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

Abstract

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 \((e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N})\) 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 l2 such that the function $$I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,$$ 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 \(\delta\in(\sqrt{2/3},1]\) and \(0<\beta<\sqrt{3\delta^{2}-2}\). Suppose that δB2⊂Z⊂B2, and let (xj:j∈ℕ) be a Riesz basis sequence for L2(Z). If \(f\in PW_{\beta B_{2}}\), then \(f=\lim_{\lambda\to 0^{+}}I_{\lambda}(f)\) 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).