Variations on a theorem of Beurling

Details Variations on a theorem of Beurling Variations on a theorem of Beurling

2012-03-06

arxiv.org/abs/1203.1098v1

Mathematics

Classical Analysis and ODEs

21 pages

Scientific paper

We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $ 0 < a < 1.$ We show that such functions are entire vectors for the Schr\"{o}dinger 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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