Nonuniform Sampling and Recovery of Multidimensional Bandlimited Functions by Gaussian Radial-Basis Functions

摘要

Let S ? Rd be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PWS, is defined to be the set of all square-integrable functions on Rd whose Fourier transforms vanish outside S. A sequence (xj:j∈N) in Rd is said to be a Riesz-basis sequence for L2(S) (equivalently, a complete interpolating sequence for PWS) if the sequence (e-i〈 xj, ·〉:j ∈ N) of exponential functions forms a Riesz basis for L2(S). Let (xj:j∈N) be a Riesz-basis sequence for L2(S). Given λ>0 and f∈PWS, there is a unique sequence (aj) in ?2 such that the function is continuous and square integrable on Rd, and satisfies the condition Iλ(f)(xn)=f(xn) for every n∈N. This paper studies the convergence of the interpolant Iλ(f) as λ tends to zero, i. e., as the variance of the underlying Gaussian tends to infinity. The following result is obtained: Let Suppose that δB2?Z?B2, and let (xj:j∈N) be a Riesz basis sequence for L2(Z). If f PW β B2, then f =limλ→0+ Iλ (f) in L2(Rd) and uniformly on Rd. If δ=1, then one may take β to be 1 as well, and this reduces to a known theorem in the univariate case. However, if d≥2, it is not known whether L2(B2) admits a Riesz-basis sequence. On the other hand, in the case when δ < 1, there do exist bodies Z satisfying the hypotheses of the theorem (in any space dimension).