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arXiv · 1302.3751

Wavelets in function spaces on cellular domains

Benjamin Scharf
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Abstract

Nowadays the theory and application of wavelet decompositions plays an important role not only for the study of function spaces (of Lebesgue, Hardy, Sobolev, Besov, Triebel-Lizorkin type) but also for its applications in signal and numerical analysis, partial differential equations and image processing. In this context it it a hard problem to construct wavelet bases for suitable function spaces on domains, e. g. the unit cube. A big step in this direction are the contributions of Hans Triebel from 2006 to 2008 where he constructed Riesz bases for classes of Besov- and Triebel-Lizorkin spaces on domains, starting with Daubechies wavelets. But there was a problem coming from the method: He had to exclude a big number of function spaces, in particular a large class of classical Sobolev spaces. The main goal of this thesis is a construction of Riesz bases of wavelet systems also for the exceptional cases using a modification of the function spaces - the so-called reinforced function spaces.

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Benjamin Scharf. 2013-02-15. Wavelets in function spaces on cellular domains. https://arxiv.org/abs/1302.3751

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