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Wiener's Tauberian theorem in $ L_1(G//K)$ and harmonic functions in the unit disk

Authors: Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit
Journal: Bull. Amer. Math. Soc. 32 (1995), 43-49
MSC: Primary 43A80; Secondary 46H99
MathSciNet review: 1273399
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Abstract: Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in $ {L^1}(G//K)$, the space of radial integrable functions on $ G = SU(1,1)$, so that $ I = {L^1}(G//K)$ or $ I = L_0^1(G//K)$--the ideal of $ {L^1}(G//K)$ functions whose integral is zero. This is then used to prove a generalization of Furstenberg's theorem which characterizes harmonic functions on the unit disk by a mean value property and a "two circles" Morera type theorem (earlier announced by Agranovskiĭ).

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Keywords: Spherical functions, $ SU(1,1)$, Resolvent transform, Spectral synthesis, Wiener's theorem, $ \mu $-harmonic functions, two circles theorems
Article copyright: © Copyright 1995 American Mathematical Society