Cubature formulas on combinatorial graphs
aa r X i v : . [ m a t h . F A ] A p r CUBATURE FORMULAS ON COMBINATORIAL GRAPHS
Isaac Z. Pesenson Meyer Z. Pesenson Hartmut F¨uhr Abstract.
The goal of the paper is to establish cubature formulas on com-binatorial graphs. Two types of cubature formulas are developed. Cubatureformulas of the first type are exact on spaces of variational splines on graphs.Since badlimited functions can be obtained as limits of variational splines weobtain cubature formulas which are ”essentially” exact on spaces of bandlim-ited functions. Cubature formulas of the second type are exact on spaces ofbandlimited functions. Accuracy of cubature formulas is given in terms ofsmoothness which is measured by means of combinatorial Laplace operator.The results have potential applications to problems that arise in data mining.
Keywords and phrases: combinatorial Laplace operator, Poincare andPlancherel-Polya inequalities, bandlimited functions, cubature formulas,splines, framesSubject classifications:
Primary: 65D32, 41A99, 42C15, 94A20; Secondary:94A12 1.
Introduction
Cubature formulas for approximate and exact evaluation of integrals of functionsdefined on Euclidean spaces or on smooth manifolds is an important and contin-uously developing subject. During last years in connection with applications toinformation theory analysis of functions defined on combinatorial graphs attractedattention of many mathematicians. The following list of a few classical and recentpapers is very far from being complete: [2], [3], [6], [16], [8], [9].In particular certain cubature formulas for functions defined on combinatorialgraphs was recently considered in the paper [3]. There, given values of a functionf on a small subset U of the set of all vertices V of a graph, the authors estimatewavelet coefficients via specific cubature formulas.In the present paper we develop a set of rules (cubature formulas) which allowfor approximate or exact evaluation of ”integrals” P v ∈ V f ( v ) of functions by usingtheir values on subsets U ⊂ V of vertices. We make extensive use of our previouswork on Shannon sampling of bandlimited functions and variational splines on Department of Mathematics, Temple University, Philadelphia, PA 19122; [email protected]. The author was supported in part by the National Geospatial-IntelligenceAgency University Research Initiative (NURI), grant HM1582-08-1-0019. CMS Department, California Institute of Technology, MC 305-16, Pasadena, CA 91125;[email protected]. The author was supported in part by the National Geospatial-IntelligenceAgency University Research Initiative (NURI), grant HM1582-08-1-0019 and by AFOSR, MURI,Award FA9550-09-1-0643 Lehrstuhl A f¨ur Mathematik, RWTH Aachen, D-52056 Aachen, Germany; [email protected] combinatorial graphs [5]-[14]. Our results can find applications to problems thatarise in connection with data filtering, data denoising and data dimension reduction.In section 2 we review our results [11] about variational interpolating spines ongraphs and describe an algorithm which allows an effective computation of vari-ational splines. In section 3 by using interpolating splines we develop a set ofcubature formulas. Theorem 3.1 shows that these formulas are exact on the set ofvariational splines. Theorem 3.4 explains that our cubature formulas are optimal .In section 4, using our result that bandlimited functions are limits of variationalsplines (see [10], [11]) we show, that cubature formulas developed in section 3 are”essentially” exact on bandlimited functions.It can be verified for example, that for a cycle graph of 1000 vertices a setof about 670 ”uniformly” distributed vertices is sufficient to have asymptoticallyexact cubature formulas for linear combinations of the first 290 eigenfunctions (outof 1000) of the corresponding combinatorial Laplace operator.It is worth to note that all results of section 3 which provide errors of