Abstract
We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.
Highlights
Introduction and ResultsIn the problem of sampling with derivatives, one tries to recover or approximate a function by sampling a number of its derivatives
We study the problem of sampling with multiplicities in the shift-invariant space
See [3] for one of the first nonuniform sampling theorems in shift-invariant spaces, [21] for nonuniform sampling with derivatives for bandlimited functions, and [2,5,23] for more recent examples of sufficient conditions for Hermite sampling in terms of the covering density
Summary
In the problem of sampling with derivatives, one tries to recover or approximate a function by sampling a number of its derivatives. See [3] for one of the first nonuniform sampling theorems in shift-invariant spaces, [21] for nonuniform sampling with derivatives for bandlimited functions, and [2,5,23] for more recent examples of sufficient conditions for Hermite sampling in terms of the covering density. We relate the zero sets of functions in different shift-invariant spaces to each other. In this way we develop a direct line of arguments and avoid the detour in [15] via the characterization of Gaussian Gabor frames. As these are essentially known, we explain only the necessary modifications
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