$L^2-$interpolation with error and size of spectra

TL;DR

This paper establishes a relationship between approximate interpolation of delta functions on a discrete set and spectral measure estimates, linking the size of the spectrum to the density of the set.

Contribution

It introduces a new connection between $L^2$-interpolation error, spectral size, and the density of discrete sets, advancing understanding in spectral analysis.

Findings

Spectral measure estimates depend on the density of the discrete set.

Approximate interpolation imposes constraints on the size of the spectrum.

The results connect spectral properties with geometric distribution of points.

Abstract

Given a compact set $S$ and a uniformly discrete sequence $\La$, we show that "approximate interpolation" of delta--functions on $\La$ by a bounded sequence of $L^2-$functions with spectra in $S$ implies an estimate on measure of $S$ through the density of $\La$.