Introduction

In this paper we will exhibit a mathematical structure, based on certain analytic functions within the unit circle of the complex plane, that can be used to represent and analyze a very wide class of real functions. These include analytic and non-analytic integrable real functions, as well as unbounded integrable real functions. All these objects will be interpreted as parts of a larger complex-analytic structure, within which they can be treated and manipulated in a robust and unified way.

In order to assemble the mathematical structure a set of mathematical objects must be introduced, and their properties established. This will be done in Section 2, in which all the eight necessary definitions will be given, and all the corresponding properties will be stated and proved. The objects to be defined are elements within complex analysis [#!CVchurchill!#], and include a general scheme for the classification of all possible singularities of analytic functions, as well as the concept of infinite integral-differential chains of functions.

As a first and important application of this complex-analytic structure, in Section 3 we will establish the relation between the complex-analytic structure and integrable real functions. There we will show that every integrable real function defined within a finite interval corresponds to an inner analytic function and can be obtained by means of the restriction of the real part of that analytic function to the unit circle of the complex plane.

This is the first of a series of papers. The discussion of some parts and aspects of this line of work will be postponed to forthcoming papers, in order to keep each paper within a reasonable length. In the second paper of the series we will extend the complex-analytic structure presented in this paper, to include the whole space of singular Schwartz distributions, also known as generalized real functions.

In the third paper of the series we will show that the whole Fourier theory of integrable real functions is contained within that same complex-analytic structure. We will show that this structure induces a very general and powerful summation rule for Fourier series, that can be used to add up Fourier series in a consistent way, even when they are explicitly and strongly divergent. The complex-analytic structure will then allow us to extend the Fourier theory beyond the realm of integrable real functions, with the use of that summation rule.

In the fourth paper of the series we will show that one can include in the same complex-analytic structure a large class of non-integrable real functions, among those that are locally integrable almost everywhere. We will see that the complex-analytic structure allows us to associate to each such function a definite set of Fourier coefficients, despite the fact that the functions are not integrable on the unit circle. There are also other applications of the structure discussed here, for example in the two-dimensional Dirichlet problem in partial differential equations, a discussion of which will be given in the fifth paper of the series.

The material contained in this paper is a development, reorganization and extension of some of the material found, sometimes still in rather rudimentary form, in the papers [#!FTotCPI!#,#!FTotCPII!#,#!FTotCPIII!#,#!FTotCPIV!#,#!FTotCPV!#].