Ultrametric pseudodifferential operators and wavelets for the case of non homogeneous measure

TL;DR

This paper introduces orthonormal bases of ultrametric wavelets for spaces with arbitrary measures, analyzes pseudodifferential operators on these spaces, and explores their diagonalization and eigenvalues, with a novel construction of ultrametric spaces via directed trees.

Contribution

The paper develops a new framework for ultrametric wavelets in spaces with arbitrary measures and characterizes pseudodifferential operators as diagonal in these bases, including a novel ultrametric space construction.

Findings

Ultrametric wavelet bases are orthonormal in spaces with arbitrary measures.

Pseudodifferential operators are diagonalized in these wavelet bases.

Eigenvalues of the operators are explicitly computed.

Abstract

A family of orthonormal bases of ultrametric wavelets in the space of quadratically integrable with respect to arbitrary measure functions on general (up to some topological restrictions) ultrametric space is introduced. Pseudodifferential operators (PDO) on the ultrametric space are investigated. We prove that these operators are diagonal in the introduced bases of ultrametric wavelets and compute the corresponding eigenvalues. Duality between ultrametric spaces and directed trees is discussed. In particular, a new way of construction of ultrametric spaces by completion of directed trees is proposed.