"Critical densities in generalized Paley-Wiener spaces"Klotz, AndreasWe prove necessary density conditions for sampling in generalized Paley-Wiener spaces for 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 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 the generalized Paley-Wiener space 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. This is joint work with Karlheinz Gröchenig |
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