Reproducing inversion formulas and uncertainty principles for the windowed Fourier transform related to the spherical mean operator

Abstract

We define and study the windowed Fourier transform, called also the Gabor transform, associated with the spherical mean operator and we prove for this transform the reproducing inversion formulas. We also prove several versions of Heisenberg type inequalities, Donoho–Stark’s uncertainty principles and finally Faris–Price’s uncertainty principle for the previous transform.