:105) = e πi (cid:104) w q + q , (cid:37) ∨ (cid:105) = e πi (cid:104) w q , (cid:37) ∨ (cid:105) e πi (cid:104) q , (cid:37) ∨ (cid:105) = e πi (cid:104) q , (cid:37) ∨ (cid:105) e πi (cid:104) q , (cid:37) ∨ (cid:105) = (cid:98) θ (cid:37) ∨ ( w aff1 ) (cid:98) θ (cid:37) ∨ ( w aff2 ) . T. CZY ˙ZYCKI AND J. HRIVN ´AK
Since the mapping is (35) a homomorphism, its value on any element of (cid:99) W aff is determined by the valuesof (35) on generators from R ∨ . For any reflection r ∨ , . . . , r ∨ n it trivially holds that (cid:98) τ ( r ∨ i ) = 0 and thus (cid:98) θ (cid:37) ∨ ( r ∨ i ) = 1 for i ∈ { , . . . , n } . Evaluating the number (cid:98) θ (cid:37) ∨ ( r ∨ ) one needs to take into account explicitlythe non-trivial admissible shifts from Table 1 and the dual highest roots η from [13]. It appears thatthe result for all cases is that for non-trivial admissible shifts it holds2 (cid:104) η, (cid:37) ∨ (cid:105)(cid:104) η, η (cid:105) = 12 . (37)Then from (8) and (37) we obtain (cid:98) θ (cid:37) ∨ ( r ∨ ) = e πi (cid:104) (cid:98) τ ( r ∨ ) , (cid:37) ∨ (cid:105) = e πi (cid:104) η, (cid:37) ∨(cid:105)(cid:104) η, η (cid:105) = − . Summarizing the results, we conclude with the following proposition.
Proposition 2.5.
The map (35) is for any admissible shift a homomorphism (cid:98) θ (cid:37) ∨ : (cid:99) W aff → {± } andfor any non-trivial admissible shift (cid:37) ∨ are the values on the generators R ∨ of (cid:99) W aff given as (cid:98) θ (cid:37) ∨ ( r ∨ i ) = (cid:40) , i ∈ { , . . . , n }− , i = 0 . (38)Similarly to the shift homomorphism (35) we define for the dual admissible shift (cid:37) the dual shifthomomorphism θ (cid:37) : W aff → {± } — for any w aff ∈ W aff , using the mapping (11), via the equation θ (cid:37) ( w aff ) = e πi (cid:104) τ ( w aff ) , (cid:37) (cid:105) . (39)Analogous relation to (36) is also valid e πi (cid:104) wq ∨ , (cid:37) (cid:105) = e πi (cid:104) q ∨ , (cid:37) (cid:105) , q ∨ ∈ Q ∨ , w ∈ W (40)and also 2 (cid:104) ξ, (cid:37) (cid:105)(cid:104) ξ, ξ (cid:105) = 12 . (41)Thus we conclude with the following proposition. Proposition 2.6.
The map (39) is for any admissible dual shift a homomorphism θ (cid:37) : W aff → {± } and for any non-trivial admissible dual shift (cid:37) are the values on the generators R of W aff given as θ (cid:37) ( r i ) = (cid:40) , i ∈ { , . . . , n }− , i = 0 . (42)Note that, excluding the case A , any homomorphism θ : W aff → {± } has to be compatible withthe generator relations (9) from the Coxeter presentation of W aff ,( θ ( r i ) θ ( r j )) m ij = 1 , i, j ∈ { , . . . , n } . Thus, except for the admissible cases C n , n ≥
2, a homomorphism θ (cid:37) defined on generators R via (42)cannot exist — the admissible cases are the only cases where the zero vertex of the extended Coxeter-Dynkin diagram is not connected to the rest of the diagram by a single vertex, i.e. m i , i ∈ { , . . . , n } are all even. Similarly, analyzing the dual extended Coxeter-Dynkin diagrams, we observe that theadmissible cases C and B n , n ≥ r , r and the dual generators r ∨ , r ∨ of A do not obey any relation betweenthem, the case A indeed admits both homomorphisms of the forms (42) and (38).2.7. Sign homomorphisms and γ − homomorphisms. Up to four classes of orbit functions are obtained using ’sign’ homomorphisms σ : W → {± } – seee.g. [11, 13]. These homomorphisms and their values on any w ∈ W are given as products of values ofgenerators r α , α ∈ ∆. The following two choices of homomorphism values of generators r α , α ∈ ∆ leadto the trivial and determinant homomorphisms ( r α ) = 1 (43) σ e ( r α ) = − , (44) ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 9 γ γ σ e γ σ s γ σ l γ (cid:37) γ σ e (cid:37) γ σ s (cid:37) γ σ l (cid:37) r − − − − r α , α ∈ ∆ l − − − − r α , α ∈ ∆ s − − − − Table 2.
