1 Introduction

Let ℝd be the real d-dimensional Euclidean space with norm \(\|x\| =(\sum_{i=1}^{d} x_{i}^{2})^{1/2}\). Denote by \(\mathbb{B}^{d}(a,r)=\{x : \|x-a\|\le r\}\) the Euclidean ball with center a and radius r. Let \(\mathbb{B}^{d}=\mathbb{B}^{d}(0,1)\) and \(\mathbb {S}^{d-1}=\{\|x\|=1\}\) be the unit ball and sphere in the space ℝd, respectively.

Let h(t) be a function defined on ℝ+. Consider the radial function h(∥x∥) on ℝd. Given a point a in ℝd, we introduce the shifted radial function h a =h(∥x+a∥) with center −a. Let \(\mathcal{A}\) be a subset in ℝd. Denote by \(R(h,\mathcal{A})\) the class of functions of the form h(∥x+a∥), where a runs over the set \(\mathcal{A}\). Consider the class of functions

$${{\mathcal {\mathcal {R}}}}(h,\mathcal{A})=\mathop{\mathrm{span}}R(h,\mathcal{A}) $$

formed by all possible finite linear combinations of functions from the set \(R(h,\mathcal{A})\).

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 )∈ℤd be any vector with nonnegative integer coordinates. We set \(x^{s}=x_{1}^{s_{1}}\cdots x_{d}^{s_{d}}\) and |s|=s 1+⋯+s d . Consider the space \({\mathcal {P}^{d}}={\rm span}\{x^{s}: s\in\mathbb{Z}_{+}^{d}\}\) of all polynomials on ℝd, i.e., linear combinations of a finite number of monomials x s. If p(x)=∑ s c s x s is a polynomial from \({\mathcal {P}^{d}}\), then the total degree of p is defined by \({\rm deg}\,p=\max\{|s|:\ c_{s}\neq0\}\). Denote by \({\mathcal {P}_{n}^{d}}\) the space of all polynomials from \({\mathcal {P}^{d}}\) of degree at most n. The total dimension of \({\mathcal {P}_{n}^{d}}\) equals \(D_{n}={n+d \choose d}\).

2 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.

1. The Gegenbauer polynomials: The Gegenbauer polynomials (see [16–18, 20]) are usually defined via the generating function

$$\bigl(1-2tz+z^2\bigr)^{-\lambda}= \sum _{k=0}^{\infty} C_k^{\lambda}(t) z^k, $$

where |z|<1, |t|<1, and λ>0. The coefficients \(C_{k}^{\lambda}(t)\) are algebraic polynomials of degree k and are termed the Gegenbauer polynomials associated with λ. The Gegenbauer polynomials possess the following properties:

(a) The family of polynomials \(\{C_{n}^{\lambda}\}_{0}^{\infty}\) is a complete orthogonal system for the weighted space L 2(ℚ,σ), ℚ=[−1,1], σ λ (t):=(1−t 2)λ−1/2, and

$$ \int_{\mathbb{Q}} C_m^{\lambda}(t) C_n^{\lambda}(t) \sigma_\lambda(t)\,dt= \left\{ \begin{array}{l@{\quad}l} 0,&m\ne n,\\ v_{n,\lambda},&m=n,\\ \end{array} \right. $$
(1)

where we use the usual notation \(v_{n,\lambda}:= \frac{\pi ^{1/2}(2\lambda)_{n} \varGamma(\lambda+1/2)}{ (n+\lambda)n! \varGamma(\lambda)}\) and (a)0:=1, (a) n :=a(a+1)⋯(a+n−1).

Let d be a natural number. We set \(U_{n}(t)= v_{n,d/2}^{-1/2} C_{n}^{d/2}(t)\). Then the family of polynomials \(\{U_{n}\}_{0}^{\infty}\) is the complete orthonormal system in the weighted space L 2(ℚ,σ), where σ(t)=σ d/2(t)=(1−t 2)(d−1)/2.

(b) Let \(\mathcal{P}_{n}^{d}\) be the set of all algebraic polynomials of total degree n in d real variables. Let ξ be any point on the sphere \(\mathbb {S}^{d-1}\). Then the polynomial U n (ξ⋅x) is in \(\mathcal{P}_{n}^{d}\) and is orthogonal to \(\mathcal{P}_{n-1}^{d}\) in L 2(B d) (see [3, 10]), i.e.,

$$ \int_{B^d} U_n(\xi\cdot x) p(x) \,dx=0, \quad\forall p\in\mathcal{P}_{n-1}^d. $$
(2)

(c) For each ξ,η∈S d−1 and n∈ℤ+, we have (see [10])

$$ \int_{B^d} U_n(\xi\cdot x) U_n(\eta\cdot x)\,dx= \frac{U_n(\xi\cdot\eta)}{U_n(1)}. $$
(3)

(d) For each polynomial \(p(x)\in\mathcal{P}_{n}\) such that p(x)=(−1)n p(−x) for all x∈ R d, we have ([10, 16])

$$ \int_{S^{d-1}} p(\xi) U_n(\xi\cdot\eta) \,d\xi= c_n p(\eta),\quad\mathrm{where}\ c_n= \frac{2(2\pi)^{d-1}U_n(1)}{(n+1)_{d-1}}. $$
(4)

2. An Orthogonal System of Polynomials on the Sphere: Let j be a nonnegative integer number. Let \(\mathcal{H}_{j}^{\mathrm{hom}}\) be the subspace in \(\mathcal{P}_{j}^{d}\) formed by all harmonic homogeneous polynomials (i.e., the spherical harmonics) of degree j. We know [16, 20] that the dimension l j of \(\mathcal {H}_{j}^{\mathrm{hom}}\) equals , j≥2, and l 0=1, dim l 1=d. It is easy to verify that the dimension \(l_{j}= (1+\frac {2}{(d-2)!}+c(s,d) )s(s+1)\cdots(s+d-3)\), where 0≤c(s,d)≤1 is some constant depending only on s and d. Let \(\boldsymbol{\varPi}_{j}=\{Y_{j,1},\ldots,Y_{j,l_{j}}\}\) be an orthonormal basis in the space \(\mathcal{H}_{j}^{\mathrm{hom}}(\mathbb {S}^{d-1})\) of functions formed by restrictions of functions from \(\mathcal{H}_{j}^{\mathrm{hom}}\) to the sphere \(\mathbb {S}^{d-1}\). Then the set \(\bigcup_{j=0}^{\infty}\boldsymbol{\varPi}_{j}\) 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 \(\bigcup_{j=0}^{\infty}\boldsymbol{\varPi}_{j}\), the following holds:

$$(Y_{j,k},Y_{j',k'})= \int_{S^{d-1}} Y_{j,k}(\xi) \overline{Y_{j',k'}(\xi)} \,d\xi= \delta_{j,j'}\delta_{k,k'}, $$

where by dξ we denote the normalized Lebesgue measure on \(\mathbb {S}^{d-1}\).

3. An Orthogonal System of Polynomials on the Ball: Consider the Hilbert space \(L_{2}(\mathbb {S}^{d-1})\) of complex-valued square-integrable functions h on the sphere \(\mathbb {S}^{d-1}\) with the inner product

$$(s_1,s_2)= \int_{\mathbb {S}^{d-1}} s_1(\xi) \overline{s_2(\xi)}\,d\xi,\quad s_1,s_2\in L_2\bigl(\mathbb {S}^{d-1}\bigr). $$

Also consider the Hilbert space L 2(ℚ,σ) of real functions on the segment ℚ=[−1,1] with the norm

$$\|g\|_{L_2(\mathbb{Q},\sigma)}= \biggl( \int_{\mathbb{Q}} \bigl|g(t)\bigr|^2 \sigma(t)\,dt \biggr)^{1/2},\quad\mathrm{where}\ \sigma(t)= \bigl(1-t^2\bigr)^{(d-1)/2}. $$

