Sparse grid approximation in weighted Wiener spaces

TL;DR

This paper investigates how sparse grid methods can efficiently approximate multivariate periodic functions within weighted Wiener spaces, providing convergence rates and characterizations using quasi-interpolation operators.

Contribution

It introduces new approximation results for sparse grid methods in weighted Wiener spaces, including convergence rates and Littlewood-Paley-type characterizations.

Findings

Established convergence rates of sparse grid methods in weighted Wiener norms.

Extended Littlewood-Paley characterizations using quasi-interpolation operators.

Unified framework for various quasi-interpolation operators in approximation.

Abstract

We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grids methods constructed with the help of quasi-interpolation operators. The class of such operators includes classical interpolation and sampling operators, Kantorovich-type operators, scaling expansions associated with wavelet constructions, and others. We obtain the rate of convergence of the corresponding sparse grids methods in weighted Wiener norms as well as analogues of the Littlewood-Paley-type characterizations in terms of families of quasi-interpolation operators.