Fourier interpolation and time-frequency localization

TL;DR

This paper establishes a fundamental lower bound on the distribution of points for Fourier interpolation formulas, closely aligning with recent advanced constructions by Radchenko, Viazovska, Bondarenko, Radchenko, and Seip.

Contribution

It proves a universal lower bound on the counting functions for interpolation formulas, connecting general conditions to specific recent interpolation constructions.

Findings

Lower bound on counting functions matches known interpolation formulas

Connects general interpolation conditions with specific recent formulas

Provides theoretical limits for Fourier interpolation point distributions

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.