A higher order Faber spline basis for sampling discretization of functions

TL;DR

This paper introduces a higher order Faber spline basis that extends classical Faber basis to functions with higher regularity, offering localized, well-supported basis functions for sampling discretization of Besov and Triebel-Lizorkin spaces.

Contribution

The paper constructs a higher order Faber spline basis using Taylor's remainder and dual wavelets, extending classical Faber basis to higher smoothness levels.

Findings

Provides sampling characterizations for Besov spaces

Overcomes smoothness restrictions of classical Faber basis

Basis functions are exponentially localized and unconditional

Abstract

This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber-Schauder basis except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel's monograph (see References [47]) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor's remainder formula to the dual Chui-Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel-Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber-Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber-Schauder situation.