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

$$ \mathbf {I}= \bigl\{I:=(i,j,k):\ i\in\mathbb{Z}_+,\,\,j\in\{ 0,\ldots,i\},\,\, j=i\,(\mathrm{mod}\,2),\,\,k=1,\ldots,l_j\bigr\}, $$
(5)

where l j is the dimension of space \(\mathcal{H}_{j}^{\mathrm{hom}}\) of harmonic homogeneous polynomials of degree j. For every index I=(i,j,k) from I, we construct the function on ℝd,

$$ P_I(x):=P_{i,j,k}(x):=\nu_{ij} \int _{\mathbb{S}^{d-1}}U_{i}(x\cdot\xi)\,Y_{j,k}(\xi)\, d \xi, $$
(6)

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 ℝd of total degree i. Consider the system of polynomials

$$ \mathbf {P}=\{P_I\}_{I\in \mathbf {I}}. $$
(7)

The system P is the complete orthonormal system of polynomials in the space \(L_{2}(\mathbb{B}^{d})\) (see [7]). In particular, for any natural n, the finite subsystem of polynomials

$$ \mathbf {P}_n=\bigl\{P_I:\,\,I=(i,j,k)\in \mathbf {I},\,\,i \le n\bigr\} $$
(8)

in P forms ([7]) an orthogonal basis in the space \({\mathcal {P}_{n}^{d}}\) of all polynomials on ℝd of total degree n.

3 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,

$${\mathbf {Q}_{2n}}=\{P_I:\ I\in \mathbf {I}_n\}. $$

Definition

Let \({\mathcal {Q}_{2 n}^{d}}=\mathop{\mathrm{span}}{\mathbf {Q}_{2n}}\) 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

$$ {\mathcal {P}_{n}^d}\subset {\mathcal {Q}_{2 n}^d}\subset\mathcal{P}_{2n}^d. $$
(9)

Denote by S n the dimension of the space \({\mathcal {Q}_{2 n}^{d}}\). From (9), we have D n ≤S n ≤D 2n . Since the collection of polynomials \(\{P_{I}\} _{I\in \mathbf {I}_{n}}\) is the complete orthonormal system in the space \({\mathcal {Q}_{2 n}^{d}}\), 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 \({\mathcal {Q}_{2 n}^{d}}\) 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 \(a_{1},\ldots,a_{S_{n}}\) from ℝd forming a uniqueness set for the space \({\mathcal {Q}_{2 n}^{d}}\).

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 \({\mathcal {A}_{n}}=\{a_{1},\ldots,a_{S_{n}}\}\) is said to be a h-basis set for the space \({\mathcal {Q}_{2 n}^{d}}\) if \(\mathcal{R}(h,{\mathcal {A}_{n}})={\mathcal {Q}_{2 n}^{d}}\); that is, every polynomial p from \({\mathcal {Q}_{2 n}^{d}}\) can be represented as

$$p(x)=c_1h\bigl(\|x+a_1\|\bigr)+\cdots+c_{S_n}h\bigl(\|x+a_{S_n}\|\bigr),\quad c_i\in {\mathbb{R}}. $$

A set of points \({\mathcal {A}_{n}}\) is called a uniqueness set for the space \({\mathcal {Q}_{2 n}^{d}}\) if, for any two polynomials p 1 and p 2 from \({\mathcal {Q}_{2 n}^{d}}\), 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∈ℝd.

Theorem 3.1

Let n be any natural number and h be any univariate even polynomial of degree 2n. Then set \({\mathcal {A}_{n}}\) is an h-basis set for the space  \({\mathcal {Q}_{2 n}^{d}}\) if and only if \({\mathcal {A}_{n}}\) is the uniqueness set for \({\mathcal {Q}_{2 n}^{d}}\).

