Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators
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Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators A.Y. Khrennikov · V.M. Shelkovich · Jan Harm van der Walt
Received: 12 February 2013 / Revised: 15 July 2013 / Published online: 14 November 2013 © Springer Science+Business Media New York 2013
Abstract In our previous paper, the Haar multiresolution analysis (MRA) {Vj }j ∈Z in L2 (A) was constructed, where A is the adele ring. Since L2 (A) is the infinite tensor product of the spaces L2 (Qp ), 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 Vj , j ∈ 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. ZúñigaGalindo. We prove that the constructed wavelet functions are eigenfunctions of this
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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A.Y. Khrennikov ( ) International Center for Mathematical Modelling in Physics and Cognitive Sciences MSI, Linnaeus University, Växjö-Kalmar, 351 95, Växjö, Sweden e-mail: [email protected] V.M. Shelkovich Department of Mathematics, St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Russia V.M. Shelkovich Mathematical Physics Department, Faculty of Physics, St. Petersburg State University, Ulianovskaya 2, 198904, St. Petersburg, Russia J.H. van der Walt Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa e-mail: [email protected]
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J Fourier Anal Appl (2013) 19:1323–1358
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. Keywords Adeles · Multiresolution analysis · Wavelets · Infinite tensor products of Hilbert spaces · Adelic fractional operator Mathematics Subject Classification Primary 11F85 · 42C40 · 47G30 · Secondary 26A33 · 46F10
1 Introduction 1.1 p-Adic and Adelic Analysis According to the well-known Ostrovsky theorem, any nontrivial valuation on the field of rational numbers Q is equivalent either to the real valuation | · | or to one of the p-adic valuations | · |p . p-Adic norm | · |p is defined as follows: |0|p = 0; if an arbitrary rational number x = 0 is represented as x = p γ m n , where γ = γ (x) ∈ Z and the integers m, n are
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