$p$-Adic Haar multiresolution analysis and pseudo-differential operators

TL;DR

This paper introduces a $p$-adic multiresolution analysis (MRA) framework, constructs Haar wavelet bases in the $p$-adic setting, and explores their eigenfunction properties for pseudo-differential operators, especially in the 2-adic case.

Contribution

It develops a $p$-adic Haar MRA, constructs multiple orthonormal wavelet bases, and establishes their eigenfunction properties for pseudo-differential operators, extending wavelet theory to $p$-adic analysis.

Findings

Constructed $p$-adic Haar orthonormal bases in ${\\cL}^2(Q_p^n)$.

Identified wavelets as eigenfunctions of pseudo-differential operators.

Demonstrated the 2-adic Haar MRA's unique periodicity property.

Abstract

The notion of {\em $p$-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. This equation reflects the fact that the characteristic function of the unit disc is a sum of $p$ characteristic functions of mutually disjoint discs of radius $p^{-1}$. This refinement equation generates a MRA. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are described. We also constructed multidimensional 2-adic Haar orthonormal bases for ${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our bases in applications.