Introduction

In a previous paper [#!CAoRFI!#] we introduced a certain complex-analytic structure within the unit disk of the complex plane, and showed that one can represent essentially all integrable real functions within that structure. In another, subsequent previous paper [#!CAoRFII!#], we showed that one can also represent within the same structure the singular objects known as distributions, loosely in the sense of the Schwartz theory of distributions [#!DTSchwartz!#], which are also known as generalized real functions. In this paper we will show that a large class of non-integrable real functions can also be represented within that same structure. All these objects will be interpreted as parts of this larger complex-analytic structure, within which they can be treated and manipulated in a robust and unified way.

The construction presented in [#!CAoRFI!#], which leads to the inclusion of all integrable real functions in the complex-analytic structure, starts from the Fourier coefficients of an arbitrarily given real function on the unit circle. Since these coefficients are defined as integrals involving the real function, there is the obvious necessity that these real functions be integrable on that circle. Therefore, that construction will not work for functions which fail to be integrable there. However, once we are in the space of inner analytic functions within the open unit disk, the concept of integral-differential chains of inner analytic functions, which was introduced in [#!CAoRFI!#], comes to our rescue. It will allow us to extend the representation within the complex-analytic structure to a large class of non-integrable real functions.

For ease of reference, we include here a one-page synopsis of the complex-analytic structure introduced in [#!CAoRFI!#]. It consists of certain elements within complex analysis [#!CVchurchill!#], as well as of their main properties.