$p$-Adic multiresolution analyses

TL;DR

This paper characterizes p-adic multiresolution analyses, showing that only 1-periodic test functions can serve as orthogonal scaling functions, all generating Haar MRA, and introduces a method for constructing p-adic wavelet frames.

Contribution

It provides a complete characterization of test functions for p-adic MRAs and a new method for constructing p-adic wavelet frames, advancing the understanding of p-adic wavelet theory.

Findings

Only 1-periodic test functions can be orthogonal scaling functions.

All such scaling functions generate Haar MRA.

Any set of wavelet functions generates a p-adic wavelet frame.

Abstract

We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame.