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 FaberâSchauder basis (Faber, 1908) 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 Triebelâs monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylorâs remainder formula to the dual ChuiâWang 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 TriebelâLizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear FaberâSchauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the FaberâSchauder situation.
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