Factorization of Singularities

One of the interesting facts that follow from the analysis in the previous paper [1] is that, since the limiting function of a DP Fourier series is always given by the limit of the corresponding inner analytic function from within the open unit disk to the unit circle, in any open subset of that circle where $w(z)$ is analytic it is also $C^{\infty}$ along $\theta$. Therefore a DP Fourier series that converges in a piecewise fashion between two consecutive singularities of $w(z)$ does so to a piecewise section of a $C^{\infty}$ function. This means that it should be possible to recover the $C^{\infty}$ function involved in each section, and also that it should be possible to represent them by series that converge at a faster rate and can thus be differentiated at least a few times. In this section we will show how one can accomplish the latter goal.

We start with a simple case, which in fact we have already demonstrated completely in the previous section. The proof of convergence of DP Fourier series with monotonic coefficients described in Subsection 5.1 can be understood as a process of factorization of the singularity of the inner analytic function $w(z)$. Interpreted in terms of $S_{z}$ we may write the relation between that series and the corresponding center series $C_{z}$ as

$$\begin{eqnarray*} S_{z} & = & \frac{1}{z-1}\, C_{z}, \\ C_{z} & = & \sum_{k=1}^{\infty} b_{k}z^{k}. \end{eqnarray*}$$

As we have shown before, since $S_{z}$ converges to an inner analytic function $w(z)$, so does $C_{z}$, and hence we have

$$w(z) = \frac{1}{z-1}\, \gamma(z),$$

where $C_{z}$ converges to $\gamma(z)$. What was done here is to factor out of $S_{z}$ a simple pole at the point $z=1$. Hence the original series $S_{v}$, which is not absolutely or uniformly convergent and is associated to an inner analytic function that has a borderline hard singularity at $z=1$, is translated into a series $C_{v}$ which is absolutely and uniformly convergent, and that is associated to an inner analytic function that has a borderline soft singularity at that point. Note that the $S_{z}\to C_{z}$ transformation does not change the maximum disk of convergence or the location of any singularities. Just like logarithmic integration, it just softens the existing singularities.