Consider the Hilbert space \(L_{2}=L_{2}(\mathbb{B}^{d})\) of complex functions on the ball B d with inner product

$$\langle f_1,f_2 \rangle= \int_{B^d} f(x)\overline{g(x)}\,dx, \quad f_1,f_2\in L_{2}. $$

Let f and G be a function and a subspace in L 2, respectively. Denote by \(E(f,G)=\inf_{g\in G}\|f-g\|_{L_{2}}\) the distance of the function f from G. Let F be a function class in L 2. By \(E(F,G)=\sup_{f\in F} \inf_{g\in G}\|f-g\|_{L_{2}}\), 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, \({\mathcal {A}_{n}}\) be a uniqueness set for \({\mathcal {Q}_{2 n}^{d}}\), and h be any univariate even polynomial of degree n. Then for any function f from the space L 2, the following inequality holds:

$$E \bigl(f,{{\mathcal {\mathcal {R}}}}(h,\mathcal{A}_{n}) \bigr)= E\bigl(f,{ \mathcal{Q}_{2 n}^d}\bigr). $$

Let \(s=(s_{1},\ldots,s_{d})\in\mathbb{Z}_{+}^{d}\) and |s|=s 1+⋯+s d . Introduce the differential operator \(D^{s}= \partial^{|s|}/ \partial^{s_{1}}x_{1}\cdots\partial^{s_{d}}x_{d}\). Let r be any natural number. In the space L 2, we consider the Sobolev class of functions

$$W_2^{r}\bigl(\mathbb{B}^d\bigr):= \Bigl\{f:\, \, \|f\|_{W_2^{r}}= \|f\|_{L_2} + \max_{|s|=r} \bigl\|D^{s} f \bigr\|_{L_2} \le1\Bigr\}. $$

Corollary 3.3

Let n be any even number, \({\mathcal {A}_{n}}\) be an uniqueness set for \({\mathcal {Q}_{2 n}^{d}}\), and h be any univariate even polynomial of degree n. Then the inequalities

$$ c_1 n^{-r}\le E\bigl(W_2^r \bigl(\mathbb{B}^d\bigr), {{\mathcal {\mathcal {R}}}}(h,\mathcal{A}_n)\bigr) \le c_2 n^{-r} $$
(10)

hold, where c 1 and c 2 depend only on r and d.

The upper bound for \(E(W_{2}^{r}(\mathbb{B}^{d}), {{\mathcal {\mathcal {R}}}}(h,\mathcal{A}_{n}))\) 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 \(W_{2}^{r}(\mathbb{B}^{d})\) (see [19]).

4 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 \(L_{2}({\mathbb{B}}^{d})\). Denote by M I (f)=〈f,P I 〉 the I-moment of the function f. Let a be any point in ℝd 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 \(M_{I}(r_{a}^{2s})\) of the shifted radial function \(r_{a}^{2s}= \|x+a\|^{2s}\).

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 \(\mathbb {S}^{d-1}\) of degree j.

Lemma 4.1

Let I=(i,j,k) be any triple index from I and a be any point from ℝd. We write

$$ u_{\beta,i}:= \bigl(t^\beta,U_i \bigr)_\mathbb{Q}:=\int_\mathbb {Q}t^\beta U_i(t)\sigma(t)\,dt $$
(11)

and consider the function on ℝd

$$ V_{i,j,k}(a):= \bigl((a\cdot\theta)^i, Y_{j,k} \bigr):=\int_{\mathbb {S}^{d-1}} (a\cdot \theta)^i \overline{Y_{j,k}}(\theta)\,d\theta. $$
(12)

Then, for every natural s, the I-moment of the function \(r_{a}^{2s}\) equals

$$M_I\bigl(r_{a}^{2s}\bigr) = \nu\sum _{\alpha=0}^{2s} {2s \choose\alpha} u_{2s-\alpha,i} V_{\alpha,j,k}(a), $$

where ν depends only on I,d, and s.

