Representation of Polynomials by Linear Combinations of Radial Basis Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal {P}_{n}^{d}}$\end{document} denote the space of polynomials on ℝd of total degree n. In this work, we introduce the space of polynomials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal {Q}_{2 n}^{d}}$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal {P}_{n}^{d}}\subset {\mathcal {Q}_{2 n}^{d}}\subset\mathcal{P}_{2n}^{d}$\end{document} and which satisfy the following statement: Let h be any fixed univariate even polynomial of degree n and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} be a finite set in ℝd. Then every polynomial P from the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal {Q}_{2 n}^{d}}$\end{document} may be represented by a linear combination of radial basis functions of the form h(∥x+a∥), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\in \mathcal{A}$\end{document}, if and only if the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} is a uniqueness set for the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal {Q}_{2 n}^{d}}$\end{document}.

The problem of the representation of functions by linear combination of shifts of radial basis functions is of current interest in different applications of nonlinear approximation, including approximation by wave functions, learning theory, and tomography. Results about density of the spaces formed by linear combinations of shifts of fixed functions were obtained by Wiener (see Edwards [4]), Pinkus [11,12], Schwartz [15], Agranovsky and Quinto [1], and many others. A series of important results on approximation by radial basis functions are obtained in [2,8,9,13,14].
Since a large class of functions may be approximated by polynomials, then the problem of the representation of polynomials by shifts of radial basis polynomials is closely connected with the above problems.
Let s = (s 1 , . . . , s d ) ∈ Z d be any vector with nonnegative integer coordinates. We set x s = x s 1 1 · · · x s d d and |s| = s 1 + · · · + s d . Consider the space P d = span{x s : s ∈ Z d + } of all polynomials on R d , i.e., linear combinations of a finite number of monomials x s . If p(x) = s c s x s is a polynomial from P d , then the total degree of p is defined by deg p = max{|s| : c s = 0}. Denote by P d n the space of all polynomials from P d of degree at most n. The total dimension of P d n equals D n = n+d d .

Orthogonal System P of Polynomials on the Ball
In this section, we introduce the polynomial orthogonal basis P = {P I } on the ball B d which we will use in proofs of the main results (for a more detailed exposition of this basis, see [7]). For the construction of the basis P, we use work by [3,5,6,10]. A series of applications of the basis P for approximation by ridge and radial functions is contained in [6,7]. First, we discuss some well-known results connected with orthogonal polynomials, which we use in this present work.
Let d be a natural number. We set U n (t) = v −1/2 n,d/2 C d/2 n (t). Then the family of polynomials {U n } ∞ 0 is the complete orthonormal system in the weighted space (b) Let P d n be the set of all algebraic polynomials of total degree n in d real variables. Let ξ be any point on the sphere S d−1 . Then the polynomial U n (ξ · x) is in P d n and is orthogonal to P d n−1 in L 2 (B d ) (see [3,10]), i.e., (c) For each ξ, η ∈ S d−1 and n ∈ Z + , we have (see [10]) 2. An Orthogonal System of Polynomials on the Sphere: Let j be a nonnegative integer number. Let H hom j be the subspace in P d j formed by all harmonic homogeneous polynomials (i.e., the spherical harmonics) of degree j . We know [16,20] that the dimension l j of H hom to the sphere S d−1 . Then the set ∞ j =0 Π j is an orthonormal basis in the space L 2 (S d−1 ), i.e., for any functions Y j,k and Y j ,k from ∞ j =0 Π j , the following holds: where by dξ we denote the normalized Lebesgue measure on S d−1 .

An Orthogonal System of Polynomials on the Ball: Consider the Hilbert
Also consider the Hilbert space L 2 (Q, σ ) of real functions on the segment Q = [−1, 1] with the norm Consider the system of normed Gegenbauer polynomials {U i (t)}, i = 0, 1, . . . , on the segment Q, forming a complete orthonormal system in the space L 2 (Q, σ ). Introduce the set of triple indices where l j is the dimension of space H hom j of harmonic homogeneous polynomials of degree j . For every index I = (i, j, k) from I, we construct the function on R d , where the coefficient ν ij is the normalizing factor such that P I L 2 = 1 and x · ξ = x 1 ξ 1 + · · · + x d ξ d is the inner product of the vectors x and ξ . From (6), we see that the function P I is the polynomial on R d of total degree i. Consider the system of polynomials The system P is the complete orthonormal system of polynomials in the space L 2 (B d ) (see [7]). In particular, for any natural n, the finite subsystem of polynomials in P forms ( [7]) an orthogonal basis in the space P d n of all polynomials on R d of total degree n.

