Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk
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.
scientific article; zbMATH DE number 944330
scientific article