Tests of Center Series

Three different classes of real functions are represented in the tests. Two of the examples chosen are of continuous functions, namely the triangular wave and the parabolic wave, the latter being also differentiable. In these two cases the Fourier series converge quite rapidly, since the Fourier coefficients $a_{k}$, with $k=1,2,3,\ldots,\infty$, behave for large values of $k$ as $1/k^{2}$ in the first case and as $1/k^{3}$ in the second, with the consequence that the improvement obtained with the use of the center series is modest. All the remaining examples involve discontinuous functions. In the case of the two forms of sawtooth wave and the two forms of square wave the functions are discontinuous, but limited. These series converge much slower that the previous ones, having coefficients that behave as $1/k$ for large values of $k$, and in this case the use of the first-order center series produces very significant improvements in the rhythm of convergence, and sometimes very large ones.

The two remaining examples are simple series concocted to be convergent almost everywhere but to have coefficients that behave as $1/\sqrt{k}$ for large values of $k$, and which due to this converge much slower than those in the previous class of discontinuous functions. The resulting functions are not limited, being in fact logarithmically divergent at a single point. In this case the use of the center series produces truly enormous advantages, to the point where it is not even possible to measure the improvement in some cases, due to the enormous time it would take to complete the standard Fourier runs, which could run up to many years of CPU time. In this case the use of the center series, and in particular of the second-order center series, represents the qualitative difference between being able to deal with these functions with numerical ease and not being able to deal with them at all by numerical means.

The numerical tests were executed on a one-dimensional regular lattice of points defining a set of $1000$ intervals between $\theta=-\pi$ and $\theta=\pi$, these two points being identified with each other by periodic boundary conditions. The special points, which correspond to the singularities of $w(z)$ on the unit circle, and where the representation of the real functions by the center series is not defined, were excluded from the set of lattice points. At each remaining point the two types of series were added up until the absolute value of the difference between the approximate value obtained and the known exact result fell below the required threshold. Some care was taken to avoid mistaking for convergence the mere accidental passages through the limiting value during oscillations. In the case of the simple piece-wise functions used as examples the exact result is known because we have piece-wise expressions in closed form for these functions. However, in some cases, and in particular for the Fourier Conjugate functions of all the functions used as examples, no such expressions in closed form are available.

In all the cases where exact expressions in closed form for the limiting functions are not available the following strategy was used. First, the limiting function was obtained numerically to a high degree of precision at all the points of the lattice to be used in the tests. The precision target for these calculations was set at $\varepsilon=10^{-16}$, while the comparison tests were performed for target precisions ranging from $\varepsilon=10^{-3}$ to $\varepsilon=10^{-8}$. Of course the operation of adding up the series to such high level of precision typically takes a long time. In fact, in most cases it is practically impossible to do this using the Fourier series. Therefore, we used the first-order center series for some of these preliminary calculations. This worked well in the case of the better-behaved examples, namely for those involving functions which are at least continuous. In all other cases, namely those involving discontinuous functions, even the first-order center series took too long to achieve the desired high level of precision, so that in these cases we used the second-order center series, which in all cases was sufficient for our ends. The derivation of these second-order series can also be found in Appendix A.

In the tables shown in Appendix B we report the number of series terms which were added in each case, for each series and for each level of precision to be achieved. We report two results in each case, the average number of terms added, considering all the valid lattice points in the interval $[-\pi ,\pi ]$, and the maximum number of terms added at any single point. This usually corresponds to the points right next to a special point, since these in turn correspond to singularities of $w(z)$ on the complex plane. The ratios reported express how much more efficient the first-order center series is, as compared to the original DP Fourier series.

For each example worked out we give explicitly, in the next few sub-sections, the DP Fourier series and the corresponding first-order center series, for both the original function and the corresponding FC function. When it is the case, the second-order center series is also given. The special points are listed explicitly, as well as the value of the original DP function at those points. In the graphs shown in this paper both the original DP function and the FC function are plotted using the results from calculations with first-order center series added up to precision $\varepsilon=10^{-6}$, on the same graph, in order to illustrate the general behavior of the functions.

Figure 1: One-cycle sawtooth wave: the original function (solid line) and its conjugate function (dashed line) plotted within the periodic interval $[-\pi ,\pi ]$. The dotted lines mark the special point.

The source code for all the programs used for the numerical calculations in this paper is freely available online on the web [3]. The compilation structure which is included with them is meant to work on Linux systems, and all the data reported was produced on a Debian-Linux distribution version $7.6$ running on $2.4$ GHz AMD64 hardware with $48$ GB of RAM and two CPUs. These were Intel quad-core Xeon CPUs with hyper-threading capabilities, but no parallelization of any kind was included in the programs. Therefore, all processing times reported are single-CPU, single-core times. Very little was done by hand in the way of optimization, which was mostly done automatically by the compiler. However, we do report the time of each run, in order to give some idea of the absolute efficiency which may be accomplished in practice with each type of series.