Conclusions and Outlook

We have shown that the complex-analytic structure within the unit disk of the complex plane established in a previous paper [#!CAoRFI!#], which leads to a close and deep relationship between integrable real functions on the unit circle and inner analytic functions within the unit disk centered at the origin of the complex plane, includes the whole structure of the Fourier theory of integrable real functions. This fact leads to the definition of a very general and powerful summation rule for Fourier series, which allows one to still use and manipulate in a consistent way divergent Fourier series, even when they are explicitly and strongly divergent. The connection of the complex-analytic structure with the usual Fourier theorems was exhibited.

The Fourier theory was then extended to include all the inner analytic functions associated to singular Schwartz distributions, which were discussed in detail in another previous paper [#!CAoRFII!#], in which the discussion of the complex-analytic structure was generalized to include those singular distributions. In fact, the Fourier theory can be extended to essentially the whole space of inner analytic functions. This includes at least some non-integrable real functions, as was pointed out in [#!CAoRFII!#]. The generalization to a much wider class of non-integrable real functions will be tackled in a future paper.

As part of this process of extension, we introduced the concept of an exponentially bounded sequence of complex coefficients $c_{k}$, and proved that any such sequence is the set of Taylor coefficients of some inner analytic function. As interesting open question is whether or not the reverse of this statement is true, that is, whether or not the criterion that the sequences of complex coefficients of the power series be exponentially bounded includes all possible inner analytic functions. At this time this seems rather unlikely, and in that case the problem poses itself of what more general condition on the coefficients could cover the whole space of inner analytic functions.

We believe that the results presented here establish a new perspective for the study of the Fourier theory of real functions and related objects. It provides a simple and complete account of all the mathematical structures involved, as well as of all the main results of that theory, including in particular a simple and solid proof of the completeness of the basis. Due to this, it might also constitute a simpler and more efficient way to teach the subject.