Gaussian radial-basis functions: Cardinal interpolation of l(p) and power-growth data
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Suppose is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function L(x) = k ck exp(-(x -k)2), x , satisfying the interpolatory conditions L(j) = 0j, j . The paper considers the Gaussian cardinal interpolation operator (y)(x) := ykL(x - k), y = (yk)k, x , k as a linear mapping from p() into LP(), 1 p < , and in particular, its behaviour as 0+. It is shown that ||||p is uniformly bounded (in ) for 1 < p < , and that ||||1 (Equivalent to) log(1/) as 0+. The limiting behaviour is seen to be that of the classical Whittaker operator W: y kyk sin (x - k)/(x - k), in that lim0+ ||y - Wy||p = 0, for every y P() and 1 < p < . 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.