Approximation and quadrature by weighted least squares polynomials on the sphere

Abstract

Given a sequence of Marcinkiewicz-Zygmund inequalities in L 2 on a usual compact space M, Gröchenig in [11] introduced the weighted least squares polynomials and the least squares quadrature from pointwise samples of a function, and obtained approximation theorems and quadrature errors. In this paper we obtain approximation theorems and quadrature errors on the sphere which are optimal. We also give upper bounds of the operator norms of the weighted least squares operators.

Function recovery on manifolds using scattered data

“…On the sphere, a special case of the manifolds we consider, Mhaskar [17] derives an asymptotically optimal upper bound in terms of zonal function networks. Wang and Sloan [29] use filtered polynomial approximation and Lu and Wang [16] rely on weighted least squares in the Hilbert space case to achieve optimal rates, improving upon Gröchenig [11]. On general manifolds, some results are obtained in Mhaskar [18,Theorem 3.3], which relies on approximation by eigenfunctions of the Laplace-Beltrami operator on the manifold.…”