The values of γ σ(cid:37) ( r ) , r ∈ R for a trivial 0 − shift, the non-trivial admissibledual shift (cid:37) and four sign homomorphisms σ .yielding for any w ∈ W ( w ) = 1 (45) σ e ( w ) = det w. (46)For root systems with two different lengths of roots, there are two other available choices. Using thedecomposition (1), these two new homomorphisms are given as follows [11]: σ s ( r α ) = (cid:40) , α ∈ ∆ l − , α ∈ ∆ s (47) σ l ( r α ) = (cid:40) , α ∈ ∆ s − , α ∈ ∆ l . (48)From [11] we also have the values for the reflections (6) and (8) from W,σ s ( r ξ ) = 1 , σ l ( r ξ ) = − , (49) σ s ( r η ) = − , σ l ( r η ) = 1 . (50)At this point we have all three key homomorphisms ready – the retraction homomorphism ψ , thedual shift homomorphism θ (cid:37) and up to four sign homomorphisms σ given by (10), (39) and (45), (46),(47), (48), respectively. We use these three homomorphisms to create the fourth and most ubiquitoushomomorphism γ σ(cid:37) : W aff → {± } given by γ σ(cid:37) ( w aff ) = θ (cid:37) ( w aff ) · [ σ ◦ ψ ( w aff )] , w aff ∈ W aff . (51)Note that, since for the affine reflection r ∈ R it holds that ψ ( r ) = r ξ , we have from (49) that σ s ◦ ψ ( r ) = 1 , σ l ◦ ψ ( r ) = − . (52)The values of the γ σ(cid:37) − homomorphism on any w aff ∈ W aff are determined by its values on the generatorsfrom R . Putting together the values from (42), (43) — (50) and (52), we summarize the values of γ σ(cid:37) ( r ) , r ∈ R for a trivial 0 − shift and the non-trivial admissible dual shift (cid:37) in Table 2.We use four sign homomorphisms σ , the dual retraction homomorphism (23) and the shift homomor-phism (35) to define a dual version of the γ σ(cid:37) − homomorphism (cid:98) γ σ(cid:37) ∨ : (cid:99) W aff → {± } by relation (cid:98) γ σ(cid:37) ∨ ( w aff ) = (cid:98) θ (cid:37) ∨ ( w aff ) · [ σ ◦ (cid:98) ψ ( w aff )] , w aff ∈ (cid:99) W aff . (53)Note that, since for the dual affine reflection r ∨ ∈ R ∨ it holds that (cid:98) ψ ( r ∨ ) = r η , we have from (50) that σ s ◦ (cid:98) ψ ( r ∨ ) = 1 , σ l ◦ (cid:98) ψ ( r ∨ ) = − . (54)The values of the (cid:98) γ σ(cid:37) ∨ − homomorphism on any w aff ∈ (cid:99) W aff are determined by its values on the generatorsfrom R ∨ . Putting together the values from (38), (43) — (50) and (54), we summarize the values of (cid:98) γ σ(cid:37) ∨ ( r ) , r ∈ R for a trivial 0 − shift and the non-trivial admissible shift (cid:37) ∨ in Table 3. (cid:98) γ (cid:98) γ σ e (cid:98) γ σ s (cid:98) γ σ l (cid:98) γ (cid:37) ∨ (cid:98) γ σ e (cid:37) ∨ (cid:98) γ σ s (cid:37) ∨ (cid:98) γ σ l (cid:37) ∨ r ∨ − − − − r α , α ∈ ∆ l − − − − r α , α ∈ ∆ s − − − − Table 3.
The values of (cid:98) γ σ(cid:37) ∨ ( r ) , r ∈ R ∨ for a trivial 0 − shift, the non-trivial admissibleshift (cid:37) ∨ and four sign homomorphisms σ .2.8. Fundamental domains.
Any sign homomorphism and any admissible dual shift determine a decomposition of the fundamentaldomain F . The factors of this decomposition are essential for the study of discrete orbit functions. Forany sign homomorphism σ and any admissible dual shift (cid:37) we introduce a subset F σ ( (cid:37) ) of FF σ ( (cid:37) ) = (cid:8) a ∈ F | γ σ(cid:37) (Stab W aff ( a )) = { } (cid:9) with γ σ(cid:37) − homomorphism given by (51). Since for all points of the interior of F the stabilizer is trivial,i.e. Stab W aff ( a ) = 1, a ∈ int( F ), the interior int( F ) is a subset of any F σ ( (cid:37) ). Let us also define thecorresponding subset R σ ( (cid:37) ) of generators R of W aff R σ ( (cid:37) ) = (cid:8) r ∈ R | γ σ(cid:37) ( r ) = − (cid:9) (55)The sets R σ ( (cid:37) ) are straightforwardly explicitly determined for any case from Table 2. In order todetermine the analytic form of the sets F σ ( (cid:37) ), we define subsets of the boundaries of FH σ ( (cid:37) ) = { a ∈ F | ( ∃ r ∈ R σ ( (cid:37) ))( ra = a ) } . (56) Proposition 2.7.
For the sets F σ ( (cid:37) ) it holds that F σ ( (cid:37) ) = F \ H σ ( (cid:37) ) . Proof.
Let a ∈ F . If a / ∈ F \ H σ ( (cid:37) ), then a ∈ H σ ( (cid:37) ), and there exists r ∈ R σ ( (cid:37) ) such that r ∈ Stab W aff ( a ). Then according to (55), we have γ σ(cid:37) ( r ) = −
1. Thus, γ σ(cid:37) (Stab W aff ( a )) = {± } andconsequently a / ∈ F σ ( (cid:37) ). Conversely, if a ∈ F \ H σ ( (cid:37) ), then the stabilizer Stab W aff ( a ) is either trivialor generated by generators from R \ R σ ( (cid:37) ) only. Then, since for any generator r ∈ R \ R σ ( (cid:37) ) it followsfrom (55) that γ σ(cid:37) ( r ) = 1, we obtain γ σ(cid:37) (Stab W aff ( a )) = { } , i.e. a ∈ F σ ( (cid:37) ). (cid:3) The explicit description of all domains F σ ( (cid:37) ) follows from (12) and Proposition 2.7. We define thesymbols y σ,(cid:37)i ∈ R , i = 0 , . . . , n in the following way: y σ,(cid:37)i ∈ (cid:40) R > , r i ∈ R σ ( (cid:37) ) R ≥ , r i ∈ R \ R σ ( (cid:37) ) . Thus, the explicit form of F σ ( (cid:37) ) is given by F σ ( (cid:37) ) = (cid:8) y σ,(cid:37) ω ∨ + · · · + y σ,(cid:37)n ω ∨ n | y σ,(cid:37) + y σ,(cid:37) m + · · · + y σ,(cid:37)n m n = 1 (cid:9) . (57)2.9. Dual fundamental domains.
