Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

TL;DR

This paper introduces ultrametric wavelet bases and pseudodifferential operators on ultrametric spaces, demonstrating their diagonalization and eigenvalues, and develops a change of variable to construct generalized wavelet bases.

Contribution

It presents a new family of orthonormal wavelet bases for ultrametric spaces and analyzes pseudodifferential operators acting on them, including eigenvalue computation and a change of variable for generalization.

Findings

Ultrametric wavelet bases form orthonormal systems in ultrametric function spaces.

Pseudodifferential operators are diagonal in these wavelet bases with explicitly computed eigenvalues.

A change of variable maps ultrametric spaces to the positive half-line, enabling generalized wavelet constructions.

Abstract

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases.