Density of Sampling and Interpolation in Reproducing Kernel Hilbert Spaces

TL;DR

This paper establishes necessary density conditions for sampling and interpolation in reproducing kernel Hilbert spaces, unifying various known results across signal processing, complex analysis, and harmonic analysis.

Contribution

It introduces a general framework for density conditions in RKHS with decay and localization properties, recovering and extending classical theorems.

Findings

Identifies a critical density D for sampling and interpolation sets.

Unifies multiple classical density theorems in a single framework.

Provides new density results for coherent frames in harmonic analysis.

Abstract

We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density $D$ with the following property: a set of sampling has density $\geq D$, whereas a set of interpolation has density $\leq D$. The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landau's fundamental density result. In complex analysis we rederive a critical density for generalized Fock spaces. In harmonic analysis we obtain the first general result about the density of coherent frames.