The dual version of the γ − homomorphism (cid:98) γ σ(cid:37) ∨ also determines a decomposition of the dual funda-mental domain F ∨ . The factors of this decomposition are essential for the study of the discretized orbitfunctions. We define subsets F σ ∨ ( (cid:37) ∨ ) of F ∨ by F σ ∨ ( (cid:37) ∨ ) = (cid:8) a ∈ F ∨ | (cid:98) γ σ(cid:37) ∨ (cid:0) Stab (cid:99) W aff ( a ) (cid:1) = { } (cid:9) (58)where (cid:98) γ σ(cid:37) ∨ is given by (53) and (cid:37) ∨ is an admissible shift. Since for all points of the interior of F ∨ thestabilizer is trivial, i.e. Stab (cid:99) W aff ( a ) = 1, a ∈ int( F ∨ ), the interior int( F ∨ ) is a subset of all F σ ∨ ( (cid:37) ∨ ) .Let us also define the corresponding subset R σ ∨ ( (cid:37) ∨ ) of generators R ∨ of (cid:99) W aff R σ ∨ ( (cid:37) ∨ ) = (cid:8) r ∈ R ∨ | (cid:98) γ σ(cid:37) ∨ ( r ) = − (cid:9) (59) ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 11
The sets R σ ∨ ( (cid:37) ∨ ) are straightforwardly explicitly determined for any case from Table 3. We definesubsets of the boundaries of F ∨ by H σ ∨ ( (cid:37) ∨ ) = (cid:8) a ∈ F ∨ | ( ∃ r ∈ R σ ∨ ( (cid:37) ∨ ))( ra = a ) (cid:9) . (60)Similarly to Proposition 2.7 we derive its dual version. Proposition 2.8.
For the sets F σ ∨ ( (cid:37) ∨ ) it holds that F σ ∨ ( (cid:37) ∨ ) = F ∨ \ H σ ∨ ( (cid:37) ∨ ) . The explicit description of all domains F σ ∨ ( (cid:37) ∨ ) follows from (25) and Proposition 2.8. We define thesymbols z σ,(cid:37) ∨ i ∈ R , i = 0 , . . . , n in the following way: z σ,(cid:37) ∨ i ∈ (cid:40) R > , r i ∈ R σ ∨ ( (cid:37) ∨ ) R ≥ , r i ∈ R ∨ \ R σ ∨ ( (cid:37) ∨ ) . Thus, the explicit form of F σ ∨ ( (cid:37) ∨ ) is given by F σ ∨ ( (cid:37) ∨ ) = (cid:110) z σ,(cid:37) ∨ ω + · · · + z σ,(cid:37) ∨ n ω n | z σ,(cid:37) ∨ + z σ,(cid:37) ∨ m ∨ + · · · + z σ,(cid:37) ∨ n m ∨ n = 1 (cid:111) . (61)3. Four types of orbit functions
Symmetries of orbit functions.
Four sign homomorphisms , σ e , σ s and σ l determine four types of families of complex orbit functionsfor root systems with two different lenghts of roots. If a given root system has only one length of rootsthere exist two homomorphism , σ e only. Within each family of special functions, determined by asign homomorphism σ , are the complex functions ϕ σb : R n → C standardly labeled by weights b ∈ P .In this article we start by allowing b ∈ R n and we have the orbit functions of the general form ϕ σb ( a ) = (cid:88) w ∈ W σ ( w ) e πi (cid:104) wb, a (cid:105) , a ∈ R n . The discretization properties of all four cases of these functions on a finite fragment of the grid M P ∨ and with b ∈ P were described in [11, 13].Here we firstly examine the discretization by taking discrete values of b ∈ (cid:37) + P , with (cid:37) an admissibledual shift. Using the γ σ(cid:37) − homomorphism, defined by (51), we are able to describe invariance propertiesof all types of orbit functions in the following compact form. Proposition 3.1.
Let (cid:37) be an admissible dual shift and b ∈ (cid:37) + P . Then for any w aff ∈ W aff and a ∈ R n it holds that ϕ σb ( w aff a ) = γ σ(cid:37) ( w aff ) ϕ σb ( a ) . (62)Moreover the functions ϕ σb are zero on the boundary H σ ( (cid:37) ), i.e. ϕ σb ( a (cid:48) ) = 0 , a (cid:48) ∈ H σ ( (cid:37) ) . (63) Proof.
Considering an element of the affine Weyl group of the form w aff a = w (cid:48) a + q ∨ , w (cid:48) ∈ W , q ∨ ∈ Q ∨ and b = (cid:37) + λ , with λ ∈ P , we firstly calculate that e πi (cid:104) wb, q ∨ (cid:105) = e πi (cid:104) λ + (cid:37), w − q ∨ (cid:105) = e πi (cid:104) (cid:37), w − q ∨ (cid:105) = e πi (cid:104) (cid:37), q ∨ (cid:105) = θ (cid:37) ( w aff )where the first equality follows from w ∈ W being an orthogonal map, the second from W − invarianceof Q ∨ and Z − duality of the lattices P and Q ∨ , the third is equation (40). Then we have that ϕ σb ( w aff a ) = (cid:88) w ∈ W σ ( w ) e πi (cid:104) wb, w (cid:48) a + q ∨ (cid:105) = (cid:88) w ∈ W σ ( w ) e πi (cid:104) wb, w (cid:48) a (cid:105) e πi (cid:104) wb, q ∨ (cid:105) = θ (cid:37) ( w aff ) (cid:88) w ∈ W σ ( w ) e πi (cid:104) wb, w (cid:48) a (cid:105) = θ (cid:37) ( w aff ) σ ( w (cid:48) ) ϕ σb ( a ) which is equation (62). Putting the generators (55) from R σ ( (cid:37) ) into (62) we immediately have for thepoints from H σ ( (cid:37) ) that ϕ σb ( a ) = ϕ σb ( ra ) = − ϕ σb ( a ) , r ∈ R σ ( (cid:37) ) , a ∈ H σ ( (cid:37) ) . (cid:3) Thus, all orbit functions are (anti)symmetric with respect to the action of the affine Weyl group W aff .This allows us to consider the values ϕ σb ( a ) only for points of the fundamental domain a ∈ F . Moreover,from (63) and Proposition 2.7 we conclude that we can consider the functions ϕ σb , b ∈ (cid:37) + P on thedomain F σ ( (cid:37) ) only .Secondly, reverting to a general b ∈ R n , we discretize the values of a by taking a ∈ M ( (cid:37) ∨ + P ∨ ) with (cid:37) ∨ an admissible shift and any M ∈ N . Using the (cid:98) γ σ(cid:37) ∨ − homomorphism, defined by (53), we describeinvariance with respect to the action of (cid:99) W aff in the following compact form. Proposition 3.2.
