Critical and asymmetric Fourier uniqueness pairs
Torgeir Keun Lysen
TL;DR
The paper develops a flexible, density-based framework for Fourier uniqueness that bridges the supercritical and subcritical regimes. By introducing subexponentially and widely admissible density functions $S$, it provides uniform and highly asymmetric criteria under which discrete sets $\Lambda$ and $M$ yield uniqueness for the Fourier–Sobolev space $\mathcal{H}$, and in a narrower subfamily, frames and Fourier interpolation for $H(S)$. The results connect to BR–S-type sparse interpolation and Kulikov’s density condition, using a novel fractional Poincaré–Wirtinger approach together with Beurling–Malliavin multiplier theory to rule out nonzero functions vanishing on these sets. Moreover, the paper proves explicit lower and upper frame bounds for sampling inequalities, yielding stable interpolation formulas in highly asymmetric settings and offering near-optimal asymptotics in the density–growth landscape. Overall, it extends Fourier uniqueness and interpolation beyond symmetric $p=q=2$ pairs toward near Shannon–Whittaker regimes while providing quantitative, robust criteria for both uniqueness and stability.
Abstract
Motivated by the recent work of Kulikov, Nazarov, and Sodin, we construct sufficient conditions for discrete subsets of $\mathbb{R}$, which lie between the supercritical and subcritical cases, to constitute Fourier uniqueness pairs. This family of critical uniqueness pairs includes pairs that are strongly asymmetric, stretching beyond those associated with zeros of zeta and L-functions, discovered by Bondarenko, Radchenko, and Seip, and getting arbitrarily close to the classical Shannon--Whittaker uniqueness pair. We also identify a somewhat more restrictive family of strongly asymmetric uniqueness pairs that yield frames and hence Fourier interpolation.