Some results for the windowed Fourier transform related to the spherical mean operator (Q2023896)

Time frequency analysis plays an important role in harmonic analysis, in particular, in signal theory. In this context and motivated by quantum mechanics, the physicist Dennis Gabor introduced the Gabor transform, in which he uses translation, convolution, and modulation operators of a single Gaussian to represent a one-dimensional signal. \textit{H. Shapiro} [``Uncertainty principles for bases in \(l^2 \mathbb{R} \)'', in: Proceedings of the Conference on Harmonic Analysis and Number Theory. CIRM. Marseille-Luminy. 16--21 (2005)] has studied the localization for an orthonormal sequence. In the paper under review, the windowed Fourier transform associated with the spherical mean operator is defined and studied. The boundedness and compactness of localization operators of this windowed one are proved. Next, some uncertainty principles for the Fourier and the windowed Fourier transforms associated with the spherical mean operator are established. More precisely, Shapiro-type uncertainty inequalities are given.