Variations on a theorem of Beurling

Rahul Garg 1  and Sundaram Thangavelu 2
  • 1 Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
  • 2 Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Abstract

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

nn|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.

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