Variations on a theorem of Beurling

TL;DR

This paper explores functions satisfying a subcritical Beurling's condition, showing they are entire vectors for Schrödinger representations and analyzing the exponential decay of their Hermite coefficients.

Contribution

It establishes a connection between Beurling's condition and entire vectors in Schrödinger representations, and characterizes Hermite coefficient decay for Fourier eigenfunctions.

Findings

Functions satisfying the subcritical Beurling's condition are entire vectors.

Eigenfunctions of the Fourier transform under this condition have exponentially decaying Hermite coefficients.

The decay rate depends on the parameter a in the Beurling's condition.

Abstract

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 $a$.