Wiener’s Tauberian theorem in 𝐿₁(𝐺//𝐾) and harmonic functions in the unit disk

Ben Natan, Y.; Benyamini, Y.; Hedenmalm, H.; Weit, Y.

doi:10.1090/S0273-0979-1995-00554-9

Wiener’s Tauberian theorem in $L_1(G//K)$ and harmonic functions in the unit disk
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by Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit PDF

Bull. Amer. Math. Soc. 32 (1995), 43-49 Request permission

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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