Multi-Resolution Analysis in Arbitrary Hilbert Spaces

TL;DR

This paper explores the concept of multi-resolution analysis within general Hilbert spaces beyond traditional function spaces, examining how classical properties translate and illustrating the approach with various examples.

Contribution

It introduces a framework for multi-resolution analysis in arbitrary Hilbert spaces and discusses the adaptation of classical properties to this general setting.

Findings

Classical properties can be partially translated to general Hilbert spaces.

The paper provides multiple examples demonstrating the proposed framework.

The approach broadens the applicability of multi-resolution analysis beyond function spaces.

Abstract

We discuss the possibility of introducing a multi-resolution in a Hilbert space which is not necessarily a space of functions. We investigate which of the classical properties can be translated to this more general framework and the way in which this can be done. We comment on the procedure proposed by means of many examples.