Strong annihilating pairs for the Fourier-Bessel transform

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 $(\text{eps},\alpha )$-thin, this extending a result of Shubin-Vakilian-Wolff. As a side result we prove that the dilation of a ${\text{cc}}_0$-function are linearly independent. We also extend Faris’s local uncertainty principle to the Fourier-Bessel transform.