Numerical Results

As preliminary information, Table 2 gives the processing times of the runs used to generate the high-precision representation of the functions for which we do not have piece-wise expressions in closed form. This was done with either the first-order or the second-order center series, depending on the case. The table makes it clear that for all the functions tested a sufficiently high-order center series can be added up to high levels of precision in short times.

Table 2: The processing time of each run used to produce the high-precision numerical representations of the functions, in seconds. The targeted precision level was $10^{-16}$. The order of the center series used in each case is recorded.

Function	Order	Run Time
Conjugate of the One-Cycle Sawtooth Wave	$2^{\rm nd}$	$35.35$
Conjugate of the Standard Square Wave	$2^{\rm nd}$	$16.78$
Conjugate of the Two-Cycle Sawtooth Wave	$2^{\rm nd}$	$16.89$
Conjugate of the Triangular Wave	$1^{\rm st}$	$12.03$
Conjugate of the Shifted Square Wave	$2^{\rm nd}$	$35.12$
Conjugate of the Parabolic Wave	$1^{\rm st}$	$0.58$
The First Slow-Converging Wave	$2^{\rm nd}$	$214.45$
The Conjugate of the Function Above	$2^{\rm nd}$	$207.91$
The Second Slow-Converging Wave	$2^{\rm nd}$	$412.91$
The Conjugate of the Function Above	$2^{\rm nd}$	$404.63$

The remaining tables can be found in the subsections that follow. They give, for each function tested, the number of added terms in each one of the two types of series, as a function of the required precision level, as well as the corresponding processing times. This is done for the average and the maximum numbers of added terms. The ratios shown represent the efficiency of the first-order center series as compared to the Fourier series. All processing times reported are for the calculation of the functions to the required precision at the complete set of valid points of the lattice within the interval $[-\pi ,\pi ]$.

The entries marked with crosses refer to the results from those runs that, as it turned out, it was not possible to execute within the limits of the available computational infrastructure. These are all runs using the Fourier series to compute the unlimited discontinuous functions. Derived numerical results that it was not possible to calculate are marked with N/A. In order to give some idea of the difficulties involved in the summation of the Fourier series, even for the relatively low precision levels involved, we may mention that the failed runs were interrupted while still incomplete after more than three months of processing time. According to the estimates that we are now able to work out, some of them would have gone up to thousands of years of processing time.