Quantitative uncertainty principles for the Gabor spherical mean transform
For \(f\in L^2(d\nu)\) and \(h\in L^2(d\nu)\setminus\{0\}\), the Gabor spherical mean operator transform \(\mathscr{G}_h\) is defined by \N\[\N\mathscr{G}_h(f)((r,x),(\xi_0,\xi))=\int_{0}^{\infty} \!\!\! \int_{\mathbb{R}}{f(s,y)\ \overline{h_{(r,x),(\xi_0,\xi)}(s,y)}}\, d\nu(s,y),\N\]\Nwhere \((r,x),(\xi_0,\xi)\in [0,\infty)\times \mathbb{R}^n\), \(h_{(r,x),(\xi_0,\xi)}=\tau_{(r,x)}(h_{(\xi_0,\xi)})\), \(\tau(r,x)\) is the translation operator associated with the spherical mean operator defined by \N\[\N\tau_{(r,x)}(f)(s,y)=\dfrac{\Gamma\left(\dfrac{n+1}{2}\right)}{\sqrt{\pi}\ \Gamma\left(\dfrac{n}{2}\right)}\int_{0}^{\pi}{f(\sqrt{r^2+s^2+2rs\cos{\theta}},x+y)\ \sin^{n-1}{(\theta)}}\ d\theta\N\]\Nand \(h_{(\xi_0,\xi)}\) is the modulation of \(h\) defined by \N\[\Nh_{(\xi_0,\xi)}=\widetilde{\mathscr{F}}\left(\sqrt{\tau_{(\xi_0,\xi)}(|\widetilde{\mathscr{F}}(h)|^2)}\right).\N\]\NThe authors continue the study of harmonic analysis for the Gabor spherical mean operator transform. A new inversion formula associated with this transform is established. They also prove some quantitative uncertainty principles including the Donoho-Stark uncertainty principle, a generalized entropy uncertainty principle, a generalized Heisenberg uncertainty principle, the \(L^p\)-generalized Heisenberg uncertainty principle and the \(L^p\)-local uncertainty principle for the Gabor spherical mean operator transform.
scientific article; zbMATH DE number 7901559
scientific article