Fourier interpolation and time-frequency localization

Abstract

We prove that under very mild conditions for any interpolation formula $f(x)={\sum }_{\lambda \in \Lambda }f(\lambda )a_{\lambda }(x)+{\sum }_{\mu \in M}\hat{f}(\mu )b_{\mu }(x)$ we have a lower bound for the counting functions $n_{\Lambda }(R_1)+n_M(R_2)\ge 4R_1{R}_2-C{\log }^{2+\epsilon }(4R_1{R}_2)$ which very closely matches interpolation formulas discovered by Radchenko and Viazovska and by Bondarenko, Radchenko and Seip.

9 pages