Variations on a theorem of Beurling

Rahul Garg; Sundaram Thangavelu

doi:10.1515/apam-2014-0048

Published by De Gruyter June 4, 2015

Variations on a theorem of Beurling

Rahul Garg and Sundaram Thangavelu

From the journal Advances in Pure and Applied Mathematics

https://doi.org/10.1515/apam-2014-0048

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Abstract

We consider functions satisfying the subcritical Beurling's condition, viz.,

∫ℝn∫ℝn|f(x)||f^(y)|ea|x·y|dxdy<∞

for some 0<a<1. We show that such functions are entire vectors for the Schrödinger representations of the Heisenberg group. If an eigenfunction f of the Fourier transform satisfies the above condition, we show that the Hermite coefficients of f have certain exponential decay which depends on a.

Keywords: Beurling's condition; Bargmann transform; Hermite coefficients; Heisenberg group; entire vectors

MSC: 42C15; 42B35; 42C10

We wish to thank the referee for carefully reading the manuscript and making useful suggestions which have been followed in this revised version.

Received: 2014-11-29

Accepted: 2015-5-7

Published Online: 2015-6-4

Published in Print: 2015-7-1

Variations on a theorem of Beurling

Abstract

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