Let (cid:37) ∨ be an admissible shift and a ∈ M ( (cid:37) ∨ + P ∨ ) with M ∈ N . Then for any w aff ∈ (cid:99) W aff and b ∈ R n it holds that ϕ σMw aff ( bM ) ( a ) = (cid:98) γ σ(cid:37) ∨ ( w aff ) ϕ σb ( a ) . (64)Moreover the functions ϕ σb are identically zero on the boundary M H σ ∨ ( (cid:37) ∨ ), i.e. ϕ σb ≡ , b ∈ M H σ ∨ ( (cid:37) ∨ ) . (65) Proof.
Considering an element of the dual affine Weyl group of the form w aff a = w (cid:48) a + q and a =( (cid:37) ∨ + s ) /M , with s ∈ P ∨ , we firstly calculate that e πi (cid:104) qM, wa (cid:105) = e πi (cid:104) w − q, (cid:37) ∨ + s (cid:105) = e πi (cid:104) (cid:37) ∨ , w − q (cid:105) = e πi (cid:104) (cid:37) ∨ , q (cid:105) = (cid:98) θ (cid:37) ∨ ( w aff )where the first equality follows from w ∈ W being an orthogonal map, the second from W − invarianceof Q and Z − duality of the lattices P ∨ and Q , the third is equation (36). Then we have that ϕ σMw aff ( bM ) ( a ) = (cid:88) w ∈ W σ ( w ) e πi (cid:104) w (cid:48) b + qM, wa (cid:105) = (cid:88) w ∈ W σ ( w ) e πi (cid:104) w (cid:48) b, wa (cid:105) e πi (cid:104) qM, wa (cid:105) = (cid:98) θ (cid:37) ∨ ( w aff ) (cid:88) w ∈ W σ ( w ) e πi (cid:104) w (cid:48) b, wa (cid:105) = (cid:98) θ (cid:37) ∨ ( w aff ) σ ( w (cid:48) ) ϕ σb ( a )which is equation (64). Putting the generators (59) from R σ ∨ ( (cid:37) ∨ ) into (64) we immediately have forthe points from M H σ ∨ ( (cid:37) ∨ ) that ϕ σb ( a ) = ϕ σMr ( bM ) ( a ) = − ϕ σb ( a ) , r ∈ R σ ∨ ( (cid:37) ∨ ) , b ∈ M H σ ∨ ( (cid:37) ∨ ) . (cid:3) Thus, all orbit functions are (anti)symmetric with respect to the action of the modified dual affineWeyl group — the group T ( M Q ) (cid:111) W . This allows us to consider the functions ϕ σb only for labels fromthe fundamental domain b ∈ M F ∨ . Moreover, from (65) and Proposition 2.8 we conclude that we canconsider the functions ϕ σb ( a ) , a ∈ M ( (cid:37) ∨ + P ∨ ) with the labels b from M F σ ∨ ( (cid:37) ∨ ) only .3.2. Discretization of orbit functions.
By discretization of orbit functions we mean formulating their discrete Fourier calculus; at this pointthis step is straightforward — we discretize both the arguments of the functions ϕ σb ( a ) to M ( (cid:37) ∨ + P ∨ )and the labels to (cid:37) + P with (cid:37) and (cid:37) ∨ being admissible. Combining propositions 3.1 and 3.2, we obtainthat the discrete values a of the functions ϕ σb ( a ) can be taken only on the set F σM ( (cid:37), (cid:37) ∨ ) = (cid:20) M ( (cid:37) ∨ + P ∨ ) (cid:21) ∩ F σ ( (cid:37) ) (66)and the labels b are then considered in the setΛ σM ( (cid:37), (cid:37) ∨ ) = ( (cid:37) + P ) ∩ M F σ ∨ ( (cid:37) ∨ ) . (67) ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 13
The explicit form of these sets is crucial for application and can be for each case of admissible shifts (cid:37) ∨ = (cid:37) ∨ ω ∨ + · · · + (cid:37) ∨ n ω ∨ n and admissible dual shifts (cid:37) = (cid:37) ω + · · · + (cid:37) n ω n derived from (57) and (61).If we define the symbols u σ,(cid:37)i ∈ (cid:40) N , r i ∈ R σ ( (cid:37) ) Z ≥ , r i ∈ R \ R σ ( (cid:37) )and the symbols t σ,(cid:37) ∨ i ∈ (cid:40) N , r i ∈ R σ ∨ ( (cid:37) ∨ ) Z ≥ , r i ∈ R ∨ \ R σ ∨ ( (cid:37) ∨ )we obtain the set F σM ( (cid:37), (cid:37) ∨ ) explicitly F σM ( (cid:37), (cid:37) ∨ ) = (cid:26) u σ,(cid:37) + (cid:37) ∨ M ω ∨ + · · · + u σ,(cid:37)n + (cid:37) ∨ n M ω ∨ n | u σ,(cid:37) + u σ,(cid:37) m + · · · + u σ,(cid:37)n m n = M (cid:27) (68)and also the set Λ σM ( (cid:37), (cid:37) ∨ )Λ σM ( (cid:37), (cid:37) ∨ ) = (cid:110) ( t σ,(cid:37) ∨ + (cid:37) ) ω + · · · + ( t σ,(cid:37) ∨ n + (cid:37) n ) ω n | t σ,(cid:37) ∨ + t σ,(cid:37) ∨ m ∨ + · · · + t σ,(cid:37) ∨ n m ∨ n = M (cid:111) . (69)The counting formulas for the numbers of elements of all F σM (0 ,
0) are derived in [11,13]. Recall from [13]that the number of elements of | F M (0 , | of C n , n ≥ ν n ( M ), where ν n ( M ) is given on oddand even values of M by, ν n (2 k ) = (cid:18) n + kn (cid:19) + (cid:18) n + k − n (cid:19) ν n (2 k + 1) =2 (cid:18) n + kn (cid:19) . Theorem 3.3.