Main Results
In this paper, we introduce another subspace of polynomials, which is defined by the following. Consider the subset of indices I n = {I = (i, j, k) ∈ I : i + j ≤ 2n} in the set I and the corresponding collection of polynomials from the system P, Definition Let Q d 2n = span Q 2n be the linear space of polynomials formed by the linear span of polynomials from Q 2n .
It is obvious that for every natural n, we have Denote by S n the dimension of the space Q d 2n . From (9), we have D n ≤ S n ≤ D 2n . Since the collection of polynomials {P I } I ∈I n is the complete orthonormal system in the space Q d 2n , then S n coincides with the cardinality of the set I n .
Let h be any fixed univariate even polynomial of degree 2n. We show that every polynomial p n from the space Q d 2n may be represented by the linear combination of some S n shifts h( x +a i ), i = 1, . . . , S n , of the radial polynomial h( x ). Moreover, we show that the representation is valid for any collection of points a 1 , . . . , a S n from R d forming a uniqueness set for the space Q d 2n . Let n be any natural number. Let h(t) = b 2n t 2n + · · · + b 0 , b 2n = 0, be any univariate even polynomial of degree 2n; that is, all coefficients b m with m odd are equal to zero.
A set of points A n = {a 1 , . . . , a S n } is said to be a h-basis set for the space Q d 2n if R(h, A n ) = Q d 2n ; that is, every polynomial p from Q d 2n can be represented as A set of points A n is called a uniqueness set for the space Q d 2n if, for any two polynomials p 1 and p 2 from Q d 2n , the relations p 1 (a i ) = p 2 (a i ), i = 1, . . . , S n , implies the equality p 1 (a) = p 2 (a) for all a ∈ R d . Consider the Hilbert space L 2 = L 2 (B d ) of complex functions on the ball B d with inner product Let f and G be a function and a subspace in L 2 , respectively. Denote by E(f, G) = inf g∈G f − g L 2 the distance of the function f from G. Let F be a function class in L 2 . By E(F, G) = sup f ∈F inf g∈G f − g L 2 , we denote the deviation of the class F from G. The following follows directly from Theorem 3.1 and the embedding (9). Corollary 3.2 Let n be any even number, A n be a uniqueness set for Q d 2n , and h be any univariate even polynomial of degree n. Then for any function f from the space L 2 , the following inequality holds: Let s = (s 1 , . . . , s d ) ∈ Z d + and |s| = s 1 + · · · + s d . Introduce the differential operator D s = ∂ |s| /∂ s 1 x 1 · · · ∂ s d x d . Let r be any natural number. In the space L 2 , we consider the Sobolev class of functions

Corollary 3.3 Let n be any even number, A n be an uniqueness set for Q d 2n , and h be any univariate even polynomial of degree n. Then the inequalities
hold, where c 1 and c 2 depend only on r and d.

The upper bound for E(W r 2 (B d ), R(h, A n )) directly follows from Corollary 3.2 and
Jackson's theorem. The lower bound follows from the estimate of the Kolmogorov n-widths of the class W r 2 (B d ) (see [19]).

Moments of Radial Functions
In this section, we establish some auxiliary results which we will use in the proof of the main theorem. Let f be a function from the space L 2 (B d ). Denote by M I (f ) = f, P I the I -moment of the function f . Let a be any point in R d and s be a natural number. Introduce the functions r(x) = x and r a (x) = x + a . In the following lemma, we calculate the moments M I (r 2s a ) of the shifted radial function r 2s a = x + a 2s .
Given triple index I = (i, j, k) from I, we consider the normed Gegenbauer polynomial U i (t) on the segment Q = [−1, 1] of degree i and the spherical harmonic Y j,k (θ ) on the sphere S d−1 of degree j . (11) and consider the function on R d

