Mexican Hat Wavelet on the Heisenberg Group

TL;DR

This paper investigates wavelets on the Heisenberg group, establishing equivalence of admissibility conditions for Schwartz functions and constructing a Mexican-Hat wavelet analogue with specific decay and vanishing moments.

Contribution

It introduces a new admissibility criterion for wavelets on the Heisenberg group and constructs a Mexican-Hat wavelet analogue with Gaussian decay and vanishing moments.

Findings

Equivalence of Calderon and usual admissibility for Schwartz functions

Construction of a Mexican-Hat wavelet on the Heisenberg group

Wavelet with Gaussian decay and 2 vanishing moments

Abstract

In this article wavelets (admissible vectors) on the Heisenberg group are studied from the point of view of Calderon's formula. Further we shall show that for the class of Schwartz functions the Calderon admissibility condition is equivalent to the usual admissibility property which will be introduced in this work. Furthermore motivated by a well-known example on the real line, the Mexican-Hat wavelet, we demonstrate the existence and construction of an analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which together with all of its derivatives has Gaussian decay.