Fourier Interpolation and Time-Frequency Localization

We prove that under very mild conditions for any interpolation formula f(x)=∑λ∈Λf(λ)aλ(x)+∑μ∈Mf^(μ)bμ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$\end{document} we have a lower bound for the counting functions nΛ(R1)+nM(R2)≥4R1R2-Clog2(4R1R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$\end{document} which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.