Wavelet analysis as a p-adic spectral analysis

TL;DR

This paper establishes a connection between p-adic spectral analysis and wavelet analysis by constructing an eigenfunction basis for the Vladimirov operator and relating it to Haar wavelets via a p-adic change of variables.

Contribution

It introduces a new orthonormal eigenfunction basis for the Vladimirov operator and links p-adic spectral analysis with wavelet analysis through a specific change of variables.

Findings

Eigenfunction basis for Vladimirov operator constructed

p-adic change of variables relates p-adic spectral analysis to wavelets

Wavelet analysis can be viewed as p-adic spectral analysis

Abstract

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 variables) is considered. This map (for p=2) provides an equivalence between the constructed basis of eigenfunctions of the Vladimirov operator and the wavelet basis in L^2(R) generated from the Haar wavelet. This means that the wavelet analysis can be considered as a p-adic spectral analysis.