NECESSARY DENSITY CONDITIONS FOR SAMPLING AND INTERPOLATION IN SPECTRAL SUBSPACES OF ELLIPTIC DIFFERENTIAL OPERATORS

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.