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

Content Provider	Springer Nature Link
Author	Khrennikov, A. Y. Walt, Jan Harm Shelkovich, V. M.
Copyright Year	2013
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.
Ending Page	1358
Page Count	36
Starting Page	1323
File Format	PDF
ISSN	10695869
e-ISSN	15315851
Journal	Journal of Fourier Analysis and Applications
Issue Number	6
Volume Number	19
Language	English
Publisher	Springer US
Publisher Date	2013-11-14
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Adeles Infinite tensor products of Hilbert spaces Signal, Image and Speech Processing Multiresolution analysis Adelic fractional operator Wavelets and other special systems $p$ -adic theory, local fields Abstract Harmonic Analysis Wavelets Mathematical Methods in Physics Pseudodifferential operators Fourier Analysis Operations with distributions Approximations and Expansions Fractional derivatives and integrals Partial Differential Equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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