Wavelets Beyond Admissibility

TL;DR

This paper explores wavelets derived from non-square-integrable group representations, extending wavelet applicability to classic spaces like Hardy space and broadening theoretical understanding.

Contribution

It introduces a framework for wavelets from non-admissible group representations, expanding the scope beyond traditional admissible wavelet constructions.

Findings

Wavelets can be constructed from non-square-integrable group representations.

Extension of wavelet theory to Hardy space and similar contexts.

Broader applicability of wavelet methods in analysis and signal processing.

Abstract

The purpose of this paper is to articulate an observation that many interesting type of wavelets (or coherent states) arise from group representations which are not square integrable or vacuum vectors which are not admissible. This extends an applicability of the popular wavelets construction to classic examples like the Hardy space. Keywords: Wavelets, coherent states, group representations, Hardy space, functional calculus, Berezin calculus, Radon transform, Moebius map, maximal function, affine group, special linear group, numerical range.