On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian

Abstract

Let $\lambda$ be a positive number, and let $(x_j:j\in \mathbb{Z})\subset \mathbb{R}$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb{R}\ni t\mapsto e^{i{x}_{j}t}:j\in \mathbb{Z}\}$ is a Riesz basis ({\it i.e.,} unconditionalbasis) for $L_2[-\pi ,\pi ]$. Given a function $f\in L_2(\mathbb{R})$ whose Fourier transform is zero almost everywhere outside the interval $[-\pi ,\pi ]$, there is a unique square-summable sequence $(a_j:j\in \mathbb{Z})$, depending on $\lambda$ and $f$, such that the function$$I_{\lambda }(f)(x):=\sum _{j\in \mathbb{Z}}{a}_j{e}^{-\lambda (x-x_j{)}^2},x\in \mathbb{R},$$ is continuous and square integrable on $(-\infty ,\infty )$, and satisfies the interpolatory conditions $I_{\lambda }(f)(x_j)=f(x_j)$, $j\in \mathbb{Z}$. It is shown that $I_{\lambda }(f)$ converges to $f$ in $L_2(\mathbb{R})$, and also uniformly on $\mathbb{R}$, as $\lambda \to 0^{+}$. A multidimensional version of this result is also obtained. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on ${\ell }_p(\mathbb{Z})$ for every $p\in [1,\infty ]$.