ON SHIFTED CARDINAL INTERPOLATION BY GAUSSIANS AND MULTIQUADRICS

A radial basis function approximation is a linear combination of trans lates of a fixed function phi: R(d) --> R. Such functions possess many useful and interesting properties when the translates are integers an d phi is radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussian phi = G(.-alpha), whe re G(x) = exp(-lambda parallel to x parallel to(2)(2)) theory of ellip tic functions to establish the invertibility of the Toeplitz operator (phi(alpha + j - k))(j, k is an element of Ld) when alpha has no half- integer components; it is singular otherwise. This implies the existen ce of a shifted Gaussian cardinal function, that is, a linear combinat ion chi of integer translates of the shifted Gaussian satisfying chi(j ) = delta(0j). We also study shifted cardinal functions when the param eter lambda tends to zero. In particular, we discover their uniform co nvergence to the sine function when the shift vector alpha possesses n o half-integer components. Our methods are based in part on similar re sults established by the first author when the basis function is the H ardy multiquadric. Several intriguing links with the theory of shifted B-spline cardinal interpolation are described in the finale. (C) 1996 Academic Press, Inc.