Fourier Interpolation and Time-Frequency Localization

Abstract

We prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ∑ λ ∈ Λ f ( λ ) a λ ( x ) + ∑ μ ∈ M f ^ ( μ ) b μ ( x ) we have a lower bound for the counting functions $$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n Λ ( R 1 ) + n M ( R 2 ) ≥ 4 R 1 R 2 - C log 2 ( 4 R 1 R 2 ) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.

From sphere packing to Fourier interpolation

“…One significant line of work [3,4] connects Fourier interpolation to uniqueness theory for the Klein-Gordon equation [26][27][28]. Other noteworthy papers examine the density of possible interpolation points [1,30,31,39] and whether they can be perturbed [35], interpolation formulas using zeros of zeta and L-functions [6], and extensions to nonradial functions [1,36,37,40]. Perhaps the most surprising development so far has been a paper on sphere packing and quantum gravity [25], which shows the equivalence of linear programming bounds with the spinless modular bootstrap bound for free bosons in conformal field theory, and which furthermore shows that certain bases of special functions constructed by Mazáč and Paulos [34] for the conformal bootstrap can be transformed into Fourier interpolation bases.…”

Uniqueness of Solutions to Nonlinear Schrödinger Equations from their Zeros

“…In the celebrated work [21], Radchenko and Viazovska recently proved that, restricted to the class of Schwartz functions, (A, B) = ( √ n, √ n) n∈N is a Fourier uniqueness pair. After the Radchenko-Viazovska breakthrough, several other results came about giving necessary and sufficient conditions on the pair (A, B) so that the Fourier uniqueness property holds, for which we refer the reader to [15,23,24] and the references therein.…”