On Shifted Cardinal Interpolation by Gaussians and Multiquadrics Original Research Article

A radial basis function approximation is a linear combination of translates of a fixed functionϕ: Rd→R. Such functions possess many useful and interesting properties when the translates are integers andϕis radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussianϕ=G(·−α), whereG(x)=exp(−λ‖x‖22) andα∈Rd. Specifically, we exploit the theory of elliptic functions to establish the invertibility of the Toeplitz operator[formula]whenαhas no half-integer components; it is singular otherwise. This implies the existence of ashifted Gaussian cardinal function, that is, a linear combinationχof integer translates of the shifted Gaussian satisfyingχ(j)=δ0j. We also study shifted cardinal functions when the parameterλtends to zero. In particular, we discover their uniform convergence to the sinc function when the shift vectorαpossesses no half-integer components. Our methods are based in part on similar results established by the first author when the basis function is the Hardy multiquadric. Several intriguing links with the theory of shifted B-spline cardinal interpolation are described in the finale.