approxi-mation of integrals of functions on V through their values on a U ⊂ V reflect1) geometry of U which is inherited into the quantity p | V | − | U | = p | S | andinto the Poincare constant Λ (see section 3 for definitions),2) smoothness of functions which is measured in terms of combinatorial Laplaceoperator.In section 4 we develop a different set of cubature formulas which are exact onappropriate sets of bandlimited functions. The results in this section are formulatedin the language of frames and only useful if it is possible to calculate dual framesexplicitly. Since in general it is not easy to compute a dual frame we finish thissection by explaining another approximate cubature formula which is based on theso-called frame algorithm.This paper is a discrete counterpart of the paper [15]. In a forthcoming paperwe are going to extend our results to weighted and infinite graphs.2. Variational (polyharmonic) splines on graphs
Let G denote an undirected weighted graph, with a finite or countable numberof vertices V ( G ) and weight function w : V ( G ) × V ( G ) → R +0 . w is symmetric, i.e., w ( u, v ) = w ( v, u ), and w ( u, u ) = 0 for all u, v ∈ V ( G ). The edges of the graph arethe pairs ( u, v ) with w ( u, v ) = 0.Let L ( G ) denote the space of all real-valued functions with the inner product h f, g i = X v ∈ V ( G ) f ( v ) g ( v )and the norm k f k = X v ∈ V ( G ) | f ( v ) | / . In the case of a finite graph and L ( G )-space the weighted Laplace operator L : L ( G ) → L ( G ) is introduced via(2.1) ( L f )( v ) = X u ∈ V ( G ) ( f ( v ) − f ( u )) w ( v, u ) . UBATURE FORMULAS ON COMBINATORIAL GRAPHS 3
This graph Laplacian is a well-studied object; it is known to be a positive-semidefiniteself-adjoint bounded operator. The notation l will be used for the Hilbert space ofall sequences of real numbers y = { y ν } , for which P ν | y ν | < ∞ . Variational splines on combinatorial graphs were developed in [11].
Variational Problem
Given a subset of vertices U = { u } ⊂ V, a sequence of real numbers y = { y u } ∈ l , u ∈ U , a natural k , and a positive ε > Find a function Y from the space L ( G ) which has the following properties: Y ( u ) = y u , u ∈ U, Y minimizes functional Y → (cid:13)(cid:13) ( εI + L ) k Y (cid:13)(cid:13) . We show that the above variational problem has a unique solution Y U,yk,ε .For the sake of simplicity we will also use notation Y yk assuming that U and ε are fixed.We say that Y yk is a variational spline of order k . It is also shown that everyspline is a linear combination of fundamental solutions of the operator ( εI + L ) k and in this sense it is a polyharmonic function with singularities. Namely it isshown that every spline satisfies the following equation(2.2) ( εI + L ) k Y yk = X u ∈ U α u δ u , where { α u } u ∈ U = n α u ( Y yk ) o u ∈ U is a sequence from l and δ u is the Dirac measureat a vertex u ∈ U . The set of all such splines for a fixed U ⊂ V and fixed k > , ε ≥ , will be denoted as Y ( U, k, ε ) . A fundamental solution F uk (cid:16) = F uk,ε (cid:17) , u ∈ V, of the operator ( εI + L ) k is thesolution of the equation(2.3) ( εI + L ) k F uk = δ u , k ∈ N , where δ u is the Dirac measure at u ∈ V ( G ). It follows from (2.2) that the followingrepresentation holds Y yk = X u ∈ U α u F u k . It is shown in [11] that for every set of vertices U = { u } , every natural k , every ε ≥ , and for any given sequence y = { y u } ∈ l , the solution Y yk of the VariationalProblem has a representation(2.4) Y yk = X u ∈ U y u L uk , where L uk is the so called Lagrangian spline, i.e. it is a solution of the same Varia-tional Problem with constraints L uk ( v ) = δ u,v , u ∈ U, where δ u,v is the Kroneckerdelta. It implies in particular, that Y ( U, k, ε ) is a linear set.Given a function f ∈ L ( G ) we will say that the spline Y fk interpolates f on U if Y fk ( u ) = f ( u ) for all u ∈ U . Algorithm for computing variational splines.
