An elementary proof of the Trombi theorem for the Fourier transform of \({\mathcal C}^ p(G:F)\)
type: Item
instance of: scholarly article
title: An elementary proof of the Trombi theorem for the Fourier transform of \({\mathcal C}^ p(G:F)\)
zbMATH Open document ID: 0666.43003
author: Masaaki Eguchi
author: Masato Wakayama
published in: Hiroshima Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let G be a real, connected, semisimple Lie group with finite center. Let \(G=KAN\) be an Iwasawa decomposition of G, \(g=k\oplus a\oplus n\), the corresponding Lie algebra decomposition. For any \(F\subset \hat K\), \(| F| <\infty\), and \(0<p\leq 2\) let \({\mathcal C}^ p(G:F)\) be the \(L^ p\)-Schwartz space of type F. The image of \({\mathcal C}^ p(G:F)\) by the Fourier transform has been characterized by \textit{P. Trombi} [J. Funct. Anal. 40, 84-125 (1981; Zbl 0465.22006)]. To prove the continuity of the inverse Fourier transform (which is the delicate part in the proof) Trombi uses the asymptotic behavior of the Eisenstein integrals at infinity [cf. \textit{P. Trombi}, ibid. 30, 83-105 (1978; Zbl 0392.43011)]. The purpose of the present paper is to give an alternative and simpler proof of the continuity of the inverse Fourier transform, by using the Gangolli estimates for the coefficients of the Harish-Chandra expansion of the Eisenstein integrals obtained by \textit{M. Eguchi}, \textit{M. Hashizume} and \textit{S. Koizumi} [Hiroshima Math. J. 17, 457-469 (1987; Zbl 0655.22007)].
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 4090121
zbMATH Keywords: real, connected, semisimple Lie group
zbMATH Keywords: Iwasawa decomposition
zbMATH Keywords: Schwartz space
zbMATH Keywords: inverse Fourier transform
zbMATH Keywords: Eisenstein integrals
zbMATH Keywords: Harish-Chandra expansion
reviewed by: R. J. Miatello
MaRDI profile type: Publication
Recommended article: Q3517447
Recommended article: Q4526043
Recommended article: Q3800516
Recommended article: Q5129703
Recommended article: An elementary proof of the Paley–Wiener theorem in Cm
Recommended article: An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups
Recommended article: A characterization theorem for the Fourier transform on \(R^ n\)
Recommended article: Q4359078
Recommended article: An analogue of the Titchmarsh theorem for the Fourier transform on the group of \(p\)-adic numbers
Recommended article: Generalization of Titchmarsh's theorem for the Fourier transform in the space \(\mathrm {L}^{2}(\mathbb {R}^{n})\)
description: scientific article; zbMATH DE number 4090121