Differentiation and Shift Property Characterisations of the Group Fourier Transform on the Heisenberg Group
Year: 2026
Type: preprint
Abstract: Recently, Krukowski characterised the Euclidean Fourier transform in terms of its behaviour with respect to differentiation. For locally compact abelian groups, he obtained a characterisation based on the behaviour of the Fourier transform under translation. Each of these results assumed that the operator (to be characterised) is an integral transform (with a kernel satisfying suitable regularity conditions), paired with a normalisation. In this paper, we obtain analogous results for the group Fourier transform on the Heisenberg group. Here, the differentiation property is formulated in terms of the behaviour of the group Fourier transform with respect to the left-invariant vector fields, and the shift property is on its behaviour under right translation. Both of these characterise the group Fourier transform on the Heisenberg group, once paired with a normalisation. Our results assume that the transform is continuous linear, and do not require it to be an integral transform a priori.
Source: arXiv (Cornell University)
Author: Lavanya, R. Lakshmi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224928116