Let (cid:37) ∨ , (cid:37) be the non-trivial admissible and the dual admissible shifts, respectively. Thenumbers of points of grids F σM of Lie algebras A , B n , C n are given by the following relations.(1) A , | F M ( (cid:37), | = | F M (0 , (cid:37) ∨ ) | = | F M ( (cid:37), (cid:37) ∨ ) | = M − , | F σ e M ( (cid:37), | = | F σ e M (0 , (cid:37) ∨ ) | = | F σ e M ( (cid:37), (cid:37) ∨ ) | = M − . (2) C , | F M ( (cid:37), | = | F M (0 , (cid:37) ∨ ) | = | F σ s M (0 , (cid:37) ∨ ) | = | F σ l M ( (cid:37), | = ν ( M − | F M ( (cid:37), (cid:37) ∨ ) | = | F σ e M ( (cid:37), (cid:37) ∨ ) | = | F σ s M ( (cid:37), (cid:37) ∨ ) | = | F σ l M ( (cid:37), (cid:37) ∨ ) | = ν ( M − , | F σ e M (0 , (cid:37) ∨ ) | = | F σ e M ( (cid:37), | = | F σ s M ( (cid:37), | = | F σ l M (0 , (cid:37) ∨ ) | = ν ( M − , (3) C n , n ≥ , | F M ( (cid:37), | = | F σ l M ( (cid:37), | = ν n ( M − | F σ e M ( (cid:37), | = | F σ s M ( (cid:37), | = ν n ( M − n + 1)(4) B n , n ≥ , | F M (0 , (cid:37) ∨ ) | = | F σ s M (0 , (cid:37) ∨ ) | = ν n ( M − | F σ e M (0 , (cid:37) ∨ ) | = | F σ l M (0 , (cid:37) ∨ ) | = ν n ( M − n + 1) . Proof.
The number of points is calculated for each case of admissible shifts using formula (68). For thecase C n , we have from Table 2 that R σ s ( (cid:37) ) = { r , r , . . . , r n − } and thus F σ s M ( (cid:37),
0) = (cid:40) u σ s ,(cid:37) M ω ∨ + · · · + u σ s ,(cid:37)n M ω ∨ n | u σ s ,(cid:37) + 2 u σ s ,(cid:37) + · · · + 2 u σ s ,(cid:37)n − + u σ s ,(cid:37)n = M (cid:41) (70) where u σ s ,(cid:37)i ∈ (cid:40) N , i ∈ { , , . . . , n − } , Z ≥ , i = n. Introducing new variables (cid:101) u σ s ,(cid:37)i ∈ Z ≥ and setting u σ s ,(cid:37)i = (cid:40)(cid:101) u σ s ,(cid:37)i + 1 , i ∈ { , , . . . , n − } , (cid:101) u σ s ,(cid:37)i , i = n. the defining equation in the set (70) can be rewritten as (cid:101) u σ s ,(cid:37) + 2 (cid:101) u σ s ,(cid:37) + · · · + 2 (cid:101) u σ s ,(cid:37)n − + (cid:101) u σ s ,(cid:37)n = M − n + 1 . This equation is the same as the defining equation of F M − n +1 (0 ,
0) of C n , hence | F σ s M ( (cid:37), | = | F M − n +1 (0 , | (71)and the counting formula follows. Similarly, we obtain formulas for the remaining cases. (cid:3) Example . For the Lie algebra C , we have its determinant of the Cartain matrix c = 2. For M = 4is the order of the group P ∨ /Q ∨ equal to 32, and according to Theorem 3.3 we calculate | F ( (cid:37), | = | F (0 , (cid:37) ∨ ) | = | F σ s (0 , (cid:37) ∨ ) | = | F σ l ( (cid:37), | = 6 | F ( (cid:37), (cid:37) ∨ ) | = | F σ e ( (cid:37), (cid:37) ∨ ) | = | F σ s ( (cid:37), (cid:37) ∨ ) | = | F σ l ( (cid:37), (cid:37) ∨ ) | = 4 , | F σ e (0 , (cid:37) ∨ ) | = | F σ e ( (cid:37), | = | F σ s ( (cid:37), | = | F σ l (0 , (cid:37) ∨ ) | = 2 , Let us denote the boundaries of the triangle F which are stabilized by the reflection r i by b i , i ∈ { , , } .Then the sets H σ ( (cid:37) ) of boundaries (56) are of the explicit form H ( (cid:37) ) = { b } , H σ e ( (cid:37) ) = { b , b } ,H σ s ( (cid:37) ) = { b , b } , H σ l ( (cid:37) ) = { b } . The coset representatives of the finite group P ∨ /Q ∨ , the shifted representatives of the set of cosets ( (cid:37) ∨ + P ∨ ) /Q ∨ and the fundamental domain F with its boundaries { b , b , b } are depicted in Figure 1.It is established in [11, 13] that the numbers of elements of F σM (0 ,
0) coincide with the numbers ofelements of Λ σM (0 , Theorem 3.4.
Let (cid:37) ∨ , (cid:37) be any admissible and any dual admissible shifts, respectively. Then it holdsthat | Λ σM ( (cid:37), (cid:37) ∨ ) | = | F σM ( (cid:37), (cid:37) ∨ ) | . Proof.
The equality | Λ σM (0 , | = | F σM (0 , | is proved in [11, 13]. For other combinations of admissibleshifts is the number of points calculated for each case using formula (69). For the case C n , we havefrom Table 3 that R σ s ∨ (0) = { r ∨ , r , . . . , r n − } and thusΛ σ s M ( (cid:37),
0) = (cid:26) t σ s , ω + · · · + (cid:18) t σ s , n + 12 (cid:19) ω n | t σ s , + t σ s , + 2 t σ s , + · · · + 2 (cid:18) t σ s , n + 12 (cid:19) = M (cid:27) (72)where t σ s , i ∈ (cid:40) N , i ∈ { , , . . . , n − } , Z ≥ , i = n. Introducing new variables (cid:101) t σ s ,(cid:37)i ∈ Z ≥ and setting t σ s , i = (cid:40)(cid:101) t σ s , i + 1 , i ∈ { , , . . . , n − } , (cid:101) t σ s ,(cid:37)i , i = n. the defining equation in the set (72) can be rewritten as (cid:101) t σ s , + (cid:101) t σ s , + 2 (cid:101) t σ s , + · · · + 2 (cid:101) t σ s , n = M − n + 1 . ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 15 b r ξ = ω ∨ α ∨ α r α = α ∨ ω = ω ∨ b b ω η Fr Figure 1.
