Discrete Uniqueness Sets for Functions with Spectral Gaps

TL;DR

This paper extends the concept of discrete uniqueness sets from entire functions with spectral constraints to broader function spaces like Sobolev spaces, highlighting the importance of the spectral set's structure.

Contribution

It demonstrates that Sobolev spaces also admit discrete uniqueness sets for spectral sets with periodic gaps, and explores non-uniqueness phenomena for randomly gapped spectral sets.

Findings

Sobolev spaces have discrete uniqueness sets for spectral sets with periodic gaps.

For spectral sets with random gaps, non-uniqueness allows interpolation of arbitrary discrete data.

Periodic gaps in spectral sets are crucial for the existence of uniqueness sets.

Abstract

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having "periodic gaps". The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $\Lambda$ satisfy a strong non-uniqueness property: Every discrete function $c(\lambda)\in l^2(\Lambda)$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.