Gaussian radial‐basis functions: Cardinal interpolation of ℓ^p and power‐growth data

Abstract

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function $L_\lambda (x) = \sum\nolimits_{k \in \mathbb{Z}} {c_k \exp ( - \lambda (x - k)^2 ),x \in \mathbb{R}} ,$ satisfying the interpolatory conditions $L_\lambda (j) = \delta _{0j} ,j \in \mathbb{Z}.$ The paper considers the Gaussian cardinal interpolation operator

$$(\mathcal{L}_\lambda {\text{y}})(x): = \sum\limits_{k \in \mathbb{Z}} {y_k L_\lambda (x - k),{\text{ y}} = (y_k )_{k \in \mathbb{Z}} ,{\text{ }}x \in \mathbb{R}} ,$$

as a linear mapping from ℓ^p(ℤ) into L ^p(ℝ), 1≤ p ∞, and in particular, its behaviour as λ→0⁺. It is shown that $\left\| {\mathcal{L}_\lambda } \right\|_p$ is uniformly bounded (in λ) for 1 < p < ∞, and that $\left\| {\mathcal{L}_\lambda } \right\|_1 \asymp \log (1/\lambda )$ as λ→0⁺. The limiting behaviour is seen to be that of the classical Whittaker operator

$$\mathcal{W}:{\text{y}} \mapsto \sum\limits_{k \in \mathbb{Z}} {y_k \frac{{\sin \pi (x - k)}}{{\pi (x - k)}}} ,$$

in that $\lim _{\lambda \to 0^ + } \left\| {\mathcal{L}_\lambda {\text{y}} - \mathcal{W}{\text{y}}} \right\|_p = 0,$ for every ${\text{y}} \in \ell ^p (\mathbb{Z}){\text{ and }}1 < p < \infty .$ It is further shown that the Gaussian cardinal interpolants to a function f which is the Fourier transform of a tempered distribution supported in (-π,π) converge locally uniformly to f as λ→0⁺. Multidimensional extensions of these results are also discussed.