Conclusions and Outlook

We have shown that the complex-analytic structure that we introduced in [#!CAoRFI!#] can be used to discuss the issue of the convergence of Fourier series. Using that structure we derived the known formulas for the partial sums of a Fourier series, in terms of Dirichlet integrals. From that same structure, and in fact as part of the same argument, we also obtained a new result, namely formulas giving the remainders of a Fourier series in terms of a similar but considerably more complex type of integral, in fact a double integral.

The introduction of a modified version of the Hilbert transform, which we named the compact Hilbert transform, had a central role to play in this development. The main result that follows from it is the necessary and sufficient condition for the convergence of a Fourier series, which is expressed in Equations (97) and (98). One might consider whether or not the integration kernel involved in this new type of Dirichlet integral, versions of which are shown in Equations (95) and (98), can be cast in some more convenient form, and possibly even calculated in close form in some useful way. So far no meaningful results of this type have been found.

A direct calculation of this integration kernel in closed form seems to be difficult, and possibly not overly useful, since it seems that such calculations tend to just take us back to the rather trivial identity

$${\cal D}_{\rm r}[N,f(\theta)] = {\cal I}[f(\theta)] - {\cal D}_{\rm s}[N,f(\theta)],$$ (114)

where ${\cal I}[f(\theta)]$ is the identity operator, whose integration kernel is a Dirac delta ``function'', an identity which is simply equivalent to

$$R_{N}^{F}(\theta) = f(\theta) - S_{N}^{F}(\theta).$$ (115)

In so far as can be currently ascertained, this identity does not provide any constructively useful information about the convergence problem.

We already knew that he convergence problem of Fourier series relates to the existence and nature of singularities of the corresponding inner analytic functions on the unit circle. The convergence condition expressed in Equations (97) and (98) reflect this relation, since the existence of non-integrable singularities at the unit circle may very well disturb the validity of the condition by making the integrals diverge or at least not go to zero in the $N\to\infty$ limit. Note that this relation is non-local because, due to the fact that the integrals are over the whole unit circle, the existence of a non-integrable singularity at a single point may prevent the convergence of the series at almost all points.

It seems to us, at this time, that the most promising possible development of the convergence analysis presented here is probably one targeted at detailing the relation between the convergence of the series and the specific classification of the singularities on the unit circle, involving the concepts of hard and soft singularities, as well as the corresponding degrees of hardness or softness that can be attributed to them.