Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk
type: Item
instance of: scholarly article
title: Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk
zbMATH Open document ID: 0858.43003
DOI: 10.1007/BF02559544
author: Yaakov Ben Natan
author: Yoav Benyamini
author: Y. Weit
author: Håkan Hedenmalm
published in: Arkiv för Matematik
publication date: 1997-03-25T00:00:00Z
review text: For any noncompact connected semisimple Lie group there has been no satisfactory analog of Wiener's Tauberian theorem. Let \(G= \text{SU} (1,1)\) be the multiplicative group of all complex matrices \(\left (\begin{smallmatrix} \alpha & \beta \\ \overline \beta & \overline\alpha \end{smallmatrix} \right)\) with determinant 1 and \(K\) be the subgroup of all ``rotations'' in \(G\). Let \(L^1(G//K) = L^1(K \backslash (G/K))\) be the convolution algebra of all integrable functions on \(G\) which are invariant under rotations from the left and the right. For \(f\in L^1 (G//K)\) and a collection \({\mathfrak S}\) of \(f\), let \[ \begin{aligned} \delta_\infty (f) & = -\lim \sup e^{-\pi t} \log \widehat f(1/2+it) \;(t\to+\infty),\;\delta_\infty({\mathfrak S}) = \inf \bigl\{\delta_\infty(f) : f \in {\mathfrak S} \bigr\},\\ \delta_0(f) & = -\lim \text{sup xlog} \widehat f(x)\;(x\to 0+),\;\delta_0({\mathfrak S}) = \inf \bigl\{\delta_\infty (f):f \in{\mathfrak S} \bigr\}. \end{aligned} \] By the method of resolvent transforms the authors obtain a genuine analog of Wiener's theorem: Let \({\mathfrak S}\) be a family of functions in \(L^1(G//K)\), and let \(I({\mathfrak S})\) be the smallest closed ideal in \(L^1(G//K)\) containing \({\mathfrak S}\). Then (1) \(I({\mathfrak S})= L^1(G//K)\) holds if and only if the Fourier transforms of the elements of \({\mathfrak S}\) have no common zeros in \(\Sigma\), and \(\delta_\infty ({\mathfrak S}) = 0\); (2) \(I({\mathfrak S}) = L^1_0 (G/K)\) holds if and only if the only common zeros in \(\Sigma\) of the Fourier transforms of the elements of \({\mathfrak S}\) are 0 and 1 and \(\delta_\infty ({\mathfrak S}) = \delta_0 ({\mathfrak S})=0\), where \(\Sigma = \{s\in\mathbb{C}:\;0\leq\text{Re } s\leq 1\}\) and \(L^1_0(G//K) = \{f\in L^1(G//K) : \widehat f(0)=0\}\). -- From this theorem the authors deduce characterizations of harmonic functions and functions holomorphic in the unit disk.
reviewed by: Yu Jiarong
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 43A30
zbMATH DE Number: 944330
zbMATH Keywords: SU(1,1)
zbMATH Keywords: Wiener's Tauberian theorem
zbMATH Keywords: convolution algebra
zbMATH Keywords: Fourier transforms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 944330