Sampling and Interpolation in Weighted L~2-Spaces of Band-Limited Functions

摘要

We consider Hilbert spaces of functions or distributions on R~dconstructed by taking the closure of the space of test functions supportedon a fixed bounded open set U with respect to a weighted L~2-norm for theirFourier transform defined by a weight ω which is both moderate and tempered.The evaluation at a point λ∈ R~d of the Fourier transform of anyelement in one of these spaces is then given by the inner product with anelement u_λ in the same space. Given a discrete set Λ ⊂ R~d, we consider thecollection {ω(λ)~(1/2)u_λ}λ∈Λ and ask whether it could form a frame (leading tostable sampling) or a Riesz sequence (leading to interpolation) for the givenspace. We show, that under certain conditions, a stable sampling result (resp.interpolation result) in the unweighted case (where the weight is identically1) implies a similar result for the weighted case and vice-versa. In particular,this allows us to formulate a generalization of the classical L~2-results ofLandau about stable sampling and interpolation in the weighted setting.