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An Approach to Wavelet Isomorphisms of Function Spaces Via Atomic Representations

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Abstract

Both wavelet and atomic decomposition techniques are essential tools in the study of function spaces nowadays, but they both have their advantages and disadvantages. The celebrated bridge between both concepts was given by the compactly supported Daubechies wavelets which can be interpreted as atoms. In this paper we deal with the converse direction, that is, we present a fairly general approach how to construct compactly supported wavelets when an atomic decomposition is known already. The main idea is Taylor’s expansion combined with our new, so-called \(\varkappa \)-convergence assumption in the admitted sequence spaces. We finally exemplify our main result and collect some known and new settings where such a wavelet decomposition is obtained, e.g., in spaces of Besov or Triebel–Lizorkin type with a doubling weight.

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Acknowledgements

The authors would like to thank the referees for their careful reading and many helpful comments, which improved the presentation of this paper.

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Authors and Affiliations

  1. Institute of Mathematics, Friedrich Schiller University Jena, 07737, Jena, Germany

    Dorothee D. Haroske, Philipp Skandera & Hans Triebel

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  1. Dorothee D. Haroske
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  2. Philipp Skandera
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  3. Hans Triebel
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Correspondence to Dorothee D. Haroske.

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Communicated by Stephan Dahlke.

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Haroske, D.D., Skandera, P. & Triebel, H. An Approach to Wavelet Isomorphisms of Function Spaces Via Atomic Representations. J Fourier Anal Appl 24, 830–871 (2018). https://doi.org/10.1007/s00041-017-9538-6

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  • DOI: https://doi.org/10.1007/s00041-017-9538-6

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