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. The construction leading to this result started with the use of the Fourier coefficients $\alpha_{k}$ and $\beta_{k}$ of the integrable real function $f(\theta)$, from which we defined a set of complex Taylor coefficients $c_{k}$, thus leading to the corresponding inner analytic function $w(z)$. It is therefore clearly apparent that there is a close relation between that complex-analytic structure and the Fourier theory [#!FSchurchill!#] of integrable real functions.

In this paper we will make that relation explicit by showing, in Sections 2--5, that all the elements of the Fourier theory of integrable real functions are contained within the complex-analytic structure. What we mean by these elements is the set of mathematical objects including the Fourier basis of functions, the Fourier series, the scalar product for integrable real functions, the relations of orthogonality and norm of the basis elements, and the completeness of the Fourier basis, including its so-called completeness relation.

The fact that one can recover the real functions from their Fourier coefficients almost everywhere, even when the corresponding Fourier series are divergent, as we showed in [#!CAoRFI!#], leads to a powerful and very general summation rule for all Fourier series. Furthermore, we will show in Section 6 that the complex-analytic structure allows us to extend the Fourier theory beyond the realm of integrable real functions, to include the singular Schwartz distributions that we examined in detail in another previous paper [#!CAoRFII!#], as well as at least some non-integrable real functions, and possibly other objects.

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.