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 :jN) 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 :jN) be a Riesz-basis sequence for L 2 (S). Given >0 and fPW S , there is a unique sequence (a j ) in 2 such that the function is continuous and square integrable on R d , and satisfies the condition I (f)(x n )=f(x n ) for every nN. This paper studies the convergence of the interpolant I (f) as tends to zero, i. e., as the variance of the underlying Gaussian tends to infinity. The following result is obtained: Let Suppose that B 2 ZB 2 , and let (x j :jN) be a Riesz basis sequence for L 2 (Z). If f PW B 2 , then f =lim0 + I (f) in L 2 (R d ) and uniformly on R d . If =1, then one may take to be 1 as well, and this reduces to a known theorem in the univariate case. However, if d2, it is not known whether L 2 (B 2 ) admits a Riesz-basis sequence. On the other hand, in the case when < 1, there do exist bodies Z satisfying the hypotheses of the theorem (in any space dimension). 2010 Springer Science+Business Media, LLC.