Fourier Interpolation and Time-Frequency Localization

Abstract We prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + _{\mu M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ? ? ? ? a + ? M ^ b we have a lower bound the counting functions $$n_\Lambda (R_1) n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n R 1 2 ? 4 - C log which closely matches recent formulas of Radchenko and Viazovska Bondarenko, Seip.