A higher order Faber spline basis for sampling discretization of functions

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 [19] 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 [46] 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.

Orthogonal Systems of Spline Wavelets as Unconditional Bases in Sobolev Spaces

“…The author would like to thank her advisor Andreas Seeger for introducing this problem, for his guidance and several illuminating discussions. She is also grateful to Dr. Nadiia Derevianko and Prof. Tino Ullrich for sharing a copy of their manuscript [7], which helped shape some of the ideas contained here. Research supported in part by NSF grant 1500162.…”