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

中文翻译：

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傅里叶插值和时频定位

我们证明在非常温和的条件下对于任何插值公式\(f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M {\hat{f}}(\mu )b_{\mu }(x)\)我们有计数函数的下界\(n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)\)与 Radchenko 和 Viazovska 以及 Bondarenko、Radchenko 和 Seip 最近的插值公式非常匹配。