The fundamental domain F and its boundaries of C . The fundamental domain F is depicted as the grey triangle containing borders b , b and b , which are depicted as the thickdashed line, dot-and-dashed and dot-dot-and-dashed lines, respectively. The coset representativesof P ∨ /Q ∨ and the shifted representatives of cosets ( (cid:37) ∨ + P ∨ ) /Q ∨ are shown as 32 white andblack dots, respectively. The six black points in F form the set F (0 , (cid:37) ∨ ). The dashed linesrepresent ’mirrors’ r , r and r . Larger circles are elements of the root lattice Q ; together withthe squares they are elements of the weight lattice P . This equation is the same as the defining equation of Λ M − n +1 (0 ,
0) of C n and taking into account (71)we obtain that | Λ σ s M ( (cid:37), | = | Λ M − n +1 (0 , | = | F M − n +1 (0 , | = | F σ s M ( (cid:37), | . Similarly, we obtain the equalities for the remaining cases. (cid:3)
Example . For the Lie algebra C and for M = 4 is the order of the group P/ Q equal to 32, andaccording to Theorem 3.4 we calculate | Λ ( (cid:37), | = | Λ (0 , (cid:37) ∨ ) | = | Λ σ s (0 , (cid:37) ∨ ) | = | Λ σ l ( (cid:37), | = 6 | Λ ( (cid:37), (cid:37) ∨ ) | = | Λ σ e ( (cid:37), (cid:37) ∨ ) | = | Λ σ s ( (cid:37), (cid:37) ∨ ) | = | Λ σ l ( (cid:37), (cid:37) ∨ ) | = 4 , | Λ σ e (0 , (cid:37) ∨ ) | = | Λ σ e ( (cid:37), | = | Λ σ s ( (cid:37), | = | Λ σ l (0 , (cid:37) ∨ ) | = 2 , Let us denote the boundaries of the triangle 4 F ∨ which are stabilized by the reflection r ∨ i by b ∨ i , i ∈ { , , } . Then the sets 4 H σ ∨ ( (cid:37) ∨ ) of boundaries (60) are the relevant boundaries of 4 F ∨ and aregiven as 4 H ∨ ( (cid:37) ∨ ) = { b ∨ } , H σ e ∨ ( (cid:37) ∨ ) = { b ∨ , b ∨ } , H σ s ∨ ( (cid:37) ∨ ) = { b ∨ } , H σ l ∨ ( (cid:37) ∨ ) = { b ∨ , b ∨ } . The coset representatives of the finite group P/ Q , the shifted representatives of the set of cosets( (cid:37) + P ) / Q and the magnified fundamental domain 4 F ∨ with its boundaries { b ∨ , b ∨ , b ∨ } are depictedin Figure 2. b ∨ α α α ∨ α α = α ∨ ω r ∨ , ω F ∨ F ∨ ξ = ω ∨ b ∨ ω = ω ∨ b ∨ r ∨ r ∨ r ∨ η Figure 2.
The magnified fundamental domain 4 F ∨ and its boundaries of C . The darkergrey triangle is the fundamental domain F ∨ and the lighter grey triangle is the domain 4 F ∨ containing borders b ∨ , b ∨ and b ∨ , which are depicted as the thick dashed line, dot-and-dashedand dot-dot-and-dashed lines, respectively. The coset representatives of P/ Q and the shiftedrepresentatives of cosets ( (cid:37) + P ) / Q are shown as 32 white and black dots, respectively. The sixblack points in 4 F ∨ form the set Λ ( (cid:37), r ∨ , r ∨ and r ∨ and the affine mirror r ∨ , is defined by r ∨ , λ = 4 r ∨ ( λ/ Discrete orthogonality and transforms of orbit functions
Discrete orthogonality of orbit functions.
To describe the discrete orthogonality of the ϕ σb functions with shifted points from F σM ( (cid:37), (cid:37) ∨ ) andshifted weights from Λ σM ( (cid:37), (cid:37) ∨ ), we need to generalize the discrete orthogonality from [11,13,23] . Recallthat basic discrete orthogonality relations of the exponentials from [13, 23] imply for any µ ∈ P that (cid:88) y ∈ M P ∨ /Q ∨ e πi (cid:104) µ, y (cid:105) = (cid:40) cM n , µ ∈ M Q , µ / ∈ M Q. (73)The scalar product of any two functions f, g : F σM ( (cid:37), (cid:37) ∨ ) → C is given by (cid:104) f, g (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = (cid:88) a ∈ F σM ( (cid:37),(cid:37) ∨ ) ε ( a ) f ( a ) g ( a ) , (74)where the numbers ε ( a ) are determined by (13). We prove that Λ σM ( (cid:37), (cid:37) ∨ ) is the lowest maximal set ofpairwise orthogonal orbit functions. Theorem 4.1.
Let (cid:37) ∨ , (cid:37) be any admissible and any dual admissible shifts, respectively, and σ be asign homomorphism. Then for any b, b (cid:48) ∈ Λ σM ( (cid:37), (cid:37) ∨ ) it holds that (cid:104) ϕ σb , ϕ σb (cid:48) (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = c | W | M n h ∨ M ( b ) δ b,b (cid:48) , (75)where c , h ∨ M ( b ) were defined by (4), (26), respectively, | W | is the number of elements of the Weyl group. ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 17
Proof.