Proof

Given s, we will use the equality

$$ \|x\|^{2s}= c_s\int_{\mathbb {S}^{d-1}} (x\cdot\theta)^{2s}\,d\theta, \quad\mathrm{where}\ c_s= \biggl(\int_{\mathbb {S}^{d-1}}(e\cdot\theta)^{2s}\,d\theta \biggr)^{-1}, $$
(13)

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 ℝd.

For a fixed point θ on \(\mathbb {S}^{d-1}\), we consider the univariate polynomial w θ (t)=(t+a⋅θ)2s of degree 2s. From (13), we have

$$ \|x+a\|^{2s}= c_s\int_{\mathbb {S}^{d-1}} (x\cdot\theta+a\cdot\theta)^{2s}\,d\theta= c_s\int _{\mathbb {S}^{d-1}} w_\theta(x\cdot\theta)\,d\theta. $$
(14)

Using (14) and (6), we can represent the I-moment of the function \(r_{a}^{2s}\) by the following:

where c sij =c s ν ij . Therefore, we have

$$ \bigl\langle r_{a}^{2s},P_I \bigr\rangle = c_{sij} \int_{\mathbb {S}^{d-1}\times \mathbb {S}^{d-1}} Y_{jk}(\xi)\,d\xi\,d \theta\int_{\mathbb{B}^d} w_\theta(x\cdot\theta)\,U_i(x\cdot\xi)\,dx. $$
(15)

Given θ, we decompose the function w θ by the orthogonal system of Gegenbauer polynomials {U α }:

$$w_\theta(t)=\sum_{\alpha=0}^{2s} (w_\theta,U_\alpha)_\mathbb{Q}\,U_\alpha(t). $$

Then the last integral in (15) equals

$$ \int_{\mathbb{B}^d} w_\theta(x\cdot\theta) \,U_i(x\cdot\xi)\,dx =\sum_{\alpha=0}^{2s} (w_\theta,U_\alpha)_\mathbb{Q} \int_{\mathbb{B}^d} U_\alpha(x\cdot\theta)\,U_i(x\cdot\xi)\,dx. $$
(16)

The following identity holds (see (3)):

$$\int_{\mathbb{B}^d} U_\alpha(x\cdot\theta)\,U_i(x \cdot\xi)\,dx = \frac{U_i(\theta\cdot\xi)}{U_i(1)}\delta_{\alpha,i}. $$

Therefore, from (16), we obtain

$$\int_{\mathbb{B}^d} w_\theta(x\cdot\theta) \,U_i(x\cdot\xi)\,dx = (w_\theta,U_i)_\mathbb{Q} \frac{U_i(\theta\cdot\xi)}{U_i(1)}. $$

Substitute this expression into (15):

By (4), we have

$$\int_{\mathbb {S}^{d-1}} Y_{j,k}(\xi) U_i(\theta\cdot \xi)\,d\xi= c_i' Y_{j,k}(\theta). $$

Hence, we obtain

$$ \bigl\langle r_{a}^{2s},P_I\bigr\rangle= \nu \int_{\mathbb {S}^{d-1}} (w_\theta,U_i )_\mathbb{Q} Y_{jk}(\theta) \,d\theta, $$
(17)

where \(\nu= c_{sij}c_{i}'/U_{i}(1)\). The equality

$$w_\theta(t)=(t+a\cdot\theta)^{2s}=\sum_{\alpha=0}^{2s} {2s \choose \alpha} t^{2s-\alpha}(a\cdot\theta)^\alpha $$

implies

Thus, we obtain from (17),

$$\bigl\langle r_{a}^{2s},P_{I} \bigr\rangle= \nu\sum _{\alpha=0}^{2s} {2s \choose\alpha} \bigl(( \bullet)^{2s-\alpha},U_i\bigr)_{\mathbb{Q}} \bigl((a\cdot \theta)^\alpha, Y_{j,k} \bigr). $$