CUBATURE FORMULAS ON COMBINATORIAL GRAPHS
The above results give a constructive way for computing variational splines.Suppose we are going to construct splines which have prescribed values on a subsetof vertices U ⊂ V .1. One has to solve the following | U | systems of linear equations of the size | V | × | V | (2.5) ( εI + L ) k F uk,ε = δ u , u ∈ U, k ∈ N , in order to determine functions F uk,ε .2. Let δ w,v be the Kronecker delta. One has to solve | U | linear system of thesize | U | × | U | to determine coefficients α wu (2.6) δ w,γ = X u ∈ U α wu F uk,ε ( γ ) , w, γ ∈ U.
3. It gives the following representation of the corresponding Lagrangian spline(2.7) L wk,ε = X u ∈ U α wu F uk,ε , w ∈ U.
4. Every spline Y ys,ε ∈ Y ( U, s, ε ) which takes prescribed values y = { y w } , w ∈ U, can be written explicitly as Y ys,ε = X w ∈ W y w L ws,ε . Cubature formulas which are exact on variational splines
We introduce the following scalars θ u = θ u ( U, k, ε ) = X v ∈ V L U,uk,ε ( v )and by applying the formula (2.4) we obtain the following fact. Theorem 3.1.
In the same notations as above for every subset of vertices U = { u } and every k ∈ N , ε > , there exists a set of weights θ u = θ u ( U, k, ε ) , u ∈ U, suchthat for every spline Y U,yk,ε that takes values Y U,yk,ε ( u ) = y u , u ∈ U, the following exactformula holds (3.1) X v ∈ V Y U,yk,ε ( v ) = X u ∈ U y u θ u For a subset S ⊂ V (finite or infinite) the notation L ( S ) will denote the spaceof all functions from L ( G ) with support in S : L ( S ) = { ϕ ∈ L ( G ) , ϕ ( v ) = 0 , v ∈ V \ S } . Definition 1.
We say that a set of vertices S ⊂ V is a Λ-set if for any ϕ ∈ L ( S )it admits a Poincare inequality with a constant Λ = Λ( S ) > k ϕ k ≤ Λ kL ϕ k , ϕ ∈ L ( S ) . The infimum of all Λ > S is a Λ-set will be called the Poincare constantof the set S and denoted by Λ( S ).The following lemma holds true [11]. UBATURE FORMULAS ON COMBINATORIAL GRAPHS 5
Lemma 3.2. If A is a self-adjoint positive definite operator in a Hilbert space H and for an ϕ ∈ H and a positive a > the following inequality holds true k ϕ k ≤ a k Aϕ k , then for the same ϕ ∈ H , and all k = 2 l , l = 0 , , , ... the following inequality holds k ϕ k ≤ a k k A k ϕ k . Proof.
By the spectral theory there exist a direct integral of Hilbert spaces X = Z k A k X ( τ ) dm ( τ )and a unitary operator F from H onto X , which transforms domain of A t , t ≥ , onto X t = { x ∈ X | τ t x ∈ X } with norm k A t f k H = Z k A k τ t k F f ( τ ) k X ( τ ) dm ( τ ) ! / and F ( A t f ) = τ t ( F f ). According to our assumption we have for a particular ϕ ∈ H Z k A k | F ϕ ( τ ) | dm ( τ ) ≤ a Z k A k τ | F ϕ ( τ ) | dm ( τ )and then for the interval B = B (0 , a − ) we have Z B | F ϕ ( τ ) | dm ( τ ) + Z [0 , k A k ] \ B | F ϕ | dm ( τ ) ≤ a Z B τ | F ϕ | dm ( τ ) + Z [0 , k A k ] \ B τ | F ϕ | dm ( τ ) ! . Since a τ < B (0 , a − )0 ≤ Z B (cid:0) | F ϕ | − a τ | F ϕ | (cid:1) dm ( τ ) ≤ Z [0 , k A k ] \ B (cid:0) a τ | F ϕ | − | F ϕ | (cid:1) dm ( τ ) . This inequality implies the inequality0 ≤ Z B (cid:0) a τ | F ϕ | − a τ | F ϕ | (cid:1) dm ( τ ) ≤ Z [0 , k A k ] \ B (cid:0) a τ | F ϕ | − a τ | F ϕ | (cid:1) dm ( τ )or a Z [0 , k A k ] τ | F ϕ | dm ( τ ) ≤ a Z R + τ | F ϕ | dm ( τ ) , which means k ϕ k ≤ a k Aϕ k ≤ a k A ϕ k . Now, by using induction one can finish the proof of the Lemma. The Lemma isproved. (cid:3)
The following Theorem gives a cubature rule that allows to compute the integral P v ∈ V f ( v ) by using only values of f on a smaller set U . CUBATURE FORMULAS ON COMBINATORIAL GRAPHS
Theorem 3.3.
