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Construction of Banach frames and atomic decompositions of anisotropic Besov spaces

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Abstract

We construct generalised shift-invariant systems of functions of several real variables for anisotropic Besov spaces that can be generated by the decomposition method using any given expansive matrix and establish the conditions on those systems under which they will constitute Banach frames or sets of atoms for the anisotropic homo- or inhomogeneous Besov spaces.

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References

  1. Besov, O.V.: Study of one family of spaces of functions in connexion with theorems of embedding and extension. Trudi MIAN SSSR, pp. 42–81 (1961)

  2. Besov, O.V., Ilin, V.P., Nikolski, S.M.: Integral Representations of Functions and Embedding Theorems. Nauka, Moscow (1975)

    Google Scholar 

  3. Borup, L., Nielsen, M.: Frame decomposition of decomposition spaces III. J. Fourier Anal. Appl. 13, 39–70 (2007)

    Article  MathSciNet  Google Scholar 

  4. Bownik, M.: Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z. 250, 539–571 (2005)

    Article  MathSciNet  Google Scholar 

  5. Bytchenkoff, D., Voigtlaender, F.: Design and properties of wave packet smoothness spaces. J. Math. Pures Appl. 133, 185–262 (2020)

    Article  MathSciNet  Google Scholar 

  6. Cheshmavar, J., Führ, H.: A classification of anisotropic Besov spaces. Appl. Harmon. Comput. Anal (2019). https://doi.org/10.1016/j.acha.2019.04.006. (in press)

    Article  MATH  Google Scholar 

  7. Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia, PA (1992)

    Book  Google Scholar 

  8. Feichtinger, H.G., Gröbner, P.: Banach spaces of distributions defined by decomposition method. Math. Nachr. 123, 97–120 (1985)

    Article  MathSciNet  Google Scholar 

  9. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)

    Article  MathSciNet  Google Scholar 

  10. Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93, 34–170 (1989)

    Article  MathSciNet  Google Scholar 

  11. Gröchenig, K.: Describing functions: atomic decompositions versus frames. Mon. Math. 112, 1–41 (1991)

    Article  MathSciNet  Google Scholar 

  12. Nikolski, S.M.: Approximation of Functions of Several Variables and Embedding Theorems. Nauka, Moscow (1969)

    Google Scholar 

  13. Sobolev, S.L.: Some Applications of Functional Analysis in Mathematical Physics. Leningrad State University, Leningrad (1950)

    Google Scholar 

  14. Triebel, H.: Theory of Function Spaces III. Birkhäuser Verlag, Basel (2006)

    MATH  Google Scholar 

  15. Voigtlaender, F.: Structured, compactly supported Banach frame decompositions of decomposition spaces. http://arxiv.org/abs/1612.08772

  16. Voigtlaender, F., Pein, A.: Analysis vs. synthesis sparsity for \(\alpha \)-shearlets. http://arxiv.org/abs/1702.03559

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Acknowledgements

I thank the Centre National de la Recherche Scientifique of France and the Deutscher Akademischer Austauschdienst of Germany for their funding and Professor Gitta Kutyniok for her support of this work.

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  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Dimitri Bytchenkoff

  2. Laboratoire d’Energétique et de Mécanique Théorique et Appliquée, Université de Lorraine, 2 avenue de la Forêt de Haye, 54505, Vandoeuvre-lès-Nancy, France

    Dimitri Bytchenkoff

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  1. Dimitri Bytchenkoff
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Correspondence to Dimitri Bytchenkoff.

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Bytchenkoff, D. Construction of Banach frames and atomic decompositions of anisotropic Besov spaces. Annali di Matematica 200, 1341–1365 (2021). https://doi.org/10.1007/s10231-020-01040-y

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  • DOI: https://doi.org/10.1007/s10231-020-01040-y

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