p-Adic Multiresolution Analysis and Wavelet Frames
TL;DR: In this article, a complete characterization of test functions generating an MRA (scaling functions) is given, and it is shown that only 1-periodic test functions may be taken as orthogonal scaling functions and all such scaling functions generate the Haar MRA.
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Abstract: We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.
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Citations
p-Adic mathematical physics : the first 30 years
Branko Dragovich,Branko Dragovich,A. Yu. Khrennikov,Sergei Vladimirovich Kozyrev,Igor Volovich,E. I. Zelenov +5 more
TL;DR: P-Adic Mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena as discussed by the authors, which emerged as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology.
135
$p$-Adic Mathematical Physics: The First 30 Years
Branko Dragovich,Branko Dragovich,A. Yu. Khrennikov,Sergei Vladimirovich Kozyrev,Igor Volovich,E. I. Zelenov +5 more
TL;DR: A brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade, mainly paid to developments with promising future prospects.
85
•Book
Ultrametric Pseudodifferential Equations and Applications
Andrei Khrennikov,Sergei Vladimirovich Kozyrev,W. A. Zúñiga-Galindo +2 more
- 25 May 2018
TL;DR: In this article, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and padic dynamical systems.
68
Step Refinable Functions and Orthogonal MRA on Vilenkin Groups
TL;DR: In this article, necessary and sufficient conditions on refinable step functions under which this function generates an orthogonal MRA in the L 2 -spaces on Vilenkin group were given.
44
Wavelet packets and wavelet frame packets on local fields of positive characteristic
Biswaranjan Behera,Qaiser Jahan +1 more
TL;DR: In this article, the concepts of multiresolution analysis (MRA) and wavelets can be generalized to a local field K of positive characteristic, and the wavelet packets associated with such MRAs are constructed.
41
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