# Wavelet analysis as a p-adic spectral analysis

@article{Kozyrev2008WaveletAA, title={Wavelet analysis as a p-adic spectral analysis}, author={Sergei Kozyrev}, journal={arXiv: Mathematical Physics}, year={2008} }

New orthonormal basis of eigenfunctions for the Vladimirov operator of p–adic fractional derivation is constructed. The map of p–adic numbers onto real numbers (p–adic change of variable) is considered. p–Adic change of variable maps the Haar measure on p–adic numbers onto the Lebesgue measure on the positive semiline. p–Adic change of variable (for p = 2) provides an equivalence between the constructed basis of eigenfunctions of the Vladimirov operator and the wavelet basis in L 2 (R…

## 143 Citations

### p-Adic wavelets and their applications

- Mathematics
- 2014

The theory of p-adic wavelets is presented. One-dimensional and multidimensional wavelet bases and their relation to the spectral theory of pseudodifferential operators are discussed. For the first…

### An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators

- Mathematics
- 2009

AbstractIn this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in $$
\mathcal{L}^2 (\mathbb{Q}_p^n )
$$ is constructed. We also study the connections between…

### Multidimensional p-adic wavelets for the deformed metric

- Mathematics
- 2010

The approach to p-adic wavelet theory from the point of view of representation theory is discussed. p-Adic wavelet frames can be constructed as orbits of some p-adic groups of transformations. These…

### Ju l 2 01 8 Enhanced Symmetry of the p-adic Wavelets

- Mathematics
- 2018

Wavelet analysis has been extended to the p-adic line Qp. The p-adic wavelets are complex valued functions with compact support. As in the case of real wavelets, the construction of the basis…

### p-Adic Multiresolution Analysis and Wavelet Frames

- Mathematics
- 2008

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…

### Multidimensional basis of p-adic wavelets and representation theory

- Mathematics, Computer Science
- 2009

It is shown that the set of products of the vectors from the constructed basis and p-roots of unity is the orbit of the corresponding p-adic group of linear transformations.

### $p$-Adic multidimensional wavelets and their application to $p$-adic pseudo-differential operators

- Mathematics
- 2006

In this paper we study some problems related with the theory of multidimensional p-adic wavelets in connection with the theory of mul- tidimensional p-adic pseudo-differential operators (in the…

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

- Mathematics
- 2006

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…

### p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators

- Mathematics
- 2007

The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function of the unit disc.…

## References

SHOWING 1-10 OF 20 REFERENCES

### The wavelet transform, time-frequency localization and signal analysis

- PhysicsIEEE Trans. Inf. Theory
- 1990

Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied and the notion of time-frequency localization is made precise, within this framework, by two localization theorems.

### p-Adic Valued Distributions in Mathematical Physics

- Mathematics
- 1994

Introduction. I. First Steps to Non-Archimedean Fields. II. The Gauss, Lebesgue and Feynman Distributions over Non-Archimedean Fields. III. The Gauss and Feynman Distributions on Infinite-Dimensional…

### THE WAVE FUNCTION OF THE UNIVERSE AND p-ADIC GRAVITY

- Mathematics, Physics
- 1991

A new approach to the wave function of the universe is suggested. The key idea is to take into account fluctuating number fields and present the wave function in the form of a Euler product. For this…

### Generalized functions over the field of p-adic numbers

- Mathematics, Psychology
- 1988

CONTENTSIntroduction § 1. p-adic numbers § 2. Characters of the field Qp § 3. Invariant measure on the field Qp § 4. Test functions and generalized functions § 5. Homogeneous generalized functions §…

### P-adic analysis and mathematical physics

- Mathematics, Physics
- 1994

P-adic numbers p-adic analysis non-Archimedean geometry distribution theory pseudo differential operators and spectral theory p-adic quantum mechanics and representation theory quantum field theory…

### Application of p-adic analysis to models of breaking of replica symmetry

- Materials Science
- 1999

Methods of p-adic analysis are applied to the investigation of spontaneous symmetry breaking in the models of spin glasses. A p-adic expression for the Parisi replica matrix is given and, moreover,…

### Application of p-adic analysis to models of spontaneous breaking of the replica symmetry

- Physics
- 2008

Application of p-adic analysis to models of spontaneous breaking of the replica symmetry Abstract Methods of p-adic analysis are applied to the investigation of the spontaneous symmetry breaking in…

### Replica Fourier Transforms on Ultrametric Trees, and Block-Diagonalizing Multi-Replica Matrices

- Computer Science
- 1997

The analysis of objects living on ultrametric trees, in particular the block-diagonalization of 4-replica matrices, is shown to be dramatically simplified through the introduction of properly chosen operations on those objects.

### Orthonormal bases of compactly supported wavelets

- Computer Science
- 1988

This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.

### On the Replica Fourier Transform

- Mathematics
- 1997

The Replica Fourier Transform introduced previously is related to the standard definition of Fourier transforms over a group. Its use is illustrated by block-diagonalizing the eigenvalue equation of…