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

Abstract

Let $S\subset\R^d$ 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 $(x_j:j\kin\N)$ 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 $(e^{-i\la x_j,\cdot\ra}:j\kin\N)$ 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 $λ>0$ and $f\in PW_S$, 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 $\R^d$, and satisfies the condition $I_λ(f)(x_n)=f(x_n)$ for every $n\kin\N$. This paper studies the convergence of the interpolant $I_λ(f)$ as $λ$ tends to zero, {\it i.e.,\} as the variance of the underlying Gaussian tends to infinity. The following result is obtained: Let $δ\in(\sqrt{2/3},1]$ and $0<β<\sqrt{3δ^2 -2}$. Suppose that $δB_2\subset Z\subset B_2$, and let $(x_j:j\in\N)$ be a Riesz basis sequence for $L_2(Z)$. If $f\in PW_{βB_2}$, then $f=\lim_{λ\to 0^+} I_λ(f)$ 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 $d\ge2$, 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).