Introduction

In this chapter, we extend the treatment we gave to power series in the preceding chapter to Fourier series and their generalizations, whether convergent or divergent. In particular, we are concerned with Fourier cosine and sine series, orthogonal polynomial expansions, series that arise from Sturm—Liouville problems, such as Fourier—Bessel series, and other general special function series.

Several convergence acceleration methods have been used on such series, with limited success. An immediate problem many of these methods face is that they do not produce any acceleration when applied to Fourier and generalized Fourier series. The transformations of Euler and of Shanks discussed in the following chapters and the d-transformation are exceptions. See the review paper by Smith and Ford [318] and the paper by Levin and Sidi [165]. With those methods that do produce acceleration, another problem one faces in working with such series is the lack of stability and acceleration near points of singularity of the functions that serve as limits or antilimits of these series. Recall that the same problem occurs in dealing with power series.

In this chapter, we show how the d-transformation can be used effectively to accelerate the convergence of these series. The approach we are about to propose has two main ingredients that can be applied also with some of the other sequence transformations.