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Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators

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Abstract

In our previous paper, the Haar multiresolution analysis (MRA) \(\{V_{j}\}_{j\in \mathbb {Z}}\) in \(L^{2}(\mathbb {A})\) was constructed, where \(\mathbb {A}\) is the adele ring. Since \(L^{2}(\mathbb {A})\) is the infinite tensor product of the spaces \(L^{2}({\mathbb {Q}}_{p})\), p=∞,2,3,…, the adelic MRA has some specific properties different from the corresponding finite-dimensional ones. Nevertheless, this infinite-dimensional MRA inherits almost all basic properties of the finite-dimensional case. In this paper we derive explicit formulas for bases in V j , \(j\in \mathbb {Z}\), and for the wavelet bases generated by the above-mentioned adelic MRA. In view of the specific properties of the adelic MRA, there arise some technical problems in the construction of wavelet bases. These problems were solved with the aid of the operator formalization of the process of generation of wavelet bases. We study the spectral properties of the fractional operator introduced by S. Torba and W.A. Zúñiga-Galindo. We prove that the constructed wavelet functions are eigenfunctions of this fractional operator. This paper, as well as our previous paper, introduces new ideas to construct different infinite-dimensional MRAs. Our results can be used in the theory of adelic pseudo-differential operators and equations over the ring of adeles and in adelic models in physics.

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Notes

  1. In particular, we get to know about the practically forgotten paper of S. Dahlke [8] who (already in 1994) constructed MRA on arbitrary locally compact Abelian groups. We remark that, however, our p-adic and adelic MRA are not covered by Dahlke’s MRA, since he used a subgroup as a set of wavelet generating shifts. The latter is not the case in our approach.

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Acknowledgements

The authors were supported by the joint grant of Swedish Research Council and South-African Research Council, Sida project “Non-Archimedean analysis: from fundamentals to applications”.

We are greatly indebted to Sergii M. Torba for fruitful discussions and extremely grateful to A.V. Kosyak, with whom we intensively discussed many questions and technical problems arisen when working on this paper.

Partially the results of this paper were presented in the talk of one of the coauthors (A. Khrennikov) at the seminar of the Numerical Harmonic Analysis Group of University of Vienna; the speaker is thankful to all participants of the seminar for fruitful discussions and for “knowledge transfer” on harmonic analysis on locally compact groups (especially zero dimensional) and history of its development.Footnote 1 The visit to Vienna was based on the visiting professor fellowship at the quantum foundation group of prof. A. Zeilinger, Austrian Academy of Science, and the grant “Non-Archimedean analysis: from fundamentals to applications”.

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Authors and Affiliations

  1. International Center for Mathematical Modelling in Physics and Cognitive Sciences MSI, Linnaeus University, Växjö-Kalmar, 351 95, Växjö, Sweden

    A. Y. Khrennikov

  2. Department of Mathematics, St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Russia

    V. M. Shelkovich

  3. Mathematical Physics Department, Faculty of Physics, St. Petersburg State University, Ulianovskaya 2, 198904, St. Petersburg, Russia

    V. M. Shelkovich

  4. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

    Jan Harm van der Walt

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  1. A. Y. Khrennikov
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  3. Jan Harm van der Walt
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Correspondence to A. Y. Khrennikov.

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Communicated by Hans G. Feichtinger.

This paper was completed a few days before sudden death of Vladimir Shelkovich in February 2013.

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Khrennikov, A.Y., Shelkovich, V.M. & van der Walt, J.H. Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators. J Fourier Anal Appl 19, 1323–1358 (2013). https://doi.org/10.1007/s00041-013-9304-3

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