Rational trigonometric approximations using Fourier series partial sums

Abstract

A class of approximation.s {SN,M} to a periodic function f which uses the ideas of Pad6, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation SNM is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of SN,M agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these "Fourier-Pad6" approximations converge point-wise to (f(x+) + f(x-))/2 more rapidly (in some cases by a factor of 1/k2M) than the Fourier series partial sums on which they are based.

High Order Accuracy Methods for the Simulations of Reactive Flows

“…Fade rational approximations have been considered as one of the popular computational methods of representing functions, especially rapidly converging functions since it was proposed by H. Pade in 1892 [8,49]. They are generally more efficient than polynomial approximations with a reduced number of operations for the same accuracy [23,25,26].…”

Fourier--Padé approximations and filtering for spectral simulations of an incompressible Boussinesq convection problem

“…With the advent of computer in the 1950s, Padé rational approximations have become a popular computational method for representing functions, especially rapidly converging functions. They are generally more efficient than polynomial approximations, with a reduced number of operations at the same accuracy [5,6,7,8,16].…”

“…In [8], Geer presents a method for implementing the rational trigonometric approximations for even or odd 2π-periodic piecewise smooth functions and the application to the solution of an initial boundary value problem for a simple heat equation. In his work, Fourier-Padé approximants are defined in a nonlinear way such that the relation between the coefficients of the rational approximations and the Fourier coefficients involves a necessary procedure of calculating the integration of rational functions, which makes the numerical scheme relatively complicated.…”

Padé-Legendre Interpolants for Gibbs Reconstruction

“…[5,6,8,9,19]. However, most of these previous efforts have dealt with exact expansions and simple functions.…”

“…An exception is [5] in which a rational approximation was used to post-process the pseudo-spectral Fourier solution of Burgers' equation and an incompressible Boussinesq convection flow. Furthermore, most previous work is based on Fourier-Padé methods [5,6,9]. Although [8,19] also deal with functions based on orthogonal polynomials as in this paper, they consider only simple functions and denominators of very low order.…”