Approximation of discrete functions and size of spectrum

TL;DR

This paper establishes a sharp lower bound on the measure of a spectral set based on the approximation of delta functions on a discrete set by functions in Paley--Wiener space, linking spectral size to discrete spectrum density.

Contribution

It provides a new sharp estimate relating the measure of the spectral set to the approximation error and density of the discrete set, extending understanding of spectral approximation constraints.

Findings

Derived a sharp lower bound for measure(S) based on approximation error and set density.

Extended the estimate to cases with moderate growth of approximating functions.

Identified growth restrictions for approximation norms to maintain sharp bounds.

Abstract

Let $\Lambda$ be a uniformly discrete set and $S$ be a compact set in $R$. We prove that if there exists a bounded sequence of functions in Paley--Wiener space $PW_S$, which approximates $\delta-$functions on $\Lambda$ with $l^2-$error $d$, then measure($S$)$\geq 2\pi(1 - d^2)D^+(\Lambda)$. This estimate is sharp for every $d$. Analogous estimate holds when the norms of approximating functions have a moderate growth, and we find a sharp growth restriction.