On Shifted Cardinal Interpolation by Gaussians and Multiquadrics
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A radial basis function approximation is a linear combination of translates of a fixed function φ: script capital R signd → script capital R sign. Such functions possess many useful and interesting properties when the translates are integers and φ is radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussian φ = G(· - α) where G(x) = exp( - λ ∥x∥22) and α ∈ script capital R signd. Specifically, we exploit the theory of elliptic functions to establish the invertibility of the Toeplitz operator (φ(α + j - k)) j, k ∈ script capital F sign d when α has no half-integer components; it is singular otherwise. This implies the existence of a shifted Gaussian cardinal function, that is, a linear combination χ of integer translates of the shifted Gaussian satisfying χ(j) = δ0j. We also study shifted cardinal functions when the parameter λ tends to zero. In particular, we discover their uniform convergence to the sinc function when the shift vector α possesses 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. © 1996 Academic Press, Inc.
author list (cited authors)
Baxter, B., & Sivakumar, N.