Absolute and Uniform Convergence

The simplest case of what we call here weak convergence is that in which there is absolute and uniform convergence at the unit circle to a function which, although necessarily continuous, is not $C^{\infty}$. If the coefficients $a_{k}$ are such that the series $S_{z}$ is absolutely convergent on a single point $z_{1}$ of the unit circle, that is, if the series of absolute values

$$\sum_{k=1}^{\infty} \vert a_{k}\vert\vert z_{1}\vert^{k} = \sum_{k=1}^{\infty} \vert a_{k}\vert,$$

where $\vert z_{1}\vert=1$, is convergent, then the series $S_{v}$ is absolutely convergent on the whole unit circle, given that the criterion of convergence is clearly independent of $\theta$ in this case. For the same reason, the series in uniformly convergent and thus converges to a continuous function over the whole unit circle. Therefore the corresponding FC Fourier series are also absolutely and uniformly convergent to continuous functions $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ over their whole domain.

In this case, although the function $w(z)$ must have at least one singularity on the unit circle, since the series $S(z)$ converges everywhere that function is still well defined over the whole unit circle, and by Abel's theorem [2] the $\rho\to 1$ limit of the function $w(z)$ from the interior of the unit disk to the unit circle exists at all points of that circle. It follows that the singularity or singularities of $w(z)$ at the unit circle must not involve divergences to infinity. We will classify such singularities, at which the complex function is still well-defined, although non-analytic, as soft singularities. Singularities at which the complex function diverges to infinity or cannot be defined at all will be called hard singularities.

Since they are convergent everywhere, in this case all DP trigonometric series are DP Fourier series of the continuous function obtained on the unit circle by the $\rho\to 1$ limit of the real and imaginary parts of $w(z)$ from within the unit disk. In short, we have established that weakly-convergent FC pairs of DP Fourier series that converge absolutely and thus uniformly must correspond to inner analytic functions that have only soft singularities on the unit circle. In fact, this would be true of any FC pair of DP Fourier series that simply converges weakly everywhere over the unit circle.

Most of the time this situation can be determined fairly easily in terms of the coefficients $a_{k}$ of the series. If there is a positive real constant $A$, a strictly positive number $\varepsilon$ and an integer $k_{m}$ such that for $k>k_{m}$ the absolute values of the coefficients can be bounded as

$$\vert a_{k}\vert \leq \frac{A}{k^{1+\varepsilon}},$$

then the asymptotic part of the series of the absolute values of the terms of $S_{v}$, which is a sum of positive terms and thus increases monotonically, can be bounded from above by a convergent asymptotic integral, as shown in Section A.1 of Appendix A, and thus that series converges. It follows that the $S_{v}$ series is absolutely and uniformly convergent on the whole unit circle, and therefore so are the two corresponding DP Fourier series.

If the coefficients tend to zero as a power when $k\to\infty$, then it is easy to see that the limiting function defined by the series cannot be $C^{\infty}$. Since every term-wise derivative of the Fourier series with respect to $\theta$ adds a factor of $k$ to the coefficients, a decay to zero as $1/k^{n+1+\varepsilon}$ with $0<\varepsilon\leq 1$ guarantees that we may take only up to $n$ term-by-term derivatives and still end up with an absolutely and uniformly convergent series. With $n+1$ differentiations the resulting series might still converge almost everywhere, but that is not certain, and typically at least one of the two DP series in the FC pair will diverge somewhere. With $n+2$ differentiations the series $S_{v}$ is sure to diverge everywhere, and the two DP series in the FC pair are sure to diverge almost everywhere, since the coefficients no longer go to zero as $k\to\infty$.

It is easy to see that the points where one of the limiting functions $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ is not differentiable, or those at which there is a singularity in one of its derivatives, must be points where $w(z)$ is singular. At any point of the unit circle where $w(z)$ is analytic it not only is continuous but also infinitely differentiable. We see therefore that the set of points of the unit circle where $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ or their derivatives of any order have singularities of any kind must be exactly the same set of points where $w(z)$ has singularities on that circle. In our present case, since the functions must be continuous, these singularities can only be points of non-differentiability of the functions or points where some of their higher-order derivatives do not exist.

We conclude therefore that the cases in which the convergence is not strong but is still absolute and uniform are easily characterized. The really difficult cases regarding convergence are, therefore, those in which the series are not absolutely or uniformly convergent on the unit circle. These are cases in which $\vert a_{k}\vert$ goes to zero as $1/k$ or slower as $k\to\infty$, as shown in Section A.1 of Appendix A. We will refer to these cases as those of very weak convergence. In this case the Fourier series can converge to discontinuous functions, and $S_{z}$ will typically diverge at some points of the unit circle, at which $w(z)$ has divergent limits and therefore hard singularities. In order to discuss this case we must first establish a few simple preliminary facts, leading to a classification of singularities according to their severity.