Introduction

In a previous paper [2] we introduced the concept of the center series. This constitutes an improved form of trigonometric expansion of real functions. Starting from the Fourier expansion of a given real function one may construct from it, even in cases where the Fourier series is outright divergent, other expressions that converge to the function that originated the Fourier coefficients, involving certain trigonometric series which we name ``center series'', and which have better convergence characteristics than the original Fourier series. Although in [2] we give explicit examples only of first-order center series, as commented on that paper it is also possible to construct higher-order center series, with even better convergence characteristics. In fact, in this paper we will have the opportunity to calculate and use some second-order center series.

The construction of center series from Fourier series can be understood and executed in the context of a correspondence, which was established in an earlier paper [1], between Definite Parity (DP) Fourier series and certain analytic functions $w(z)$ on the unit disk of the complex plane, which we refer to as ``inner analytic functions''. In this context the construction of center series is associated to an operation of factorization of the singularities of $w(z)$ in the complex plane, as explained in [2]. Since center series are currently rather unfamiliar objects, we present in an appendix of this paper short but complete derivations of all the center series used, in any role, for the tests that were performed.

In this paper we present several comparisons between the speed of convergence of DP Fourier series and that of the corresponding first-order center series, in order to evaluate the relative efficiency with which each series in the pair represents their common limiting function. Corresponding comparisons are made also for the corresponding Fourier Conjugate (FC) series and their limiting functions. All the concepts and results involved, as well as the underlying theory, were developed in the two aforementioned papers [1] and [2]. We direct the reader to the first one for the definition and discussion of concepts and notations, while most of the center series tested in this paper were first derived in the appendices of the second one. The short derivations of all the center series used in this paper can be found in Appendix A.

The center series evaluated here are those obtained from the corresponding DP Fourier series by a single factorization of the dominant singularities of $w(z)$ on the unit circle, which we will refer to as the first-order center series. However, in order to execute the comparative test of the first-order center series and the corresponding DP Fourier series, in some cases it was necessary to use the second-order center series, obtained by a double factorization of the dominant singularities, as tools to produce high-precision numerical representations of the limiting functions. These high-precision representations were then used as gauges for the tests with both types of series.

The efficiency measured and reported is meant in a mathematical sense rather than a technically computational sense. It is measured in terms of the number of terms of the series which must be added up to yield a certain predetermined level of precision in the results. We do, however, report the approximate processing time spent by each run of the programs used for the measurements. The technical optimization of the computer code involved was not at all an issue in the numerical tests. The optimization was left to be done automatically by the compiler, in the most usual and standard way.