Since due to (63) any ϕ σb vanishes on H σ ( (cid:37) ) and it holds that F σM ( (cid:37), (cid:37) ∨ ) ∪ (cid:20) M ( (cid:37) ∨ + P ∨ ) ∩ H σ ( (cid:37) ) (cid:21) = (cid:20) M ( (cid:37) ∨ + P ∨ ) (cid:21) ∩ F, we have (cid:104) ϕ σb , ϕ σb (cid:48) (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = (cid:88) a ∈ F σM ( (cid:37),(cid:37) ∨ ) ε ( a ) ϕ σb ( a ) ϕ σb (cid:48) ( a ) = (cid:88) a ∈ [ M ( (cid:37) ∨ + P ∨ ) ] ∩ F ε ( a ) ϕ σb ( a ) ϕ σb (cid:48) ( a )Combining the relations (14) and (62), we observe that the expression ε ( a ) ϕ σb ( a ) ϕ σb (cid:48) ( a ) is W aff − invariant;the shift invariance with respect to Q ∨ , together with (18), implies that (cid:88) a ∈ [ M ( (cid:37) ∨ + P ∨ ) ] ∩ F ε ( a ) ϕ σb ( a ) ϕ σb (cid:48) ( a ) = (cid:88) x ∈ [ M ( (cid:37) ∨ + P ∨ ) /Q ∨ ] ∩ F (cid:101) ε ( x ) ϕ σb ( x ) ϕ σb (cid:48) ( x )and taking into account also W − invariance of (cid:101) ε ( x ) ϕ σb ( x ) ϕ σb (cid:48) ( x ) together with (15) and (16), we obtain (cid:88) x ∈ [ M ( (cid:37) ∨ + P ∨ ) /Q ∨ ] ∩ F (cid:101) ε ( x ) ϕ σb ( x ) ϕ σb (cid:48) ( x ) = (cid:88) y ∈ M ( (cid:37) ∨ + P ∨ ) /Q ∨ ϕ σb ( y ) ϕ σb (cid:48) ( y ) . Using the W − invariance of M ( (cid:37) ∨ + P ∨ ) /Q ∨ , which follows from (19), we obtain (cid:104) ϕ σb , ϕ σb (cid:48) (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = (cid:88) w (cid:48) ∈ W (cid:88) w ∈ W (cid:88) y ∈ M ( (cid:37) ∨ + P ∨ ) /Q ∨ σ ( ww (cid:48) ) e π i (cid:104) wb − w (cid:48) b (cid:48) , y (cid:105) = | W | (cid:88) w (cid:48) ∈ W (cid:88) y ∈ M ( (cid:37) ∨ + P ∨ ) /Q ∨ σ ( w (cid:48) ) e π i (cid:104) b − w (cid:48) b (cid:48) , y (cid:105) = | W | (cid:88) w (cid:48) ∈ W σ ( w (cid:48) ) e π i (cid:104) b − w (cid:48) b (cid:48) , (cid:37) ∨ M (cid:105) (cid:88) y ∈ M P ∨ /Q ∨ e π i (cid:104) b − w (cid:48) b (cid:48) , y (cid:105) . Having the labels b and b (cid:48) of the form b = (cid:37) + λ and b (cid:48) = (cid:37) + λ (cid:48) , with λ, λ (cid:48) ∈ P , we obtain fromProposition 2.3 that b − w (cid:48) b (cid:48) = (cid:37) − w (cid:48) (cid:37) + λ − λ (cid:48) ∈ P. Thus, the basic orthogonality relation (73) can be used. Taking into account that (28) together with b, b (cid:48) ∈ M F ∨ and b − w (cid:48) b (cid:48) ∈ M Q , for some w (cid:48) ∈ W , forces b = b (cid:48) , we observe that if b (cid:54) = b (cid:48) thenfor all w (cid:48) ∈ W it has to hold that b − w (cid:48) b (cid:48) / ∈ M Q . We conclude that if b (cid:54) = b (cid:48) then (73) implies (cid:104) ϕ σb , ϕ σb (cid:48) (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = 0. On the other hand if b = b (cid:48) then (73) forces all summands for which b − w (cid:48) b / ∈ M Q ,i.e. w (cid:48) / ∈ (cid:98) ψ (Stab (cid:99) W aff (cid:0) bM (cid:1) ) to vanish. Therefore we finally obtain (cid:104) ϕ σb , ϕ σb (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = c | W | M n (cid:88) w (cid:48) ∈ (cid:98) ψ [ Stab (cid:99) W aff ( bM )] σ ( w (cid:48) ) e π i (cid:104) b − w (cid:48) b, (cid:37) ∨ M (cid:105) (76)= c | W | M n (cid:88) w aff ∈ Stab (cid:99) W aff ( bM ) σ ( (cid:98) ψ [ w aff ]) e π i (cid:104) (cid:98) τ [ w aff ] , (cid:37) ∨ (cid:105) = c | W | M n (cid:88) w aff ∈ Stab (cid:99) W aff ( bM ) (cid:98) γ σ(cid:37) ∨ ( w aff )and taking into account (58) and definition (26), we have (cid:88) w aff ∈ Stab (cid:99) W aff ( bM ) (cid:98) γ σ(cid:37) ∨ ( w aff ) = h ∨ M ( b ) . (cid:3) Example . Consider the root system of C — its highest root ξ and the highest dual root η , depictedin Figure 1, are given by ξ = 2 α + α , η = α ∨ + 2 α ∨ . The Weyl group has eight elements, | W | = 8, and the determinant of the Cartan matrix is c = 2. We fixa sign homomorphism to σ = σ s and take a parameter with coordinates in ω − basis ( a, b ) and a pointwith coordinates in α ∨ -basis ( x, y ). Then the S s − functions are of the following explicit form [11], ϕ σ s ( a,b ) ( x, y ) =2 { cos(2 π (( a + 2 b ) x − by )) + cos(2 π ( ax + by )) − cos(2 π (( a + 2 b ) x − ( a + b ) y )) − cos(2 π ( ax − ( a + b ) y )) } . Fixing both admissible shifts to the non-zero values from Table 1, the grid F σ s M ( (cid:37), (cid:37) ∨ ) is of the form F σ s M ( (cid:37), (cid:37) ∨ ) = (cid:40) u + M ω ∨ + u M ω ∨ | u , u ∈ Z ≥ , u ∈ N , u + 2 u + u = M (cid:41) (77)and the grid Λ σ s M ( (cid:37), (cid:37) ∨ ) is of the formΛ σ s M ( (cid:37), (cid:37) ∨ ) = (cid:26) t ω + (cid:18) t + 12 (cid:19) ω | t , t ∈ Z ≥ , t ∈ N , t + t + 2 t = M (cid:27) . (78)For calculation of the coefficients ε ( x ), h ∨ M ( b ), which appear in (74) and (75), a straightforward gen-eralization of the calculation procedure, described in § a ∈ F σ s M ( (cid:37), (cid:37) ∨ )and each label b ∈ Λ σ s M ( (cid:37), (cid:37) ∨ ) are assigned the coordinates u , u , u and t , t , t from (77) and (78),respectively, and the triples which enter the algorithm in [13] are [ u , u + , u ] and [ t , t , t + ]. Thus,the values ε ( a ) of the point represented by u , u , u are ε ( a ) = 8 if u (cid:54) = 0 and ε ( a ) = 4 if u = 0. Thevalues h ∨ M ( b ) of the point represented by t , t , t are h ∨ M ( b ) = 1 if t (cid:54) = 0 and h ∨ M ( b ) = 2 if t = 0 andtherefore the relations orthogonality (75) are of the form (cid:104) ϕ σ s b , ϕ σ s b (cid:48) (cid:105) F σsM ( (cid:37),(cid:37) ∨ ) = 16 M h ∨ M ( b ) δ b,b (cid:48) . Discrete trigonometric transforms.
