Abstract
Let S⊂ℝd 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 ℝd whose Fourier transforms vanish outside S. A sequence (x j :j∈ℕ) in ℝd is said to be a Riesz-basis sequence for L 2(S) (equivalently, a complete interpolating sequence for PW S ) if the sequence \((e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N})\) 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 ℝd, 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 \(\delta\in(\sqrt{2/3},1]\) and \(0<\beta<\sqrt{3\delta^{2}-2}\). Suppose that δ B 2⊂Z⊂B 2, and let (x j :j∈ℕ) be a Riesz basis sequence for L 2(Z). If \(f\in PW_{\beta B_{2}}\), then \(f=\lim_{\lambda\to 0^{+}}I_{\lambda}(f)\) in L 2(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 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).
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Communicated by K. Gröchenig.
The research of the second author was supported by the National Science Foundation.
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Bailey, B.A., Schlumprecht, T. & Sivakumar, N. Nonuniform Sampling and Recovery of Multidimensional Bandlimited Functions by Gaussian Radial-Basis Functions. J Fourier Anal Appl 17, 519–533 (2011). https://doi.org/10.1007/s00041-010-9141-6
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DOI: https://doi.org/10.1007/s00041-010-9141-6