Covariant Transform

TL;DR

The paper introduces the covariant transform, extending wavelet theory to broader contexts like Hardy spaces and Banach spaces, enabling new applications in functional calculus and group representations.

Contribution

It develops a generalized covariant transform framework that applies wavelet-like constructions to non-square integrable groups and non-admissible vectors, broadening wavelet theory's scope.

Findings

Extends wavelet applicability to Hardy space H_2 and Banach spaces.

Connects covariant transform with various functional calculi.

Provides a unified approach to wavelets, coherent states, and group representations.

Abstract

The paper develops theory of covariant transform, which is inspired by the wavelet construction. It was observed that many interesting types of wavelets (or coherent states) arise from group representations which are not square integrable or vacuum vectors which are not admissible. Covariant transform extends an applicability of the popular wavelets construction to classic examples like the Hardy space H_2, Banach spaces, covariant functional calculus and many others. Keywords: Wavelets, coherent states, group representations, Hardy space, Littlewood-Paley operator, functional calculus, Berezin calculus, Radon transform, Moebius map, maximal function, affine group, special linear group, numerical range, characteristic function, functional model.