A Levin-type algorithm for accelerating the convergence of Fourier series

Abstract

A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincare-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq∈ℂ, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm.