Polynomial approximation on spheres - generalizing de la Vallée-Poussin

Abstract

For trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.