Conclusions

The sometimes complicated questions of convergence of Fourier series can be mapped onto the convergence of Taylor series of analytic functions. The modes of convergence of DP Fourier series can be classified according to the singularity structure of the corresponding inner analytic functions. The extreme cases of strong convergence and strong divergence are easily identified and classified, as was shown in the previous paper [1]. Simple tests can be used to identify these cases.

Weakly convergent DP Fourier series present a much more delicate and difficult problem. It can be shown that all these cases are translated, in the complex formalism, into the behavior of inner analytic functions and their Taylor series at the rim of the maximum disk of convergence of these series, which in this case is the unit circle. The treatment of this case required the introduction of a classification scheme for singularities of inner analytic functions. These were classified as either soft or hard, depending on the behavior of the inner analytic functions near them, and subsequently by integer indices giving the degrees of either softness or hardness of the singularities. Of particular importance are the degrees which we named as borderline soft and borderline hard.

This classification scheme led to the concept of integral-differential chains of inner analytic functions, which were needed in order to relate the classification of singularities with a corresponding classification of modes of convergence of the series associated to the functions. Definite results were obtained only for a certain subset of all possible such chains, consisting of regular chains in which the series $S_{v,h0}$ is an extended monotonic series. This defines a certain class of series and corresponding functions. With the limitation that we must stay within this class, it was shown that the existence and level of convergence of DP Fourier series is ruled by the nature of the dominant singularities of the inner analytic functions which are located at the rim of their maximum disk of convergence, which is the unit circle.

As a result of this classification, by constructing the inner analytic function of a given DP Fourier series in this class, one can determine the convergence of the series via the examination of the singularities of that function. At a very basic level, one can determine whether the series is strongly divergent or strongly convergent by simply determining the position of possible singularities of the inner analytic function. This leads to the three-pronged basic decision process: if there is a singularity of any type within the open unit disk, then the $S_{v}$ series is divergent everywhere; if there are no singularities within the closed unit disk, then the $S_{v}$ series converges everywhere to a $C^{\infty}$ function; if there are one or more singularities on the unit circle, but none within the open unit disk, then the convergence of the $S_{v}$ series is determined by the degree of hardness or of softness of the dominant singularities on the unit circle.

The last alternative leads to another three-pronged decision process, this time based on the type of the dominant singularities of the inner analytic function found on the unit circle. According to the underlying structure that was uncovered, at this finer level the decision structure leads to the following basic alternatives: if the dominant singularities are soft singularities, then the $S_{v}$ series converges absolutely and uniformly everywhere; if they are borderline hard singularities, then the $S_{v}$ series converges point-wise almost everywhere, but does not converge absolutely; if they are hard singularities with degree of hardness $1$ or greater, then the series $S_{v}$ diverges everywhere.

This classification scheme for all the modes of convergence, and of the corresponding degrees of softness or hardness of the dominant singularities, is illustrated in Table 1, which gives also some additional information. By and large, as the singularities become softer the series become more convergent over the unit circle, and converge to smoother functions. Given the convergence mode of the $S_{v}$ series, one can then derive the corresponding mode for the DP Fourier series, which are also included in the table.

Table 1: A table showing the proposed classification of modes of convergence, singularity structure and limiting function properties, within the class of regular integral-differential chains in which the series $S_{v,h0}$ is an extended monotonic series. The abbreviation ``ew'' stands for ``everywhere'' and ``aew'' for ``almost everywhere''. The abbreviations ``cont'' and ``diff'' stand respectively for ``continuous'' and ``differentiable''. Integration goes downward through the lines, differentiation goes upward.

