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.