Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators

Gröchenig, Karlheinz; Klotz, Andreas

doi:10.2140/apde.2024.17.587

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Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators

Karlheinz Gröchenig and Andreas Klotz

Vol. 17 (2024), No. 2, 587–616

DOI: 10.2140/apde.2024.17.587

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 Landau’s 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 $d = 1$ , 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.

Keywords

spectral subspace, Paley–Wiener space, bandwidth, Beurling density, sampling, interpolation, elliptic operator, regularity theory, slow oscillation, Higson compactification

Mathematical Subject Classification

Primary: 35J99, 46E22, 47B32, 54D35, 94A20

Secondary: 42C40

Milestones

Received: 20 September 2021

Revised: 7 June 2022

Accepted: 11 July 2022

Published: 6 March 2024

Authors

Karlheinz Gröchenig
Faculty of Mathematics University of Vienna Vienna Austria

Andreas Klotz
Faculty of Mathematics University of Vienna Vienna Austria

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Vol. 17, No. 2, 2024