Lemma 4.1 is complete. □

5 The Proof of Theorem 3.1

Let n be any natural number and \(h(t)=\sum_{s=0}^{n} c_{s} t^{2s}\), t∈ℝ, be an univariate even polynomial of degree 2n. Given a point a from ℝd, 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 . Lemma 4.1 implies

$$M_I(g_a)= \sum_{s=0}^{n} c_s M_I \bigl(r_a^{2s} \bigr) = \nu \sum_{s=0}^{n} \sum _{\alpha=0}^{2s} c_s{2s\choose\alpha} u_{2s-\alpha,i} V_{\alpha,j,k}(a). $$

Set \(c_{s,\alpha}= \nu c_{s} {2s\choose\alpha}\). By properties of orthogonality of the polynomials U i , we have from (11),

$$u_{2s-\alpha,i}=\int_\mathbb {Q}t^{2s-\alpha} U_i(t)\sigma(t)\,dt=0,\quad 2s-\alpha<i. $$

Analogously, by properties of orthogonality of spherical harmonics Y jk , we have from (12),

$$V_{\alpha,j,k}(a)= \int_{\mathbb {S}^{d-1}} (a\cdot\theta)^\alpha \overline{Y_{j,k}}(\theta)\,d\theta= 0,\quad\alpha<j. $$

Also, we see that V α,j,k (a)=0 if α≠j (mod 2). Thus,

$$M_I(g_a)= \nu\sum_{s=\frac{i+j}{2}}^{n} \sum_{\alpha=j}^{2s-i} c_{s,\alpha} u_{2s-\alpha,i} V_{\alpha,j,k}(a). $$

Note (see (5)) that i=j (mod 2). Therefore, i+j is always an even number. Interchanging the order summation, we obtain

$$ M_I(g_a)= \sum _{\alpha=j}^{2n-i} \biggl(\sum_s c_{s,\alpha} u_{2s-\alpha,i} \biggr) V_{\alpha,j,k}(a), $$
(18)

where the index s runs over the set \(\{\frac{\alpha+i}{2},\frac{\alpha +i}{2}+1,\ldots,n\}\). 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:

$$\mathbf{M}_n=\bigl\{M_{i,j,k}(g_a):\ (i,j,k)\in \mathbf {I}_{n}\bigr\} \quad\mathrm{and}\quad\mathbf{V}_n=\bigl \{V_{i,j,k}(a):\ (i,j,k)\in \mathbf {I}_{n}\bigr\}. $$

Lemma 5.1

The linear spans

$$\widehat{\mathbf{M}}_n=\mathop{\mathrm{span}} \mathbf{M}_n\quad\mathrm{and}\quad \widehat{\mathbf{V}}_n=\mathop{\mathrm{span}} \mathbf{V}_n $$

of sets M n and V n coincide.

Proof

Let I=(i,j,k)∈I n be any index. From (18), we see that

that is, \(\widehat{\mathbf{M}}_{n}\subseteq\widehat{\mathbf{V}}_{n}\). We show that \(\widehat{\mathbf{V}}_{n}\subseteq\widehat{\mathbf{M}}_{n}\). We fix j and k. The equality (18) we rewrite as

$$ M_{i,j,k}(g_a)= \sum _{\alpha=j}^{2n-i} C_{i}^\alpha V_{\alpha,j,k}(a),\quad\mathrm{where}\ C_{i}^\alpha= \sum _{s} c_{s,\alpha} u_{2s-\alpha,i}. $$
(19)

We show that the functions V α,j,k for every α=j,…,2n belong to \(\widehat{\mathbf{M}}_{n}\). We use induction on α. Let α=j. We have from (19) with i=2n−j,

$$ M_{n-j,j,k}(g_a)= C_{2n-j}^j V_{j,j,k}(a). $$
(20)

Hence, \(V_{j,j,k}\in\widehat{\mathbf{M}}_{n}\). Now assume \(V_{j,j,k},\ldots,V_{j+\beta,j,k}\in\widehat{\mathbf{M}}_{n}\) with β≥0. Show that V j+β+1,j,k also belongs to \(\widehat{\mathbf{M}}_{n}\). We have from (19) with i=2n−j−β−1,

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 \(\widehat{\mathbf{M}}_{n}\), we obtain that V j+β+1,j,k also belongs to \(\widehat{\mathbf{M}}_{n}\). Thus, the functions V i,j,k (a), for all i≥j, belong to \(\widehat{\mathbf{M}}_{n}\); that is, \(\widehat{\mathbf{V}}_{n}\subseteq\widehat{\mathbf{M}}_{n}\). Lemma 5.1 is proved. □

Lemma 5.2

The spaces \(\widehat{\mathbf{V}}_{n}\) and \({\mathcal {Q}_{2 n}^{d}}\) coincide.

