Introduction

In a previous paper [#!CAoRFI!#] we introduced a certain complex-analytic structure within the unit disk of the complex plane, and showed that it is possible to represent within that structure essentially all integrable real functions defined in a compact interval. In a subsequent paper [#!CAoRFIII!#] we showed that all the elements of the Fourier theory [#!FSchurchill!#] of integrable real functions are contained within that complex-analytic structure. However, in that paper we did not discuss in any depth the question of the convergence of Fourier series.

The fact that it is possible to recover the real functions from their Fourier coefficients almost everywhere, even when the corresponding Fourier series are divergent, as we showed in [#!CAoRFI!#], led to a powerful and very general summation rule for all Fourier series, which was presented in [#!CAoRFIII!#]. This summation rule allows one to add up a regularized version of the Fourier series, in a meaningful way, and therefore allows one to simply circumvent the fact that the original Fourier series may be divergent. However, the complex-analytic structure actually does allow for a direct discussion of the convergence problem.

In this paper we will present a more complete analysis of the convergence of Fourier series. In order to do this we will first introduce what we will name the compact Hilbert transform, which is a version of the Hilbert transform which is appropriate for functions defined on a compact interval. This will lead not only to the known explicit expression for the partial sums of the Fourier series in terms of Dirichlet integrals, but also to an explicit expression for the remainder of the Fourier series, in terms of a similar type of integral.

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.