Conclusions and Outlook

We have extended the close and deep relationship established in a previous paper [#!CAoRFI!#], between integrable real functions and complex analytic functions in the unit disk centered at the origin of the complex plane, to include singular distributions. This close relationship between, on the one hand, real functions and related objects, and on the other hand, 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 that previous paper, played a central role in the analysis presented.

One does not usually associate non-differentiable, discontinuous and unbounded real functions with single analytic functions. Therefore, it may come as a bit of a surprise that, as was established in [#!CAoRFI!#], all integrable real functions are given by the real parts of certain inner analytic functions on the open unit disk when one approaches the unit circle. This surprise is now compounded by the fact that inner analytic functions can represent singular distributions as well and, in fact, can represent what may be understood as a complete set of such singular objects.

There are many more inner analytic functions within the open unit disk than those that were examined here and in [#!CAoRFI!#], in relation to integrable real functions and singular distributions. Therefore, it may be possible to further generalize the relationship between real objects on the unit circle and inner analytic functions. For example, we have observed in this paper that there are inner analytic functions whose real parts converge to non-integrable real functions on the unit circle. Simple examples are the inner analytic functions given by

$$\bar{w}_{\delta^{n\prime}}(z,z_{1}) = -\mbox{\boldmath$\imath$} w_{\delta^{n\prime}}(z,z_{1}),$$ (75)

for $n\in\{0,1,2,3,\ldots,\infty\}$, that correspond to the non-integrable real functions which are the Fourier-conjugate functions of the Dirac delta ``function'' and its derivatives. This suggests that we consider the question of how far this can be generalized, that is, of what is the largest set of non-integrable real functions that can be represented by inner analytic functions. This issue will be discussed in the fourth paper of this series.

The singular distributions are integrable real objects associated to non-integrable singularities of the corresponding inner analytic functions, a fact which is made possible by the orientation of the singularities with respect to the direction along the unit circle. This suggests that the most general definition of such singular distributions may be formulated in terms of the type and orientation of the singularities present on the unit circle. In this case one would expect that singular distributions would be associated to inner analytic functions with hard singularities that are oriented in a particular way, so that the integrals of their real parts can be defined via limits from the open unit disk to the unit circle⁵.

We believe that the results presented here establish a new perspective for the representation and manipulation of singular distributions. It might also constitute a simpler and more straightforward way to formulate and develop the whole theory of Schwartz distributions within a compact domain.