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 $$n_\Lambda(R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log^{2+\varepsilon}(4R_1R_2)$$ which very closely matches interpolation formulas discovered by Radchenko and Viazovska and by Bondarenko, Radchenko and Seip.