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 this paper we will show that one can represent within the same structure the singular objects known as distributions, loosely in the sense of the Schwartz theory of distributions [#!DTSchwartz!#,#!DTLighthill!#], which are also known as generalized real functions. 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.

In Sections 2 and 3 we will establish the relation between the complex-analytic structure and the singular distributions. There we will show that one obtains these objects through the properties of certain limits to the unit circle, involving a particular set of inner analytic functions, which will be presented explicitly. Following what was shown in [#!CAoRFI!#] for integrable real functions, each singular distribution will be associated to a corresponding inner analytic function. In fact, we will show that the entire space of all singular Schwartz distributions defined within a compact domain is contained within this complex-analytic structure. In Section 4 we will analyze a certain collection of integrable real functions which are closely related to the singular distributions, through the concept of infinite integral-differential chains of 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.