Necessary Density Conditions for Sampling and Interpolation in Spectral Subspaces of Elliptic Differential Operators

TL;DR

This paper establishes necessary density conditions for sampling in spectral subspaces of elliptic differential operators, extending classical bandlimited function results to more general operators and variable bandwidth scenarios.

Contribution

It introduces a critical sampling density for spectral subspaces of elliptic operators, generalizing Landau's conditions to variable bandwidth functions in higher dimensions.

Findings

Identifies a critical density for sampling in spectral subspaces.

Extends Landau's density conditions to elliptic differential operators.

Provides a new density criterion for variable bandwidth functions in 1D.

Abstract

We prove necessary density conditions for sampling in spectral subspaces of a second order uniformly elliptic differential operator on $R^d$ with slowly oscillating symbol. For constant coefficient operators, these are precisely Landaus necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists. Our results prove the existence of a suitable critical sampling density and compute it in terms of the geometry defined by the elliptic operator. In dimension 1, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and we obtain a new critical density for variable bandwidth. The methods are a combination of the spectral theory and the regularity theory of elliptic partial differential operators, some elements of limit operators, certain compactifications of $R^d $, and the theory of reproducing kernel Hilbert spaces.