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Multiresolution expansions and wavelets in Gelfand–Shilov spaces

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Abstract

We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand–Shilov spaces. Important examples of multiresolution analyses for which our results apply arise in particular from Dziubański–Hernández construction of band-limited wavelets with subexponential decay. Our results are twofold. Firstly, we obtain approximation properties of multiresolution expansions of Gelfand–Shilov functions and (ultra)distributions. Secondly, we establish convergence of wavelet series expansions in the same regularity framework.

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Notes

  1. The result is stated in [18] under the extra assumption \(\rho _1>0\), but the proof given there actually works for \(\rho _1=0\) as well. The same comment applies to Theorem 1.

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Acknowledgements

S. Pilipović and N. Teofanov were supported by Serbian Ministry of Education and Science through Project 174024, project 19/6-020/961-47/18 MNRVOID of the Republic of Srpska and ANACRES. D. Rakić was supported by Serbian Ministry of Education and Science through Project III44006 and by Provincial Secretariat for Higher Education and Scientific Research through Project 142-451-2102/2019. J. Vindas was supported by Ghent University through the BOF-grants 01J11615 and 01J04017.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, 21000, Serbia

    Stevan Pilipović & Nenad Teofanov

  2. Faculty of Technology, University of Novi Sad, Bul. cara Lazara 1, Novi Sad, 21000, Serbia

    Dušan Rakić

  3. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    Jasson Vindas

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  1. Stevan Pilipović
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  2. Dušan Rakić
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  4. Jasson Vindas
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Correspondence to Dušan Rakić.

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Pilipović, S., Rakić, D., Teofanov, N. et al. Multiresolution expansions and wavelets in Gelfand–Shilov spaces. RACSAM 114, 66 (2020). https://doi.org/10.1007/s13398-020-00789-4

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