On an orthogonal bivariate trigonometric Schauder basis for the space of continuous functions

Derevianko, Nadiia; Myroniuk, Vitalii; Prestin, Jürgen

doi:10.1016/j.jat.2017.05.007

Cited by 2 publications

(1 citation statement)

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“…Therefore, we also give equivalent representations for the (quasi-)norm of Besov-Triebel-Lizorkin spaces in terms of Chui-Wang wavelet coefficients. Note also that for the periodic case very well time-localized basis functions were constructed in [30] and [18] for one-and two-dimensional cases. The corresponding characterization of Besov spaces (for the univariate case so far) was obtained in [17].…”

Section: Introductionmentioning

confidence: 99%

A higher order Faber spline basis for sampling discretization of functions

Derevianko,

Ullrich

2019

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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.

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Section: Introductionmentioning

confidence: 99%