A Fourier error analysis for radial basis functions and the Discrete Singular Convolution on an infinite uniform grid, Part 1: Error theorem and diffusion in Fourier space

Abstract

On an infinite grid with uniform spacing h, the cardinal basis Cj(x; h) for many spectral methods consists of translates of a “master cardinal function”, Cj(x; h) = C(x/h − j). The cardinal basis satisfies the usual Lagrange cardinal condition, Cj(mh) = δjm where δjm is the Kronecker delta function. All such “shift-invariant subspace” master cardinal functions are of “localized-sinc” form in the sense that C(X) = sinc(X)s(X) for a localizer function s which is smooth and analytic on the entire real axis and the Whittaker cardinal function is sinc(X) ≡ sin (πX)/(πX). The localized-sinc approximation to a general f(x) is flocalized−sinc(x;h)≡∑j=−∞∞f(jh)s([x−jh]/h)sinc([x−jh]/h). In contrast to most radial basis function applications, matrix factorization is unnecessary. We prove a general theorem for the Fourier transform of the interpolation error for localized-sinc bases. For exponentially-convergent radial basis functions (RBFs) (Gaussians, inverse multiquadrics, etc.) and the basis functions of the Discrete Singular Convolution (DSC), the localizer function is known exactly or approximately. This allows us to perform additional error analysis for these bases. We show that the error is similar to that for sinc bases except that the localizer acts like a diffusion in Fourier space, smoothing the sinc error.