On Cardinal Interpolation by Gaussian Radial-basis Functions: Properties of Fundamental Functions and Estimates for Lebesgue Constants

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 (k) = 0k , k 2 Z. One objective of this paper is to derive several additional properties of L. For example, it is shown that L possesses the sign-regularity property sgnnL (x)] = sgnnsin(x)=(x)], x 2 R, and that jL (x)j 2e 8 minf(bjxjc + 1) ?1 ; exp(?bjxjc)g, x 2 R. The analysis is based on a simple representation formula for L , and employs some methods from classical function theory. A second consideration in the paper is the Gaussian cardinal-interpolation operator L , deened by the equation (L y)(x) := P k2Z y k L (x ? k), x 2 R, y = (y k) k2Z. On account of the exponential decay of the cardinal function L , L is a well-deened linear map from`1 (Z) into L 1 (R). Its associated operator-norm kL k is called the Lebesgue constant of L. The latter half of the paper establishes the following asymptotic estimates for the Lebesgue constant: kL k 1, ! 1, and kL k log(1==), ! 0 +. Suitable multidimensional analogues of these results are also given.