Critical densities in spectral subspaces

01.12.2021 10:45 - 12:15

Andreas Klotz (University of Vienna)

We discuss 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 Landaus necessary density conditions for band-limited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists.

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, certain compactifications of R^d , and the theory of reproducing kernel Hilbert spaces.

This is joint work with Karlheinz Gröchenig.

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K. Gröchenig, J. L. Romero, M. Ehler