Periodized radial basis functions, part I: Theory

We extend the theory of periodized RBFs. We show that the imbricate series that define the Periodic Gaussian (PGA) and Sech (PSech) basis functions are Jacobian theta functions and elliptic functions “dn”, respectively. The naive periodization fails for the Multiquadric and Inverse Multiquadric RBFs, but we are able to define periodic generalizations of these, too, by proving and exploiting a generalization of the Poisson Summation Theorem. Although applications of periodic RBFs are mostly left for another day, we do illustrate the flaws and potential by solving the Mathieu eigenproblem on both uniform and highly-adapted grids. The terms of a Fourier basis can be grouped into four classes, depending upon parity with respect to both the origin and x=π/2, and so, too, the Mathieu eigenfunctions. We show how to construct symmetrized periodic RBFs and illustrate these by solving the Mathieu problem using only the periodic RBFs of the same symmetry class as the targeted eigenfunctions. We also discuss the relationship between periodic RBFs and trigonometric polynomials with the aid of an explicit formula for the nonpolynomial part of the Periodic Inverse Quadratic (PIQ) basis functions. We prove that the rate of convergence for periodic RBFs is geometric, that is, the error can be bounded by exp⁡(−Nμ) for some positive constant μ. Lastly, we prove a new theorem that gives the periodic RBF interpolation error in Fourier coefficient space. This is applied to the “spectral-plus” question. We find that periodic RBFs are indeed sometimes orders of magnitude more accurate than trigonometric interpolation even though it has long been known that RBFs (periodic or not) reduce to the corresponding classical spectral method as the RBF shape parameter goes to 0. However, periodic RBFs are “spectral-plus” only when the shape parameter α is adaptively tuned to the particular f(x) being approximated and even then, only when f(x) satisfies a symmetry condition.