A theorem of the Wiener-Tauberian type for \(L^ 1(H^ n)\)
type: Item
instance of: scholarly article
title: A theorem of the Wiener-Tauberian type for \(L^ 1(H^ n)\)
zbMATH Open document ID: 0883.43006
DOI: 10.1007/BF02837694
author: Rama Rawat
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1997-08-05T00:00:00Z
review text: Let \(H^n= C^n \times R\) denote the \(n\)-dimensional Heisenberg group and \(HM(n) =H^n \times U(n)\) the Heisenberg motion group, where \(U(n)\) is the compact group of \(n\times n\) unitary matrices with entries in \(C\). The group \(HM(n)\) acts on \(H^n\) in the following way: \((\sigma z,t) (w,s)= (\sigma w+z\), \(s+ t+1/2I_m \text{ w.z})\). The author proves a Wiener-Tauberian theorem for \(L^1(H^n)\) with this \(HM(n)\) action on \(H^n\), i.e., he provides conditions on the ``group theoretic'' Fourier transform of a function \(f\in L^1(H^n)\) in order that the linear span of \(\{g_f: g\in HM(n)\}\) is dense in \(L^1(H^n)\), where \(g_f(z,t) =f(g. (z,t))\) for \(g\in HM(n)\), \((z,t) \in H^n\).
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 1043987
zbMATH Keywords: Heisenberg group
zbMATH Keywords: unitary matrices
zbMATH Keywords: Wiener-Tauberian theorem
zbMATH Keywords: Fourier transform
reviewed by: Harkrishan L. Vasudeva
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02837694
OpenAlex ID: W2057438496
description: scientific article; zbMATH DE number 1043987