Gaussian radial-basis functions: Cardinal interpolation of lp and power-growth data

Suppose is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function L (x) = P k2Z c k exp(?(x ? k) 2), x 2 R, satisfying the interpolatory conditions L (j) = 0j , j 2 Z. The paper considers the Gaussian cardinal interpolation operator as a linear mapping from`p (Z) into L p (R), 1 p < 1, and in particular, its behaviour as ! 0 +. It is shown that kL k p is uniformly bounded (in) for 1 < p < 1, and that kL k 1 log(1==) as ! 0 +. The limiting behaviour is seen to be that of the classical Whittaker operator W : y 7 ! X k2Z y k sin (x ? k) (x ? k) , in that lim !0 + kL y?Wyk p = 0, for every y 2 ` p (Z) and 1 < p < 1. It is further shown that the Gaussian cardinal interpolants to a function f which is the Fourier transform of a tempered distribution supported in (?;) converge locally uniformly to f as ! 0 +. Multidimensional extensions of these results are also discussed.