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

2013/03/05 by Francesca Astengo, Astengo, Francesca, Bianca Di Blasio +3
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #advanced mathematical theories #math.FA

paper · pdf · doi:10.48550/arxiv.1303.0997

arxiv created 2013/03/05 · openalex publication_date 2013/03/05 · arxiv updated 2013/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove several Paley--Wiener-type theorems related to the spherical transform on the Gelfand pair (H_n\rtimes U(n),U(n)), where H_n is the 2n+1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in \mathbb R², we prove that spherical transforms of U(n)--invariant functions and distributions with compact support in H_n admit unique entire extensions to \mathbb C², 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.

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