Dominant	Convergence	Behavior of	Convergence	Character
Singularities	of $S_{v}$	Coefficients	of $S_{\rm c}$ and $S_{\rm s}$	of $f(\theta)$
$n$-hard	divergent	$\vert a_{k}\vert\propto k^{p}$	divergent	currently
$n\geq 2$	ew	$n-1\leq p<n$	aew	unknown
$1$-hard	divergent	$\vert a_{k}\vert\propto k^{p}$	divergent	$\delta$-``function''
	ew	$0\leq p<1$	aew	for $p=0$
borderline	point-wise	$\vert a_{k}\vert\propto 1/k^{p}$	point-wise	cont aew
hard	aew	$0<p\leq 1$	aew	diff aew
borderline	absolute and	$\vert a_{k}\vert\propto 1/k^{p}$	absolute and	cont ew
soft	uniform ew	$1<p\leq 2$	uniform ew	diff aew
$1$-soft	absolute and	$\vert a_{k}\vert\propto 1/k^{p}$	absolute and	diff ew
	uniform ew	$2<p\leq 3$	uniform ew	$C^{2}$ aew
$n$-soft	absolute and	$\vert a_{k}\vert\propto 1/k^{p}$	absolute and	$C^{n}$ ew
$n\geq 2$	uniform ew	$n+1<p\leq n+2$	uniform ew	$C^{n+1}$ aew

In terms of the analytic character of the limiting functions, at the most basic level strongly convergent DP Fourier series converge to restrictions to the whole unit circle of complex $C^{\infty}$ analytic functions, while weakly convergent DP Fourier series converge to globally $C^{n}$ functions which are also sectionally $C^{n+1}$, where $n$ is the degree of softness of the dominant singularities on the unit circle. In the case in which the dominant singularities are borderline hard, the series converge to sectionally continuous and differentiable functions, which however are not globally continuous. In addition to this, in the case in which the dominant singularities are simple poles one may have representations of singular objects such as the Dirac delta ``function'', as was shown in the previous paper [1]. However, this last alternative has not yet been explored in much detail.

Moreover, we presented the process of singularity factorization, through which, given an arbitrary DP Fourier series, which can even be divergent almost everywhere, one can construct from it other expressions involving trigonometric series, that converge to the function that gave origin to the given DP Fourier series. This works by the construction of a new complex power series from the Taylor series $S_{z}$ of the corresponding inner analytic function. We call these new series $C_{z}$ the center series of the series $S_{z}$ that converges to the inner analytic function associated to the original DP Fourier series. If the original series was very weakly convergent, then this new series will have much better convergence characteristics. Even if the original series is divergent one can still construct expressions involving center series that converge to the original function, in a piecewise fashion. In this way, a more practical means of recovery of the original function is provided, if compared to the explicit determination of the inner analytic function $w(z)$ in closed form, in order to enable one to take its limit to the unit circle explicitly.

Several points are left open and represent interesting possibilities for further development of the subject. One was presented in the previous paper [1] and consists of the question of whether or not there are real functions which generate strongly divergent DP Fourier series. The conjecture is that there are none, in which case taking limits of inner analytic functions from within the open unit disk would be established as a process for the generation, almost everywhere on the unit circle, of all real functions from which it is possible to define the coefficients of a DP Fourier series. Another interesting question, posed in this paper, is whether or not there are $S_{v}$ series with coefficients $a_{k}$ that behave as $\vert a_{k}\vert\propto 1/k^{p}$ with $0<p\leq 1$ and that converge everywhere on the unit circle. If there are, it would be necessary to consider the extension of our classification scheme to other classes of DP Fourier series.

Since an absolutely and uniformly convergent DP Fourier series usually converges much faster than a non-absolutely and non-uniformly convergent one, doing the $S_{v}\to C_{v}$ transformation can be of enormous numerical advantage. One verifies that, the slower the convergence of $S_{v}$, caused by a value of $p$ closer to zero when $a_{k}\propto 1/k^{p}$, the more advantageous is the use of the series $C_{v}$. Near the special points the gain is more limited, but it still exists. For simple well-known series such as the square wave, with coefficients that go to zero as $1/k$, which is the fastest possible approach within the $S_{v,h0}$ class, on average over the whole domain, and for the higher levels of numerical precision required of the results, the speedup can be as high as $1000$ or more, as we will show elsewhere [3].