For every set of vertices U ⊂ V for which S = V \ U is a Λ -setand for any ε > , k = 2 l , l ∈ N , there exist weights θ u = θ u ( U, k, ε ) such that forevery function f ∈ L ( G ) , (3.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V f ( v ) − X u ∈ U f ( u ) θ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p | S | Λ k (cid:13)(cid:13)(cid:13) ( εI + L ) k f (cid:13)(cid:13)(cid:13) . Proof. If f ∈ L ( G ) and Y U,fk,ε is a variational spline which interpolates f on a set U = V \ S then(3.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V f ( v ) − X v ∈ V Y U,fk,ε ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X v ∈ S (cid:12)(cid:12)(cid:12) f ( v ) − Y U,fk,ε ( v ) (cid:12)(cid:12)(cid:12) ≤ p | S | (cid:13)(cid:13)(cid:13) f − Y U,fk,ε (cid:13)(cid:13)(cid:13)
Since S is a Λ- set we have(3.5) k f − Y U,fk,ε (cid:13)(cid:13)(cid:13) ≤ Λ kL (cid:16) f − Y U,fk,ε (cid:17)(cid:13)(cid:13)(cid:13) . For any g ∈ L ( G ) the following inequality holds true(3.6) kL g k ≤ k ( εI + L ) g k . Thus one obtains the inequality(3.7) (cid:13)(cid:13)(cid:13) f − Y U,fk,ε (cid:13)(cid:13)(cid:13) ≤ Λ (cid:13)(cid:13)(cid:13) ( εI + L ) (cid:16) f − Y U,fk,ε (cid:17)(cid:13)(cid:13)(cid:13) . We apply Lemma 3.2 with A = εI + L , a = Λ and ϕ = f − Y U,fk,ε . It gives theinequality(3.8) (cid:13)(cid:13)(cid:13) f − Y U,fk,ε (cid:13)(cid:13)(cid:13) ≤ Λ k (cid:13)(cid:13)(cid:13) ( εI + L ) k (cid:16) f − Y U,fk,ε (cid:17)(cid:13)(cid:13)(cid:13) for all k = 2 l , l = 0 , , , ... Using the minimization property of Y U,fk,ε we obtain (cid:13)(cid:13)(cid:13) f − Y U,fk,ε (cid:13)(cid:13)(cid:13) ≤ k (cid:13)(cid:13)(cid:13) ( εI + L ) k f (cid:13)(cid:13)(cid:13) , k = 2 l , l ∈ N . Together with (3.4) it gives(3.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V f ( v ) − X v ∈ V Y U,fk,ε ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p | S | Λ k k ( εI + L ) k f k , k = 2 l , l ∈ N . By applying the Theorem 3.1 we finish the proof. (cid:3)
It is worth to note that the above formulas are optimal in the sense it is describedbelow.
Definition 2.
For the given U ⊂ V, f ∈ L ( G ) , k ∈ N , ε ≥ , R > , the notation Q ( U, f, k, ε, R ) will be used for a set of all functions h in L ( G ) such that1) h ( u ) = f ( u ) , u ∈ U, and2) (cid:13)(cid:13) ( εI + L ) k h (cid:13)(cid:13) ≤ R. UBATURE FORMULAS ON COMBINATORIAL GRAPHS 7
It is easy to verify that every set Q ( U, f, k, ε, R ) is convex, bounded, and closed.It implies that the set of all integrals of functions in Q ( U, f, k, ε, R ) is an interval i.e.(3.10) [ a, b ] = (X v ∈ V h ( v ) : h ∈ Q ( U, f, k, ε, R ) ) The optimality result is the following.
