Wavelets on ultrametric spaces

Khrennikov, Andrei; Козырев, С. В.

doi:10.1016/j.acha.2005.02.001

Cited by 78 publications

(26 citation statements)

References 22 publications

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“…To the ultrametric spaces, (H, d H ) and (G, d), there corresponds, respectively, a tree T (H) with finite ramification index, and a tree T (G) whose endspace is identified with G. Consequently, L 2 (G) is a separable Hilbert space (see [CE1], [KK1] and [KK2]). In addition, the topology of the locally constant functions of compact support D(Q S ) is expressed by the inductive limits and D(Q S ) is a locally convex complete topological algebra and a nuclear space.…”

Section: Final Remarksmentioning

confidence: 99%

A heat equation on some adic completions of ℚ and ultrametric analysis

Aguilar-Arteaga

Cruz-López

Estala-Arias

2017

P-Adic Num Ultrametr Anal Appl

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The product ring Q S = p∈S Q p , where S is a finite set of prime numbers and Q p is the field of p-adic numbers, is a second countable, locally compact and totally disconnected topological ring. This work introduces a natural ultrametric on Q S that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2 (Q S ). The fundamental solution of this equation is a normal transition function of a Markov process on Q S . As a result, the techniques developed here provides a general framework of these problems on other related ultrametric groups.

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Section: Final Remarksmentioning

confidence: 99%

A heat equation on some adic completions of ℚ and ultrametric analysis

Aguilar-Arteaga

Cruz-López

Estala-Arias

2017

P-Adic Num Ultrametr Anal Appl

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“…Further development and generalization of the theory of such type wavelets can be found in the articles by S. V. Kozyrev [36,37], A. Yu. Khrennikov, and S. V. Kozyrev [28,29], J. J. Benedetto, and R. L. Benedetto [13], and R. L. Benedetto [14].…”

Section: Contents Of the Articlementioning

confidence: 99%

Harmonic Analysis in the p-Adic Lizorkin Spaces: Fractional Operators, Pseudo-Differential Equations, p-Adic Wavelets, Tauberian Theorems

Albeverio

Khrennikov

Shelkovich

2006

J Fourier Anal Appl

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In this article the p-adic Lizorkin spaces of test functions and distributions are introduced. Multi-dimensional Vladimirov's and Taibleson's fractional operators, and a class of p-adic pseudo-differential operators are studied on these spaces. Since the p-adic Lizorkin spaces are invariant under these operators, they can play a key role in considerations related to fractional operator problems. Solutions of pseudo-differential equations are also constructed. Some problems of spectral analysis of pseudo-differential operators are studied. p-Adic multidimensional Tauberian theorems connected with these pseudo-differential operators for the Lizorkin distributions are proved.It is well known that apart from the "usual" mathematical physics ("C-case," where all functions and distributions are complex or real valued defined on spaces with real or complex Math Subject Classifications. Primary 11F85, 47G30, 40E05; secondary 26A33, 46F12.

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“…[4,9,10,22,27,35,36]); to work on wavelets on locally compact fields [1,2,5,8,13,[17][18][19] and to work on quantum mechanics on p-adic numbers [6, 12, 14-16, 23, 26, 32, 33]. Most of this work studies Fourier transforms on Q p which is relevant to the Heisenberg-Weyl group HW[Q p , Q p , Q p ].…”

Section: Introductionmentioning

confidence: 99%