Variations on a theorem of Beurling

Abstract

We consider functions satisfying the subcritical Beurling's condition, viz., $${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}|f(x)||\hat{f}(y)|e^{a|x\cdot y|}dxdy<\infty$$ 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 $a$.

21 pages

References in corpus (3)

• On the Hermite expansions of functions from Hardy class
• Hermite expansions and Hardy's theorem
• On the structure of analytic vectors for the schrodinger representation