Theorem 3.4.
For every set of vertices U ⊂ V and for any ε > , k = 2 l , l ∈ N , if θ u = θ u ( U, k, ε ) are the same weights that appeared in the previous statements,then for any g ∈ Q ( U, f, k, ε, R )(3.11) X u ∈ U g ( u ) θ u = a + b , where [ a, b ] is defined in (3.10).Proof. We are going to show that for a given function f the interpolating spline Y U,fk,ε is the center of the convex, closed and bounded set Q ( U, f, k, ε, R ) for any R ≥ (cid:13)(cid:13)(cid:13) ( εI + L ) k Y U,fk,ε (cid:13)(cid:13)(cid:13) . In other words it is sufficient to show that if Y U,fk,ε + h ∈ Q ( U, f, k, ε, R )for some function h then the function Y U,fk,ε − h also belongs to the same intersection.Indeed, since h is zero on the set U then according to (2.2) one has D ( εI + L ) k Y U,fk,ε , ( εI + L ) k h E = D ( εI + L ) k Y U,fk,ε , h E = 0 . But then (cid:13)(cid:13)(cid:13) ( εI + L ) k ( Y U,fk,ε + h ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ( εI + L ) k (cid:16) Y U,fk,ε − h (cid:17)(cid:13)(cid:13)(cid:13) . In other words, (cid:13)(cid:13)(cid:13) ( εI + L ) k (cid:16) Y U,fk,ε − h (cid:17)(cid:13)(cid:13)(cid:13) ≤ R and because Y U,fk,ε + h and Y U,fk,ε − h take the same values on U the function Y U,fk,ε − h belongs to Q ( U, f, k, ε, R ) . From here the Theorem follows. (cid:3)
Corollary 3.1.
Fix a function f ∈ L ( G ) and a set of vertices U ⊂ V for which S = V \ U is a Λ -set. Then for any ε > , k = 2 l , l ∈ N , for the same set ofweights θ u = θ u ( U, k, ε ) ∈ R that appeared in the previous statements the followinginequalities hold for every function g ∈ Q ( U, f, k, ε, R ) , (3.12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V g ( v ) − X u ∈ U f ( u ) θ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p | S | Λ k diam Q ( U, f, k, ε, R ) . Proof.
Since f and g coincide on U from (3.4) and (3.8) we obtain the inequality(3.13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V g ( v ) − X v ∈ V Y U,fk,ε ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p | S | Λ k (cid:13)(cid:13)(cid:13) ( εI + L ) k (cid:16) f − Y U,fk,ε (cid:17)(cid:13)(cid:13)(cid:13)
CUBATURE FORMULAS ON COMBINATORIAL GRAPHS
By the Theorem 3.4 the following inequality holds (cid:13)(cid:13)(cid:13) ( εI + L ) k (cid:16) Y U,fk,ε − g (cid:17)(cid:13)(cid:13)(cid:13) ≤ diam Q ( U, f, k, ε, R )for any g ∈ Q ( U, f, k, ε, R ). The last two inequalities imply the Corollary. (cid:3) Approximate cubature formulas for bandlimited functions
Operator(matrix) L is symmetric and positive definite. Let E ω ( L ) be the span ofeigenvectors of L whose corresponding eigenvalues are ≤ ω . The invariant subspace E ω ( L ) is the space of all vectors in L ( G ) on which L has norm ω . In other words f belongs to E ω ( L ) if and only if the following Bernstein-type inequality holds(4.1) kL s f k ≤ ω s k f k , s ≥ . The Bernstein inequality (5.9), the Lemma 3.2, and the Theorem 3.3 imply thefollowing result.
Corollary 4.1.
