Abstract
Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral 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 Agranovskii).
Funding Statement
The second author's work was partially supported by the fund for the promotion of research at the Technion-Israel Institute of Technology. The third author's work was partially supported by the Swedish Natural Science Research Council, and by the 1992 Wallenberg Prize from the Swedish Mathematical Society.
Citation
Yaakov Ben Natan. Yoav Benyamini. H»kan Hendenmalm. Yitzhak Weit. "Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk." Ark. Mat. 34 (2) 199 - 224, October 1996. https://doi.org/10.1007/BF02559544
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