Constructive polynomial approximation on the sphere
The authors deal with the problem of polynomial approximations of a continuous function on the unit sphere \(S^{r-1}\subseteq\mathbb{R}^r\) from the set of all polynomials in \(r\) variables of degree at most \(n\) restricted to \(S^{r-1}\). The hyperinterpolation [see \textit{I. H. Sloan}, J. Approximation Theory 83, No. 2, 238-254 (1995; Zbl 0839.41006)] operator \(L_n\) as a map from \(C(S^{r-1})\) to \(C(S^{r-1})\) is here studied and the results in this setting extend other known theoretical ones for hyperinterpolation. Two main results are the following: The norm of the hyperinterpolation operator in the setting \(C\) to \(C\) is bounded by \(\|L_n\|\leq d_n^{1\over 2}\) \((d_n=\) dimension of the space of spherical polynomials of degree at most \(n)\). For the important case \(r=3\), this theorem yields \(\|L_n\|_{C\to C}\leq n+1\), which is an improvement on the result \(\|L_n\|_{C\to C}\leq cn^2\) obtained by \textit{M. Ganesh}, \textit{I. G. Graham}, and \textit{J. Sivaloganathan} [SIAM J. Numer. Anal. 31, No. 5, 1378-1414 (1994; Zbl 0815.41008)]. Furthermore, for this case \(r=3\) and under a mild additional assumption on underlying quadrature rule, one has: \(\|L_n\|_{C\to C}\asymp n^{1\over 2}\), which is optimal with respect to order.
scientific article; zbMATH DE number 1423076