Nonuniform sampling and recovery of multidimensional bandlimited functions by Gaussian radial-basis functions

Abstract

Let $S\subset {\mathbb{R}}^d$ be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to $S$, $P{W}_S$, is defined to be the set of all square-integrable functions on ${\mathbb{R}}^d$ whose Fourier transforms vanish outside $S$. A sequence $(x_j:j\kin\N)$ in ${\mathbb{R}}^d$ is said to be a Riesz-basis sequence for $L_2(S)$ (equivalently, a complete interpolating sequence for $P{W}_S$) if the sequence $(e^{-i\la x_j,\cdot\ra}:j\kin\N)$ of exponential functions forms a Riesz basis for $L_2(S)$. Let $(x_j:j\kin\N)$ be a Riesz-basis sequence for $L_2(S)$. Given $\lambda >0$ and $f\in P{W}_S$, there is a unique sequence $(a_j)$ in ${\ell }_2$ such that the function $$ I_λ(f)(x):=\sum_{j\in\N}a_je^{-λ\|x-x_j\|_2^2}, \qquad x\kin\R^d, $$ is continuous and square integrable on ${\mathbb{R}}^d$, and satisfies the condition $I_{\lambda }(f)(x_n)=f(x_n)$ for every $n\kin\N$. This paper studies the convergence of the interpolant $I_{\lambda }(f)$ as $\lambda$ tends to zero, {\it 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 $\delta B_2\subset Z\subset B_2$, and let $(x_j:j\in \mathbb{N})$ be a Riesz basis sequence for $L_2(Z)$. If $f\in P{W}_{\beta B_2}$, then $f={\lim }_{\lambda \to 0^{+}}{I}_{\lambda }(f)$ in $L_2({\mathbb{R}}^d)$ and uniformly on ${\mathbb{R}}^d$. If $\delta =1$, then one may take $\beta$ to be 1 as well, and this reduces to a known theorem in the univariate case. However, if $d\ge 2$, it is not known whether $L_2(B_2)$ admits a Riesz-basis sequence. On the other hand, in the case when $\delta <1$, there do exist bodies $Z$ satisfying the hypotheses of the theorem (in any space dimension).