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

Karlheinz Gröchenig, Andreas Klotz

TL;DR

The 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 d = 1, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and a new critical density for variable bandwidth is obtained.

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.

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

TL;DR

The 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 d = 1, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and a new critical density for variable bandwidth is obtained.

Abstract

We prove necessary density conditions for sampling in spectral subspaces of a second order uniformly elliptic differential operator on 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 , and the theory of reproducing kernel Hilbert spaces.

Paper Structure

This paper contains 19 sections, 30 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The generalized Paley-Wiener Space and its basic properties
  5. Sampling and interpolation in $PW_{\Omega}(H_{a})$ and the Beurling Densities.
  6. Classes of symbols, limit operators
  7. Compact orbits
  8. Slowly oscillating symbols.
  9. Statement of the Density Theorem
  10. Proof of Weak Localization of the kernel
  11. Proof of the Homogeneous Approximation Property (HAP)
  12. Geometric Beurling Densities
  13. The basic computation: constant coefficients
  14. The Higson compactification
  15. Geometric densities for slowly oscillating symbols.
  16. ...and 4 more sections

Key Result

Theorem 1

(i) Assume that $S$ is uniformly separated and that then (ii) If for all $a\in\ell^{2}(S)$ there exists $f\in PW_{\Omega}$, such that $f(s)=a_{s},s\in S$, then

Theorems & Definitions (62)

  • Theorem : A
  • Theorem : B
  • Theorem : C
  • Corollary : D
  • Corollary : E
  • Remark
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 52 more