For every set of vertices U ⊂ V for which S = V \ U is a Λ -setand for any ε > , k = 2 l , l ∈ N , there exist weights θ u = θ u ( U, k, ε ) ∈ R such thatfor every function f ∈ E ω ( L ) , the following inequality holds (4.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V f ( v ) − X u ∈ U f ( u ) θ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ γ k p | S | k f k , where γ = Λ( ω + ε ) , k = 2 l , l ∈ N . If in addition the following condition holds0 < ω < − ε and f ∈ E ω ( L ) then this Corollary imply the following Theorem. Theorem 4.1. If U is a subset of vertices for which S = V \ U is a Λ -set then forany < ε < / Λ , k = 2 l , l ∈ N , there exist weights θ u = θ u ( U, k, ε ) ∈ R such thatfor every function f ∈ E ω ( L ) , where < ω < − ε, the following relation holds (4.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V f ( v ) − X u ∈ U f ( u ) θ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → , when k = 2 l → ∞ . Example 1.
Consider the unweighted cycle graph C of 1000 vertices. The Laplace op-erator L has one thousand eigenvalues which are given by the formula λ k =2 − πk , k = 0 , , ...,
999 (see [1]).It is easy to verify that every single vertex in C is a Λ = √ -set. It is alsoeasy to understand that if closures of two vertices do not intersect i. e. UBATURE FORMULAS ON COMBINATORIAL GRAPHS 9 ( v j ∪ ∂v j ) ∩ ( v i ∪ ∂v i ) = ∅ , v j , v i ∈ C , (here ∂ is the vertex boundary operator) then their union v j ∪ v i is also a Λ = √ -set. It implies, that one can remove from C every third vertex and on theremaining set of 670 the formula (4.3) will be true for the span of about 290 firsteigenfunctions of L . Example 2.
One can show [10] that if S = { v , v , ..., v N } consists of | S | successive verticesof the graph C then it is a Λ-set withΛ = 12 (cid:18) sin π | S | + 2 (cid:19) − . It implies for example that on a set of 100 uniformly distributed vertices of C the formula (4.3) will be true for every function in the span of about 40 firsteigenfunctions of L .5. Another set of exact and approximate cubature formulas forbandlimited functions
We introduce another set of cubature formulas which are exact on some sets ofbandlimited functions.
Theorem 5.1. If U is a subset of vertices for which S = V \ U is a Λ -set thenthere exist weights σ u = σ u ( U ) ∈ R , u ∈ U, such that for every function f ∈ E ω ( L ) , where < ω < , the following exact formula holds (5.1) X v ∈ V f ( v ) = X u ∈ U f ( u ) σ u , U = V \ S, Proof.
First, we show that the set U is a uniqueness set for the space E ω ( L ), i.e. for any two functions from E ω ( L ) the fact that they coincide on U implies thatthey coincide on V .If f, g ∈ E ω ( L ) then f − g ∈ E ω ( L ) and according to the inequality (5.9) thefollowing holds true(5.2) kL ( f − g ) k ≤ ω k f − g k . If f and g coincide on U = V \ S then f − g belongs to L ( S ) and since S is a Λ-setthen we will have k f − g k ≤ Λ kL ( f − g ) k , f − g ∈ L ( S ) . Thus, if f − g is not zero and ω < / Λ we have the following inequalities(5.3) k f − g k ≤ Λ kL ( f − g ) k ≤ Λ ω k f − g k < k f − g k , which contradict to the assumption that f − g is not identical zero. Thus, the set U is a uniqueness set for the space E ω ( L ). It implies that there exists a constant C = C ( U, ω ) for which the followingPlancherel-Polya inequalities hold true(5.4) X u ∈ U | f ( u ) | ! / ≤ k f k ≤ C X u ∈ U | f ( u ) | ! / for all f ∈ E ω ( L ). Indeed, the functional ||| f ||| = X u ∈ U | f ( u ) | ! / defines another norm on E ω ( L ) because the condition ||| f ||| = 0 , f ∈ E ω ( L ), impliesthat f is identical zero on entire graph. Since in finite-dimensional situation anytwo norms are equivalent we obtain existence of a constant C for which (5.4) holdstrue.Let δ v ∈ L ( G ) be a Dirac measure supported at a vertex v ∈ V . The notation ϑ v will be used for a function which is orthogonal projection of the function1 p d ( v ) δ v on the subspace E ω ( L ). If ϕ , ϕ , ..., ϕ j ( ω ) are orthonormal eigenfunctions of L which constitute an orthonormal basis in E ω ( L ) then the explicit formula for ϑ v is(5.5) ϑ v = j ( ω ) X j =0 ϕ j ( v ) ϕ j . In these notations the Plancherel-Polya inequalities (5.4) can be written in the form(5.6) X u ∈ U | h f, ϑ u i | ≤ k f k ≤ C X u ∈ U | h f, ϑ u i | , where f, ϑ u ∈ E ω ( L ) and h f, ϑ u i is the inner product in L ( G ). These inequalitiesmean that if U is a uniqueness set for the subspace E ω ( L ) then the functions { ϑ u } u ∈ U form a frame in the subspace E ω ( L ) and the tightness of this frame is1 /C . This fact implies that there exists a frame of functions { Θ u } u ∈ U in the space E ω ( L ) such that the following reconstruction formula holds true for all f ∈ E ω ( L )(5.7) f ( v ) = X u ∈ U f ( u )Θ u ( v ) , v ∈ V. By setting σ u = P v ∈ V Θ u ( v ) one obtains (5.1). (cid:3) Unfortunately this approach does not give any information about constant C in(5.6) and it make realization of the Theorem 5.1 problematic. We are going to utilizeanother approach to the Plancherel-Polya-type inequality which was developed inour paper [14] and which produces explicit constant.We will use the following notion of the relative degree. Given any subset A ⊂ V ( G ) and v ∈ V ( G ), we let w A ( v ) = X u ∈ A w ( u, v ) . We note that w A ( v ) = 0 iff there is no edge connecting v and some element of A .Let’s introduce the following notations UBATURE FORMULAS ON COMBINATORIAL GRAPHS 11 U = S , ∂ ( S ) = S , ∂ ( S ∪ S ) = S , ..., ∂ ( S ∪ .... ∪ S n − ) = S n . Clearly, { S m } n is a disjoint cover of V ( G ). We let D m = D m ( S ) = sup v ∈ S m w S m +1 ( v )and K m = K m ( S ) = inf v ∈ S m +1 w S m ( v ) . For 0 ≤ m < n let b K m ( b S ) = b K m = inf v ∈ S m w S m +1 ( v ) , as well as b D m ( b S ) = b D m = sup v ∈ S m +1 w S m ( v ) . The set U = S is called inital set of the partition S , it is of primary importancefor the following results.We define δ U = n X m =1 m X k =1 K k − m − Y i = k D i K i !!! / ,a U = n X m =0 m − Y j =0 D j K j / , ˆ δ U = n ′ X m =1 m − X k =0 b K k m − Y i = k b K i b D i !! / , ˆ a U = n ′ X m =0 m − Y j =0 b K j b D j / . The weighted gradient norm of a function f on V ( G ) is defined by k∇ w f k = X u,v ∈ V ( G ) | f ( u ) − f ( v ) | w ( u, v ) / . It is known [5], [7], that(5.8) kL / f k = k∇ f k . Note, that if f belongs to the space E ω ( L ) then the Bernstein inequality gives(5.9) k∇ f k = kL / f k ≤ √ ω k f k . After all these preparations we can formulate the following statement which followsfrom [5].
Theorem 5.2.
If the inequality (5.10) δ U √ ω < , ω > , is satisfied, then the following Plancherel-Polya-type equivalence holds for all f ∈ E ω ( L ) : (5.11) 1 − δ U √ ωa U k f k ≤ k f | U k ≤ δ U √ ω ˆ a U k f k . Using this result we prove existence of exact cubature formulas on spaces ofbandlimited functions.
