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

On an infinite grid with uniform spacing h, the cardinal basis C-j(x; h) for many spectral methods consists of translates of a "master cardinal function", C-j(x; h) = C(x/h - j). The cardinal basis satisfies the usual Lagrange cardinal condition, C-j(mh) = delta(jm) where delta(jm) is the Kronecker delta function. All such "shift-invariant subspace" master cardinal functions are of "localized-sinc" form in the sense that C(X) equivalent to 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) equivalent to sin (pi X)/(pi X). The localized-sine approximation to a general f(x) is f(localized-sinc) (x; h) equivalent to Sigma(infinity)(j=-infinity) 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-sine 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 sine bases except that the localizer acts like a diffusion in Fourier space, smoothing the sine error. (C) 2015 Elsevier Inc. All rights reserved.