Spectral analysis for radial sections of some homogeneous vector bundles on certain noncompact Riemannian symmetric spaces
type: Item
instance of: scholarly article
title: Spectral analysis for radial sections of some homogeneous vector bundles on certain noncompact Riemannian symmetric spaces
zbMATH Open document ID: 1230.43008
author: Sanjoy Pusti
author: Rudra P. Sarkar
published in: Israel Journal of Mathematics
publication date: 2011-08-25T00:00:00Z
review text: Schwartz's theorem on \(\mathbb R\) states that if \(f\in C^\infty(\mathbb R)\), then the closure of the set \(\{W*f\mid W\in C^\infty(\mathbb R)^\prime\}\) contains the function \(x\mapsto e^{i\lambda x}\) for some \(\lambda\in \mathbb C\). An analogue of Schwartz's theorem for \(K\)-biinvariant functions of a rank one symmetric space was obtained by \textit{S. C. Bagchi} and \textit{A. Sitaram} [Pac. J. Math. 84, 241--250 (1979; Zbl 0442.43017)] by replacing \(x\mapsto e^{i\lambda x}\) with zonal spherical functions. In this paper S. Pusti and P. Sarkar obtain Schwartz's theorem for \(\tau\)-radial functions on \(G=\text{Spin}_0(n,1)\). By using the slice projection theorem for the Abel transform and the Paley-Wiener theorem for \(\tau\)-radial compactly supported distributions on \(G\), they reduce the proof to the Euclidean case. As applications, they show that \(\tau\)-radial \(C^\infty\) functions in Lorentz spaces on \(G\) are not mean periodic and a closed \(\tau\)-radial-translation invariant subspace of Lorentz spaces does not contain \(\tau\)-spherical functions.
reviewed by: Takeshi Kawazoe
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 5942623
zbMATH Keywords: symmetric space
zbMATH Keywords: spin group
zbMATH Keywords: Schwartz's theorem
zbMATH Keywords: spherical function
zbMATH Keywords: periodic function
zbMATH Keywords: invariant subspace
zbMATH Keywords: Lorentz space
Wikidata QID: Q115377702
MaRDI profile type: Publication
full work available at URL: s11856-011-0057-x
OpenAlex ID: W2128109872
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DOI: 10.1007/S11856-011-0057-X
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description: scientific article; zbMATH DE number 5942623