Weak Convergence

In the previous paper [1] we discussed the issue of the convergence of FC pairs of DP Fourier series such as

$$\begin{eqnarray*} S_{\rm c} & = & \sum_{k=1}^{\infty} a_{k}\cos(k\theta), \\ S_{\rm s} & = & \sum_{k=1}^{\infty} a_{k}\sin(k\theta), \end{eqnarray*}$$

with real Fourier coefficients $a_{k}$, for $\theta$ in the periodic interval $[-\pi,\pi]$, and of the corresponding complex power series

$$\begin{eqnarray*} S_{v} & = & \sum_{k=1}^{\infty} a_{k}v^{k}, \\ S_{z} & = & \sum_{k=1}^{\infty} a_{k}z^{k}, \end{eqnarray*}$$

where $v=\exp(\mbox{\boldmath$\imath$}\theta)$ and $z=\rho v$, in two extreme cases, those of strong divergence and strong convergence. We established that, in these extreme cases, either the convergence/divergence of $S_{z}$ or the existence/absence of singularities of the analytic function $w(z)$ that $S_{z}$ converges to can be used to determine the convergence or divergence of $S_{v}$ and of the corresponding DP Fourier series over the whole unit circle of the complex plane.

Specifically, we established that the divergence of $S_{z}$, or the existence of a singularity of $w(z)$, at any single point strictly within the open unit disk, implies that $S_{v}$ diverges everywhere on the unit circle, and that the corresponding FC pair of DP Fourier series diverge almost everywhere on the periodic interval. This is what we call strong divergence. To complement this we established that the convergence of $S_{z}$ at a single point strictly outside the closed unit disk, or the absence of singularities of $w(z)$ on that disk, implies that $S_{v}$ converges to a $C^{\infty}$ function everywhere on the unit circle, and that the corresponding FC pair of DP Fourier series converge to $C^{\infty}$ real functions everywhere on the periodic interval. This is what we call strong convergence.

If the series $S_{z}$ constructed from an FC pair of DP trigonometric series converges on at least one point on the unit circle, but does not converge at any points strictly outside the closed unit disk, then the situation is significantly more complex. We will see, however, that in essence it can still be completely determined, with some limitations on the set of series involved when establishing the results. We will refer to this situation as the case of weak convergence. In this situation the maximum disk of convergence of the complex power series $S_{z}$ is the open unit disk, and therefore this series converges strongly to an analytic function $w(z)$ strictly within that disk. As is well known, it can be shown that in this case the function $w(z)$ must have at least one singularity on the unit circle.

In terms of the convergence of the series, however, the situation on the unit circle remains undefined. The series $S_{z}$ may converge or diverge at various points on that circle. The function $w(z)$ cannot be analytic on the whole unit circle, since there must be at least one point on that circle where it has a singularity, in the sense that it is not analytic at that point. This singularity can be of various types, and does not necessarily mean that the function $w(z)$ is not defined at its location. In fact, it is still possible for $w(z)$ to exist everywhere and to be analytic almost everywhere on the unit circle. No general convergence theorem is available for this case, and the analysis must be based on the behavior of the coefficients $a_{k}$ of the series. It is, therefore, significantly harder to obtain definite results.

While the case of strong divergence is characterized by the fact that the coefficients $a_{k}$ typically diverge exponentially with $k$, and the case of strong convergence by the fact that they typically go to zero exponentially with $k$, in the case of weak convergence the typical behavior is that the coefficients go to zero as a negative power of $k$. In most of this paper we will in fact limit the discussion to series in which the coefficients go to zero as a negative but not necessarily integer power of $k$.