An analogue of the Wiener-Tauberian theorem for spherical transforms on semisimple Lie groups

Let G be a semi-simple connected noncompact Lie group with finite center and K a fixed maximal compact subgroup of G. Fix a Haar measure dx on G and let I1(G) denote those functions in L1(G,dx) which are biinvariant under K. The purpose of this paper is to prove that if f ∈ I1(G) is such that its spherical Fourier transform (i.e., Gelfand transform) f is nowhere vanishing on the maximal ideal space of I1(G) and f “does not vanish too fast at ∞”, then the ideal generated by f is dense in I1(G). This generalizes earlier results of Ehrenpreis-Mautner for G = SL(2,R) and R. Krier for G of real rank one

On an analogue of the Wiener Tauberian theorem for symmetric spaces of the noncompact type

Let X be a symmetric space and G the connected component of the group of isometries of X. If ƒ ∈ L1(X), we consider conditions under which Sp{gƒ:g ∈ G} is dense in L1(X) in terms of the "Fourier transform" of ƒ. This continues earlier work on this kind of problem by L. Ehrenpreis and F. I. Mautner, R. Krier and the author

Spherical mean periodic functions on semisimple Lie groups

Let G be a connected semisimple noncompact Lie group with finite center. We define the notion of a smooth spherical mean periodic function (with respect to a fixed maximal compact subgroup K of G) and show that the classical results of L. Schwartz for mean periodic functions on the real line hold in this context

Some uncertainty principles in abstract harmonic analysis

The first part of this article is an introduction to uncertainty principles in Fourier analysis, while the second summarizes some recent work by the authors and also by Michael Cowling and the authors.The following (rather vague) principle is well known to every student of classical Fourier analysis: If a function f is 'concentrated' then its Fourier transform f is 'spread out' and vice-versa. After reviewing three precise (and different) formulations of this principle in classical Fourier analysis on Rn, we will describe how it extends to LCA groups and certain nonabelian Lie groups - for instance, semisimple Lie groups and Heisenberg groups

A qualitative uncertainty principle for semisimple Lie groups

Recently M. Benedicks showed that if a function f∈L2 (Rd) and its Fourier transform both have supports of finite measure, then f=0 almot everywhere. In this paper we give a version of this result for all noncompact semisimple connected Lie groups with finite centres

A theorem of Cramér and Wold revisited

Let H={( x,y):x>0)⊆R2 and let E be a Borel subset of H of positive Lebesgue measure We prove that if μ and ν are two probability measures on R2 such that μ(σ(E))=ν{σ(E)) for all rigid motions σ of R2. then μ=ν This generalizes a well-known theorem of Cramér and Wold

A note on L²(Γ∖SL(2, R))

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On the isomorphism of discrete subgroups of SL(2, R)

The purpose of this paper is to give a sufficient condition for the isomorphism of two discrete subgroups Γ and Γ' of SL(2, R) in terms of the norm of the primitive hyperbolic element of Γ and Γ'. The proof exploits some well-known properties of the Selberg zeta function [3]

Fourier analysis and determining sets for Radon measures on R<SUP>n</SUP>

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