Paley--Wiener Theorems for the U(n)--spherical transform on the Heisenberg group

TL;DR

This paper establishes Paley--Wiener theorems for the spherical transform on the Heisenberg group with U(n) symmetry, providing unique entire extensions and real-variable characterizations for functions and distributions with compact support.

Contribution

It proves Paley--Wiener theorems for the U(n)-spherical transform on the Heisenberg group, including characterizations of transforms and their inverses in terms of entire functions and real-variable conditions.

Findings

Spherical transforms of compactly supported functions extend to entire functions on a2^2.

Characterizations of transforms involve real-variable conditions.

Inverse transforms of compactly supported functions are characterized similarly.

Abstract

We prove several Paley--Wiener-type theorems related to the spherical transform on the Gelfand pair $\big(H_n\rtimes U(n),U(n)\big)$, where $H_n$ is the $2n+1$-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in ${\mathbb R}^2$, we prove that spherical transforms of $ U(n)$--invariant functions and distributions with compact support in $H_n$ admit unique entire extensions to ${\mathbb C}^2$, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.