Introduction

In this paper we will discuss the question of the convergence of DP Fourier series, in the light of the correspondence between FC pairs of DP Fourier series and Taylor series of inner analytic functions, which was established in a previous paper [1]. This discussion was started in that paper, and will be continued here in order to include the more complex and difficult cases. We will assume that the reader is aware of the contents of that paper, and will use without too much explanation the concepts, the definitions and the notations established there. We will limit ourselves here to the explanation of a few abbreviations, such as DP, which stands for Definite Parity, and FC, which stands for Fourier Conjugate, and to the restatement of the more basic concepts.

We refer as the basic convergence theorem of complex analysis to the result that, if a complex power series around a point $z_{0}$ converges at a point $z_{1}\neq z_{0}$, then it is convergent and absolutely convergent in the interior of a disk centered at $z=z_{0}$ with its boundary passing through $z_{1}$. In addition to this, it converges uniformly on any closed set contained within this open disk. The concept of inner analytic function makes reference to a complex function that is analytic at least on the open unit disk, assumes the value zero at $z=0$, and is the analytic continuation of a real function on the interval $(-1,1)$ of the real axis. The restriction of this complex function to the unit circle results in a FC pair of DP real functions. The corresponding Taylor series around $z=0$ converges at least on the open unit disk, assumes the value zero at $z=0$, and has real coefficients. The restriction of this complex power series to the unit circle results in a FC pair of DP Fourier series.

The concept of Definite Parity or DP Fourier series refers to a Fourier series which has only the sine terms, or only the cosine terms, without the constant term, and therefore has a definite parity on the periodic interval $[-\pi,\pi]$. In the same way, DP real functions are those that have definite parity on the periodic interval. The concept of Fourier Conjugate or FC trigonometric series refers to a conjugate series that is built from a given DP trigonometric series by the exchange of cosines by sines, or vice-versa. The concept of FC real functions refers to the real functions that FC Fourier series converge to, or more generally to the pair of real functions that generates the common Fourier coefficients of a FC pair of DP trigonometric series, even if the series do not converge.

The concepts of the degrees of hardness and of the degrees of softness of singularities in the complex plane, which were introduced in the previous paper [1], will be discussed and defined in more precise terms in this paper. The basic concept is that a soft singularity is one where the complex function is still well defined, although not analytic, when one takes the limit to the singular point. On the other hand, a hard singularity is one where the complex function diverges to infinity or does not exist when one takes that limit. A borderline hard singularity is the least hard type of hard singularity, while a borderline soft singularity is the least soft type of soft singularity.

We will consider, then, the convergence of DP Fourier series and of the corresponding complex power series. For organizational reasons this discussion must be separated into several parts. In the previous paper [1] we tackled the extreme cases which we qualify as those of strong divergence and strong convergence, for which results can be established in full generality. In this paper we will deal with the remaining cases, which we will qualify as those of weak convergence and very weak convergence. In the case of very weak convergence we will be able to present definite results only for a certain class of DP Fourier series.