Similarly to ordinary Fourier analysis, interpolating functions I σM ( (cid:37), (cid:37) ∨ ) I σM ( (cid:37), (cid:37) ∨ )( a ) = (cid:88) b ∈ Λ σM ( (cid:37),(cid:37) ∨ ) c σb ϕ σb ( a ) , a ∈ R n (79)are given in terms of expansion functions ϕ σb and unknown expansion coefficients c σb . For some function f sampled on the grid F σ s M ( (cid:37), (cid:37) ∨ ), the interpolation of f consists of finding the coefficients c σb in theinterpolating functions (79) such that it coincides with I σM ( (cid:37), (cid:37) ∨ ) at all gridpoints, i.e. I σM ( (cid:37), (cid:37) ∨ )( a ) = f ( a ) , a ∈ F σM ( (cid:37), (cid:37) ∨ ) . The coefficients c σb are due to Theorem 3.4 uniquely determined via the standard methods of calculationof Fourier coefficients c σb = (cid:104) f, ϕ σb (cid:105) F σM ( (cid:37),(cid:37) ∨ ) (cid:104) ϕ σb , ϕ σb (cid:105) F σM ( (cid:37),(cid:37) ∨ ) = ( c | W | M n h ∨ M ( b )) − (cid:88) a ∈ F σM ( (cid:37),(cid:37) ∨ ) ε ( a ) f ( a ) ϕ σb ( a ) (80)and the corresponding Plancherel formulas are of the form (cid:88) a ∈ F σM ( (cid:37),(cid:37) ∨ ) ε ( a ) | f ( a ) | = c | W | M n (cid:88) b ∈ Λ σM ( (cid:37),(cid:37) ∨ ) h ∨ M ( b ) | c σb | . Concluding Remarks • The discrete orthogonality relations of the orbit functions in Theorem 4.1 and the correspondingdiscrete transforms (80) contain as a special case a compact formulation of the previous results— definition (66) corresponds for zero shifts to the definitions of the four sets F M , (cid:101) F M , F sM , F lM in [11, 13] and the same holds for the sets of weights. Except for Theorem 3.4, the proofs arealso developed in a uniform form. However, the approach to the proof of Theorem 3.4, which ENERALIZED DISCRETE ORBIT FUNCTION TRANSFORMS 19 states the crucial fact of the completeness of the obtained sets of functions, still relies on thecase by case analysis of the numbers of points which is done for trivial shifts in [11, 13]. • To any complex function f , defined on F σ ( (cid:37) ), is by relations (79) and (80) assigned a functionalseries { I σM ( (cid:37), (cid:37) ∨ ) } ∞ M =1 . Existence of conditions for convergence of these functional series togetherwith an estimate of the interpolation error (cid:82) | f − I σM ( (cid:37), (cid:37) ∨ ) | poses an open problem. • The four standard discrete cosine transforms DCT–I,..., DCT–IV and the four sine transformsDST–I,..., DST–IV from [5] are included as a special cases of (80) corresponding to the algebra A with its two sign homomorphisms and two admissible shifts. The case of C appears tohave exceptionally rich outcome of the shifted transforms — the twelve new transforms will bedetailed in a separated article. Note that Chebyshev polynomials of one variable of the thirdand fourth kinds [22] are induced using the dual admissible shifts of A . This indicates thatsimilar families of orthogonal polynomials might be generated for higher-rank cases. • Since the four types of orbit functions generate special cases of Macdonald polynomials, it canbe expected that the orthogonality relations (75), parametrized by two independent admissibleshifts, will translate into a generalization of discrete polynomial orthogonality relations from [8].The most ubiquitous of the four types of orbit functions is the family of ϕ σ e − functions — theseantisymmetric orbit functions are constituents of the Weyl character formula and generate Schurpolynomials. Their discrete orthogonality, discussed e. g. in [7, 16, 17], is also generalized forthe admissible cases. Discrete orthogonality relations induce the possibility of deriving so calledcubatures formulas [24], which evaluate exactly multidimensional integrals of polynomials of acertain degree, and generates polynomial interpolation methods [25]. Especially in the case of C cubature formulas, which are developed in [26], several new such formulas may be expected. • Another family of special functions, which can be investigated for having similar shifting prop-erties, is the set of so called E − functions [14, 21]. In this case, the condition (19) needs to beweakened to the even subgroup of W only and can be expected to produce discrete Fourieranalysis on new types of grids; indeed this requirement is trivial for the case of A and evenallows an arbitrary shift. • A natural question arises if the restriction (19) could be weakened to obtain even richer outcomeof the shifted transforms for the present case. For the analysis on the dual weights grid P ∨ this,however, does not seem to be straightforwardly possible — condition of W − invariance (19) isequivalent to the existence of the shift homomorphism (35), which enters the definitions of thefundamental domain. By inducing sign changes the shift homomorphism controls the boundaryconditions on the affine boundary of the fundamental domain and allows to evaluate expression(76). Thus, the approach in the present work may serve as a starting point for further researchon developing the discrete Fourier analysis of orbit functions on different types of grids. Acknowledgments
The authors are grateful for partial support by the project 7AMB13PL035-8816/R13/R14. JH grate-fully acknowledges the support of this work by RVO68407700.
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