Reproducing kernel Hilbert spaces of CR functions for the Euclidean Motion group

TL;DR

This paper investigates the structure of a specific reproducing kernel Hilbert space linked to the wavelet transform on the Euclidean Motion group, revealing its complex regularity and connections to the Bargmann transform.

Contribution

It characterizes the RKHS associated with wavelet transforms on SE(2) using CR structures, extending understanding beyond square-integrable representations.

Findings

The RKHS can be described via CR regularity.

A natural Hilbert norm makes the wavelet transform an isometry.

Relations with the Bargmann transform are established.

Abstract

We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the Euclidean Motion $SE(2)$. A natural Hilbert norm for functions on the group is constructed that makes the wavelet transform an isometry, but since the considered representations are not square integrable the resulting Hilbert space will not coincide with $L^2(SE(2))$. The reproducing kernel Hilbert subspace generated by the wavelet transform, for the case of a minimal uncertainty mother wavelet, can be characterized in terms of the complex regularity defined by the natural $CR$ structure of the group. Relations with the Bargmann transform are presented.