On best approximation of classes by radial functions (Q1867260)

Given a natural number \(n\) and some fixed \( a_1, \dots, a_n \in {\mathbb R }^d\), let \({\mathcal R}(a_1, \dots, a_n)\) be the set of all functions \(g:{\mathbb R }^d \to {\mathbb R }\) of the form \[ g(x)=\sum_{k=1}^n g_k(|x-a_k|), \qquad x \in {\mathbb R }^d, \] where \(g_k:{\mathbb R } \to {\mathbb R }\) are any continuous functions. For a compact \(D \subset {\mathbb R }^d\), let \(W_2^{r,d}\) denote the Sobolev class of functions defined on \(D\) whose norms in \(L_2(D)\), along with the norms of their partial derivatives of orders \(\leq r\), are bounded by one. The author proves that for \(d \geq 2\) and any \(n\) there exist \( a_1, \dots, a_n \) on the unit sphere \(S^{d-1}\) such that every \(f \in W_2^{r,d}\) can be approximated in \(L_2(D)\) by some \(g \in {\mathcal R}(a_1, \dots, a_n)\) with an accuracy \(O(n^{-{r \over d-1}})\). A weaker statement, with unrestricted \(a_1, \dots, a_n\), follows from the author's earlier result on approximation by the ridge functions since every ridge function can be approximated on \(D\) arbitrarily close by a function of the form \(\phi (|x-a|)\) with large \(a\). The author also proves that the stated order of approximation is the best possible even with the unrestricted \(a_1, \dots, a_n\) . The upper bound is established via polynomial approximations for which purpose a special orthonormal polynomial basis is constructed. A role is played by the Gegenbauer polynomials. The lower bound is obtained by comparing entropy numbers.