Abstract
We consider Hilbert spaces of functions or distributions on Rd constructed by taking the closure of the space of test functions supported on a fixed bounded open set U with respect to a weighted L2-norm for their Fourier transform defined by a weight w which is both moderate and tempered. The evaluation at a point λ € Rd of the Fourier transform of any element in one of these spaces is then given by the inner product with an element uλ in the same space. Given a discrete set Λ C Rd , we consider the collection {√w(λ) uλ}λϵΛ and ask whether it could form a frame (leading to stable sampling) or a Riesz sequence (leading to interpolation) for the given space. We show, that under certain conditions, a stable sampling result (resp. interpolation result) in the unweighted case (where the weight is identically 1) implies a similar result for the weighted case and vice-versa. In particular, this allows us to formulate a generalization of the classical L2-results of Landau about stable sampling and interpolation in the weighted setting.
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Gabardo, JP. Sampling and Interpolation in Weighted L2-Spaces of Band-Limited Functions. STSIP 17, 197–224 (2018). https://doi.org/10.1007/BF03549664
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DOI: https://doi.org/10.1007/BF03549664