A Frequency Space for the Heisenberg Group

TL;DR

This paper introduces a novel Fourier transform framework for the Heisenberg group using Hermite functions, enabling explicit asymptotic analysis and extension of classical Fourier results to this non-commutative setting.

Contribution

It defines a new Fourier transform on the Heisenberg group as a uniformly continuous map, extending classical Fourier analysis techniques to this non-commutative context.

Findings

Extended Fourier transform for smooth, vertical-variable independent functions.

Explicit asymptotic description as vertical frequency approaches zero.

Potential adaptation of classical Euclidean Fourier results to the Heisenberg group.

Abstract

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by taking advantage of basic properties of Hermite functions) the Fourier transform f\_H of f to be a uniformly continuous mapping on the set N^d x N^d xR \ {0} endowed with a suitable distance. This enables us to extend f\_H to the completion of that space, and to get an explicit asymptotic description of the Fourier transform when the 'vertical' frequency tends to 0. We expect our approach to be relevant for adapting to the Heisenberg framework a number of classical results for the Euclidean case that are based on Fourier analysis. As an example, we here establish an explicit extension of the Fourier transform for smooth functions on H^d that are independent of the vertical variable.