Representation of polynomials by linear combinations of radial basis functions
Let \(d\in\mathbb{N}\), and denote by \(C_n^{d/2}\) the Gegenbauer polynomials of degree \(n\in \mathbb{N}_0\) associated with the parameter \(d/2\). The properly normalized Gegenbauer polynomials form a complete orthogonal system \(\{U_n : n\in\mathbb{N}_0\}\) for \(L^2([-1,1],\sigma)\), where \(\sigma\) is the weight function \((1-t^2)^{(d-1)/2}\). Let \(\mathbf{I} :=\{I := (i,j,k) : i \in\mathbb{N}_0, j \in \{0,1,\dotsc, i\},\;j = i (\mod 2),\;k = 1, \dotsc, l_j\}\), where \(l_j\) is the dimension of the space of spherical harmonics of degree \(j\). For \(I\in \mathbf{I}\), define \[ P_I(x) := \nu_{ij}\,\int_{\mathbb{S}^{d-1}} U_i (x\cdot \xi) Y_{j,k}(\xi)d\xi, \] where \(Y_{j,k}\) is a spherical harmonic polynomial and \(\nu_{ij}\) is chosen so that \(\|P_I\|_{L^2} = 1\). The author defines a new space of polynomials \(\mathcal{Q}_{2n}^d\) as \[ \mathcal{Q}_{2n}^d := \operatorname{span} \{P_I : I\in \mathbf{I}_n\}, \] where \(\mathbf{I}_n := \{I = (i,j,k)\in \mathbf{I} : i+j\leq 2n\}\). The main result of the paper is the following statement: Let \(\mathcal{A}\) be a finite set of points in \(\mathbb{R}^d\) and \(h\) a fixed univariate even polynomial of degree \(n\). Then, every polynomial \(P\in \mathcal{Q}_{2n}^d\) can be represented as a linear combination of shifted radial basis functions of the form \(h(\|\bullet + a\|)\), \(a\in \mathcal{A}\), if and only if \(\mathcal{A}\) is a uniqueness set for \(\mathcal{Q}_{2n}^d\). (Here, a uniqueness set for \(\mathcal{Q}_{2n}^d\) is any finite set \(\mathcal{A}\) such that, for any pair of polynomials \(p_1,p_2\in \mathcal{Q}_{2n}^d\) with \(p_1(a_i) = p_2 (a_i)\), \(i = 1, \dotsc, \operatorname{card}(\mathcal{A})\), the implication \(p_1(a) = p_2(a)\) for all \(a\in \mathbb{R}^d\) holds.)
scientific article; zbMATH DE number 6150047