Proof

The space \(\widehat{\mathbf{V}}_{n}\) is the linear span of polynomials

$$ V_{i,j,k}(a)=\bigl((a\cdot\theta)^i, Y_{jk}\bigr),\quad(i,j,k)\in \mathbf {I}_{n}. $$
(21)

Every polynomial 1,t,…,t n is some linear combination of Gegenbauer polynomials \(\{U_{i}(t)\}_{i=0}^{n}\). Therefore, \(\widehat{\mathbf{V}}_{n} \subseteq {\mathcal {Q}_{2 n}^{d}}\). We now prove that \({\mathcal {Q}_{2 n}^{d}}\subseteq\widehat{\mathbf{V}}_{n}\). We show that the set V n forms a linear basis in the space \({\mathcal {Q}_{2 n}^{d}}\). Indeed, consider the system of functions Q 2n consisting of polynomials (6)

$$ P_{i,j,k}(a)=\bigl(U_i(a\cdot\theta), Y_{jk}\bigr)=\sum_{\alpha=o}^i b_{i,\alpha} \bigl((a\cdot\theta)^\alpha, Y_{jk}\bigr), \quad(i,j,k)\in \mathbf {I}_{n}, $$
(22)

where b i,α are the coefficients of the Gegenbauer polynomial U i . The system Q 2n is a linear basis in the space \({\mathcal {Q}_{2 n}^{d}}\). On the other hand, from (21) and (22), we see that \(P_{i,j,k}\in\widehat{\mathbf{V}}_{n}\) for every index (i,j,k)∈I n . Thus, V n is a linear basis in \({\mathcal {Q}_{2 n}^{d}}\). □

Proof of Theorem 3.1

By Lemmas 5.1 and 5.2, the collection of functions of a-variable M n ={M I (g a )}, I∈I n , forms a basis in the space \({\mathcal {Q}_{2 n}^{d}}\). Therefore, the following obvious statement holds. □

Proposition 5.3

A set \(\mathcal{A}_{n}=\{a_{1},\ldots,a_{S_{n}}\}\), where \(S_{n}=\mathrm{dim}\,{\mathcal {Q}_{2 n}^{d}}\), is a uniqueness set for the space of polynomials \({\mathcal {Q}_{2 n}^{d}}\) if and only if the square matrix

$$\bigl(M_I(g_{a_j})\bigr), \quad I\in \mathbf {I}_{n},\ {j=1, \ldots,S_n}, $$

is nondegenerate.

Proof

Assume that a set \(\mathcal{A}_{n}\) is a uniqueness set for the space \({\mathcal {Q}_{2 n}^{d}}\). We show that any polynomial p from \({\mathcal {Q}_{2 n}^{d}}\) may be represented by

$$ p(x)=c_1h\bigl(\|x+a_1\|\bigr)+\cdots+c_{S_n}h \bigl(\|x+a_{S_n}\|\bigr). $$
(23)

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

$$ M_I(p)= c_1 M_I(g_{a_1})+ \cdots+c_{S_n} M_I(g_{a_{S_n}}),\quad I\in \mathbf {I}_{n}, $$
(24)

with respect to the unknowns coefficients \(c_{1},\ldots,c_{S_{n}}\). 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 \({\mathcal {Q}_{2 n}^{d}}\) 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 \(\mathcal{A}_{n}\) is a uniqueness set for the space \({\mathcal {Q}_{2 n}^{d}}\). □