Uniform approximation on the sphere by least squares polynomials

Abstract

The paper concerns the uniform polynomial approximation of a function f , continuous on the unit Euclidean sphere of R 3 and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from n − m up to n + m, being m = θn for any fixed parameter 0 < θ < 1. As n tends to infinity, we prove that these polynomials uniformly converge to f at the near-best polynomial approximation rate. Moreover, for fixed n, by using the same data points we can further improve the approximation by suitably modulating the action ray m determined by the parameter θ. Some numerical experiments are given to illustrate the theoretical results.keywords: polynomial approximation on the sphere, least squares approximation, uniform approximation, Lebesgue constant, de la Vallée Poussin type mean.

Fast Cross-validation in Harmonic Approximation

“…The base for approximation from scattered data is formed by positive quadrature rules, Marcinkiewicz-Zygmund inequalities which are investigated in the papers [47,32,30,4], and by bounds for best approximations [39,46,18]. Based on these result the relationship between the mesh norm, the separation distance of the sampling points, and optimal approximation rates has been analyzed in the papers [10,28,25,42]. Approximation from noisy data has been considered in [1] and a priori and a posteriori estimates of the approximation error with respect to the regularization parameter have been proven in [34].…”

Projecting Lipschitz functions onto spaces of polynomials

“…The particular case of projections onto spaces of polynomials is also very well studied (particularly in the trigonometric case), for its connections with harmonic ([Wo, III.B], [GM, OL]) and numerical ( [FL,TV,TV2]) analysis. Let us stress that, in this context, one typically fixes the dimension of the space and lets the degree of the polynomials grow, while in our paper we are interested in the opposite situation.…”