We consider functions $f$ on $\mathbb{R}^n$ for which both $f$ and their Fourier transforms $\hat{f}$ are bounded by the Gaussian $e^{-\frac{a}{2}|x|^2}$ for some $0<a<1$. Using the Bargmann transform, we show that their Fourier-Hermite coefficients have exponential decay. This is an extension of the one dimensional result of M. K. Vemuri, in which sharp estimates were proved. In higher dimensions, we obtain the analogous result for functions $f$ which are $O(n)$-finite. Here by an $O(n)$-finite function we mean a function whose restriction to the unit sphere $S^{n-1}$ has only finitely many terms in its spherical harmonic expansion. Some partial results are proved for general functions. As a corollary to these results, we obtain Hardy’s uncertainty principle. An analogous problem is studied in the case of Beurling’s uncertainty principle.

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Last updated: 13 Jun 2024