Arkiv för Matematik
- Ark. Mat.
- Volume 34, Number 2 (1996), 199-224.
Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk
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).
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.
Ark. Mat., Volume 34, Number 2 (1996), 199-224.
Received: 23 August 1995
First available in Project Euclid: 31 January 2017
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1996 © Institut Mittag-Leffler
Natan, Yaakov Ben; Benyamini, Yoav; Hendenmalm, H»kan; Weit, Yitzhak. Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk. Ark. Mat. 34 (1996), no. 2, 199--224. doi:10.1007/BF02559544. https://projecteuclid.org/euclid.afm/1485898511