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

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

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

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

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

Department of Mathematics

Research output: Contribution to journal › Comment/Debate

Abstract

Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in L¹(G//K), the space of radial integrable functions on G = SU (1, 1), so that I = L¹(G//K) or (Equation presented)—the ideal of L¹(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 Agranovskii).

Original language	English
Pages (from-to)	43-49
Number of pages	7
Journal	Bulletin of the American Mathematical Society
Volume	32
Issue number	1
DOIs	https://doi.org/10.1090/S0273-0979-1995-00554-9
State	Published - Jan 1995

Keywords

Resolvent transform
SU(1, 1)
Spectral synthesis
Spherical functions
Two circles theorems
Wiener’s theorem
μ-harmonic functions

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1090/S0273-0979-1995-00554-9

Cite this

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title = "Wiener{\textquoteright}s tauberian theorem in L1(G//k) and harmonic functions in the unit disk",

abstract = "Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in L1(G//K), the space of radial integrable functions on G = SU (1, 1), so that I = L1(G//K) or (Equation presented)—the ideal of L1(G//K) functions whose integral is zero. This is then used to prove a generalization of Furstenberg{\textquoteright}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).",

keywords = "Resolvent transform, SU(1, 1), Spectral synthesis, Spherical functions, Two circles theorems, Wiener{\textquoteright}s theorem, μ-harmonic functions",

author = "Natan, {Y. Ben} and Y. Benyamini and H. Hedenmalm and Y. Weit",

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T1 - Wiener’s tauberian theorem in L1(G//k) and harmonic functions in the unit disk

AU - Natan, Y. Ben

AU - Benyamini, Y.

AU - Hedenmalm, H.

AU - Weit, Y.

PY - 1995/1

Y1 - 1995/1

N2 - Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in L1(G//K), the space of radial integrable functions on G = SU (1, 1), so that I = L1(G//K) or (Equation presented)—the ideal of L1(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 Agranovskii).

AB - Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in L1(G//K), the space of radial integrable functions on G = SU (1, 1), so that I = L1(G//K) or (Equation presented)—the ideal of L1(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 Agranovskii).

KW - Resolvent transform

KW - SU(1, 1)

KW - Spectral synthesis

KW - Spherical functions

KW - Two circles theorems

KW - Wiener’s theorem

KW - μ-harmonic functions

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U2 - 10.1090/S0273-0979-1995-00554-9

DO - 10.1090/S0273-0979-1995-00554-9

M3 - Comment/Debate

AN - SCOPUS:0040681366

SN - 0273-0979

VL - 32

SP - 43

EP - 49

JO - Bulletin of the American Mathematical Society

JF - Bulletin of the American Mathematical Society

IS - 1

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