Nonuniform Sampling and Recovery of Multidimensional Bandlimited Functions by Gaussian Radial-Basis Functions
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Let S ⊂ R d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW S , is defined to be the set of all square-integrable functions on R d whose Fourier transforms vanish outside S. A sequence (x j :j∈N) in R d is said to be a Riesz-basis sequence for L 2 (S) (equivalently, a complete interpolating sequence for PW S ) if the sequence (e -i〈 xj, ·〉 :j ∈ N) of exponential functions forms a Riesz basis for L 2 (S). Let (x j :j∈N) be a Riesz-basis sequence for L 2 (S). Given λ>0 and f∈PW S , there is a unique sequence (a j ) in ℓ 2 such that the function is continuous and square integrable on R d , and satisfies the condition I λ (f)(x n )=f(x n ) for every n∈N. This paper studies the convergence of the interpolant I λ (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 δB 2 ⊂Z⊂B 2 , and let (x j :j∈N) be a Riesz basis sequence for L 2 (Z). If f PW β B 2 , then f =limλ→0 + Iλ (f) in L 2 (R d ) and uniformly on R d . 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 L 2 (B 2 ) 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). © 2010 Springer Science+Business Media, LLC.
author list (cited authors)
Bailey, B. A., Schlumprecht, T., & Sivakumar, N.