Fourier interpolation and time-frequency localization

We prove that under very mild conditions for any interpolation formula $f(x) = \sum_{λ\in Λ} f(λ)a_λ(x) + \sum_{μ\in M} \hat{f}(μ)b_μ(x)$ we have a lower bound for the counting functions $n_Λ(R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log^{2+\varepsilon}(4R_1R_2)$ which very closely matches interpolation formulas discovered by Radchenko and Viazovska and by Bondarenko, Radchenko and Seip.

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