Lemma 4.1 Let I = (i, j, k) be any triple index from I and a be any point from
Then, for every natural s, the I -moment of the function r 2s a equals where ν depends only on I, d, and s.
Proof Given s, we will use the equality and where by e we denote the point e = (0, . . . , 0, 1). Equality (13) directly follows from the homogeneity of the polynomials x 2s , (x · θ) 2s , and the invariance of the measure dθ with respect to the rotation operator in R d .
For a fixed point θ on S d−1 , we consider the univariate polynomial w θ (t) = (t + a · θ) 2s of degree 2s. From (13), we have Using (14) and (6), we can represent the I -moment of the function r 2s a by the following: where c sij = c s ν ij . Therefore, we have r 2s a , P I = c sij Given θ , we decompose the function w θ by the orthogonal system of Gegenbauer polynomials {U α }: Then the last integral in (15) equals The following identity holds (see (3)): Therefore, from (16), we obtain (1) . (15):

Substitute this expression into
By (4), we have Hence, we obtain r 2s a , P I = ν where ν = c sij c i /U i (1). The equality Thus, we obtain from (17), Lemma 4.1 is complete.

The Proof of Theorem 3.1
Let n be any natural number and h(t) = n s=0 c s t 2s , t ∈ R, be an univariate even polynomial of degree 2n. Given a point a from R d , consider the polynomials g(x) = h( x ) and g a (x) = g(x + a). Let I be any index from the set I n . Calculate the moments M I (g a ) of the function g a . Set c s,α = νc s 2s α . By properties of orthogonality of the polynomials U i , we have from (11), Analogously, by properties of orthogonality of spherical harmonics Y jk , we have from (12), Also, we see that V α,j,k (a) = 0 if α = j (mod 2). Thus, Note (see (5)) that i = j (mod 2). Therefore, i + j is always an even number. Interchanging the order summation, we obtain where the index s runs over the set { α+i 2 , α+i 2 + 1, . . . , n}. Since V α,j,k (a) = 0 if α = j (mod 2), then factually, α runs only over the set {j, . . . , 2n − i}. Therefore, the equality α = j = i (mod 2) is always true; that is, α + i is always an even number.
Consider two collections of polynomials of the variable a: that is, M n ⊆ V n . We show that V n ⊆ M n . We fix j and k. The equality (18) we rewrite as We show that the functions V α,j,k for every α = j, . . . , 2n belong to M n . We use induction on α. Let α = j . We have from (19) Hence, V j,j,k ∈ M n . Now assume V j,j,k , . . . , V j +β,j,k ∈ M n with β ≥ 0. Show that V j +β+1,j,k also belongs to M n . We have from (19) Then, taking into consideration that the functions V j,j,k , . . . , V j +β,j,k and M 2n−j −β−1,j,k (g a ) belong to M n , we obtain that V j +β+1,j,k also belongs to M n . Thus, the functions V i,j,k (a), for all i ≥ j , belong to M n ; that is, V n ⊆ M n . Lemma 5.1 is proved.

Lemma 5.2
The spaces V n and Q d 2n coincide.
Proof The space V n is the linear span of polynomials Every polynomial 1, t, . . . , t n is some linear combination of Gegenbauer polynomials {U i (t)} n i=0 . Therefore, V n ⊆ Q d 2n . We now prove that Q d 2n ⊆ V n . We show that the set V n forms a linear basis in the space Q d 2n . Indeed, consider the system of functions Q 2n consisting of polynomials (6) where b i,α are the coefficients of the Gegenbauer polynomial U i . The system Q 2n is a linear basis in the space Q d 2n . On the other hand, from (21) and (22), we see that P i,j,k ∈ V n for every index (i, j, k) ∈ I n . Thus, V n is a linear basis in Q d 2n .
Proof of Theorem 3.1 By Lemmas 5.1 and 5.2, the collection of functions of avariable M n = {M I (g a )}, I ∈ I n , forms a basis in the space Q d 2n . Therefore, the following obvious statement holds. Proof Assume that a set A n is a uniqueness set for the space Q d 2n . We show that any polynomial p from Q d 2n may be represented by p(x) = c 1 h x + a 1 + · · · + c S n h x + a S n .

Proposition 5.3 A set
Using moments M I (p) and M I (g a ) of functions p and g a = h(r a ), we can construct the system of linear equations M I (p) = c 1 M I (g a 1 ) + · · · + c S n M I (g a Sn ), I ∈ I n , with respect to the unknowns coefficients c 1 , . . . , c S n . By Proposition 5.3, the system (24) has a unique solution; that is, the polynomial p may be represented by (23).
Conversely, assume that every polynomial p from Q d 2n may be represented by (23). Then the matrix (M I (g a j )), I ∈ I n , j = 1, . . . , S n , is nondegenerate. Hence, by Proposition 5.3, the set A n is a uniqueness set for the space Q d 2n .