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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
Abstract

We prove necessary density conditions for sampling in spectral subspaces of a second-order uniformly elliptic differential operator on 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 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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