The Beurling and Malliavin Theorem in Several Dimensions

TL;DR

This paper extends the classical Beurling and Malliavin Theorem to multiple dimensions, providing new sufficient conditions for functions to serve as majorants in Fourier analysis and answering a question posed by H"ormander.

Contribution

It introduces a novel multidimensional generalization of the theorem with a simple method, including a sharp nonradial case, and addresses an open question in the field.

Findings

Established a sufficient condition for radial functions to be Beurling and Malliavin majorants in multiple dimensions.

Derived a new sharp condition for nonradial functions, extending the radial result.

Provided different proofs for odd and even dimensions, utilizing classical Bessel function formulas.

Abstract

The present paper is devoted to a new multidimensional generalization of the Beurling and Malliavin Theorem, which is a classical result in the Uncertainty Principle in Fourier Analysis. In more detail, we establish by an elegant but simple new method a sufficient condition for a radial function to be a Beurling and Malliavin majorant in several dimensions (this means that the function in question can be minorized by the modulus of a square integrable function which is not zero identically and which has the support of the Fourier transform included in an arbitrary small ball). As a corollary of the radial case, we also get a new sharp sufficient condition in the nonradial case. The latter result provides an answer to a question posed by L. H\"ormander. Our proof is different in the cases of odd and even dimensions. In the even dimensional case we make use of one classical formula from the theory of Bessel functions due to N. Ya. Sonin.