Strong annihilating pairs for the Fourier-Bessel transform

TL;DR

This paper establishes new uncertainty principles for the Fourier-Bessel transform, showing limitations on simultaneous support and thinness of a function and its transform, and extends related classical results.

Contribution

It introduces two novel uncertainty principles for the Fourier-Bessel transform, extending classical results and proving linear independence of dilations of certain functions.

Findings

Functions and their Fourier-Bessel transforms cannot both have finite measure support.

Supports of functions and their transforms cannot both be $( ext{eps}, ext{alpha})$-thin.

Dilation of $ ext{C}_0$-functions are linearly independent.

Abstract

The aim of this paper is to prove two new uncertainty principles for the Fourier-Bessel transform (or Hankel transform). The first of these results is an extension of a result of Amrein-Berthier-Benedicks, it states that a non zero function $f$ and its Fourier-Bessel transform $\mathcal{F}_\alpha (f)$ cannot both have support of finite measure. The second result states that the supports of $f$ and $\mathcal{F}_\alpha (f)$ cannot both be $(\eps,\alpha)$-thin, this extending a result of Shubin-Vakilian-Wolff. As a side result we prove that the dilation of a $\cc_0$-function are linearly independent. We also extend Faris's local uncertainty principle to the Fourier-Bessel transform.