Conclusions and Outlook

We have extended the close and deep relationship established in previous papers [#!CAoRFI!#,#!CAoRFII!#], between, on the one hand, integrable real functions and singular Schwartz distributions, and, in the other hand, complex analytic functions in the unit disk centered at the origin of the complex plane, to include a large class of non-integrable real functions. This close relationship between real functions and related objects, and complex analytic functions, allows one to use the powerful and extremely well-known machinery of complex analysis to deal with the real functions and related objects in a very robust way, even if these objects are very far from being analytic. The concept of integral-differential chains of proper inner analytic functions, which we introduced in [#!CAoRFI!#], played a central role in the analysis presented.

One does not usually associate non-differentiable, discontinuous and unbounded real functions, as well as singular distributions, with single analytic functions. Therefore, it may come as a bit of a surprise that, as was established in [#!CAoRFI!#,#!CAoRFII!#], essentially all integrable real functions, as well as all all singular Schwartz distributions, are given by the real parts of certain inner analytic functions on the open unit disk, in the limit in which one approaches the unit circle. This surprise is now further compounded by the fact that inner analytic functions can represent a large class of non-integrable real functions as well.

There are still more inner analytic functions within the open unit disk than those that were examined here and in [#!CAoRFI!#,#!CAoRFII!#], in relation to integrable real functions, singular distributions and non-integrable real functions. One important limitation in the arguments presented here is that requiring that there be only a finite number of non-integrable hard singularities. It may be possible, perhaps, to lift this limitation, allowing for a denumerably infinite set of such non-integrable singularities. It is probably not possible, however, to allow for a densely distributed set of such singularities. Possibly, the limitation that the number of non-integrable hard singularities be finite may be exchanged for the limitation that the number of accumulation points of a denumerably infinite set of singular points with non-integrable hard singularities be finite. We may conjecture that the following condition might work: consider the set of all closed intervals contained in the unit circle on which the function is integrable; consider the point-set infinite union of all such intervals; if the resulting set has measure $2\pi$, then it may be possible to show that the real function can still be represented by an inner analytic function $w(z)$, and thus by a definite set of Fourier coefficients. This condition certainly holds for the cases discussed in this paper, and may even turn out to be the most general possible condition leading to the results.

One interesting aspect of the work presented here is that we obtain a definite set of Fourier coefficients associated to the non-integrable real function $f(\theta)$. It is particularly interesting to note the fact that, when we determine $\alpha_{0}$ during the construction of the corresponding inner analytic function $w(z)$, we are in effect defining, by analytic means, the average value, over the unit circle, of the non-integrable function $f(\theta)$, for which such a concept was not previously defined at all. A similar remark can be made for all the other Fourier coefficients as well. Just as was the case for integrable real functions and singular Schwartz distributions, the non-integrable real functions examined here can be said to be represented directly by their sequences of Fourier coefficients.

One way to interpret the structure presented here is that, although the non-integrable real function $f(\theta)$ is defined within separate sections, with no predetermined relations among them, it is always possible to define an inner analytic function that reproduces the non-integrable real function correctly, strictly within each one of these sections, and therefore connects them all to one another in an analytic way, just as the interior of the unit disk connects the arcs of the unit circle that correspond to the sections.

We believe that the results presented here enlarge the new perspective for the analysis of real functions which was established in [#!CAoRFI!#]. This development confirms the opinion expressed there that the use of the theory of complex analytic functions makes it a rather deep and powerful point of view. Since complex analysis and analytic functions constitute in fact such a powerful tool, with so many applications in almost all areas of mathematics and physics, it is to be hoped that further applications of the ideas explored here will in due time present themselves.