Theorem 5.3. If U is a subset of vertices for which the inequality ( 5.10) is satisfiedthen there exists a set of weights µ u ∈ R , u ∈ U, such that for any f ∈ E ω ( L ) , where ω satisfies (5.10) the following exact formula holds (5.12) X v ∈ V f ( v ) = X u ∈ U f ( u ) µ u . Proof.
The previous Theorem shows that U is a uniqueness set for the space E ω ( L ),which means that every f in E ω ( L ) is uniquely determined by its values on U .Let us denote by θ v , where v ∈ U , the orthogonal projection of the Dirac measure δ v , v ∈ U , onto the space E ω ( L ). Since for functions in E ω ( L ) one has f ( v ) = h f, θ v i , v ∈ U , the inequality (5.11) takes the form of a frame inequality in theHilbert space H = E ω ( L )(5.13) (cid:18) − ǫδ U a U (cid:19) k f k ≤ X v ∈ U | h f, θ v i | ≤ ǫ ˆ δ U ˆ a U ! k f k , ǫ = √ ω, for all f ∈ E ω ( L ). According to the general theory of Hilbert frames [4] the lastinequality implies that there exists a dual frame (which is not unique in general) { Θ v } , v ∈ U, Θ v ∈ E ω ( L ), in the space E ω ( L ) such that for all f ∈ E ω ( L ) thefollowing reconstruction formula holds(5.14) f = X v ∈ U f ( v )Θ v . By setting P v ∈ V Θ v ( u ) = µ u we obtain (5.12). (cid:3) To be more specific we consider unweighted case for which(5.15) U = U ∪ ∂ ( U ) = V ( G ) . In other words, we consider a bipartite graph V ( G ) with components S = U and S = ∂ ( U ). Keeping the same notations as above we compute a U = (cid:18) D K (cid:19) / , δ U = 1 K / . Thus, we have k f k ≤ (cid:18) D K (cid:19) / k f k + 1 K / k∇ f k . By applying (5.9) along with assumption(5.16) ω < K we obtain the following estimate(5.17) k f k ≤ (cid:18) − r ωK (cid:19) (cid:18) D K (cid:19) / k f k , f = f | U . On the other hand ˆ a U = K ˆ D ! / , ˆ δ U = 1ˆ D / . This yields the norm estimate k f k + 1ˆ D / k∇ f k ≥ K ˆ D ! / k f k , f = f | U . If (5.9) holds, then(5.18) K ˆ D ! / k f k ≤ k f k + 1ˆ D / k∇ f k ≤ (cid:18) r ω ˆ D (cid:19) k f k . After all, for functions f in E ω ( L ) with ω < K we obtain the following frameinequality(5.19) A k f k ≤ X v ∈ U | < f, θ v > | ≤ B k f k , f = f | U , where(5.20) A = (cid:16) − q ωK (cid:17) D K , B = (cid:18) q ω c D (cid:19) c K c D . It shows that if the condition ω < K , K = K ( U ) , is satisfied then the set U is asampling set for the space E ω ( L ) and a reconstruction formula (5.14) holds whichleads to (5.12).Using the same notations as above we summarize these observations on thefollowing statement. Theorem 5.4. If G is a bipartite graph with components S and S , K = inf v ∈ S w S ( v ) then (1) S is a uniqueness set for E ω ( L ) for any ω < K ; (2) if θ u is orthogonal projection of δ u onto E ω ( L ) , ω < K , then { θ u } u ∈ S isa frame in E ω ( L ) with constants (5.20); (3) if { Θ u } is a frame dual in E ω ( L ) , ω < K , to the frame { θ u } u ∈ S and X v ∈ V Θ u ( v ) = µ u , u ∈ S , then for any f ∈ E ω ( L ) the following exact formula holds (5.21) X v ∈ V f ( v ) = X u ∈ S f ( u ) µ u . Note, that if a bipartite graph G = S ∪ S is complete and | S | = N > M = | S | then the spectrum of L consists of 0 , M, N, M + N, where M has multiplicity N − N has multiplicity M −
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