Convergence and Singularities

Let us examine the effects on the corresponding series and functions of each one of these two related types of differentiation and integration operations in turn. First the effect of differentiations and integrations with respect to $\theta$ acting on the DP Fourier series, and then the effect of logarithmic differentiations and integrations acting on the corresponding inner analytic functions $w(z)$.

On the Fourier side of this discussion, it is clear from the structure of the DP Fourier series that each differentiation with respect to $\theta$ adds a factor of $k$ to the $a_{k}$ coefficients, and thus makes them go to zero slower, or not at all, as $k\to\infty$. This either reduces the rate of convergence of the series or makes them outright divergent. Integration with respect to $\theta$, on the other hand, has the opposite effect, since it adds to the $a_{k}$ coefficients a factor of $1/k$, and thus makes them go to zero faster as $k\to\infty$. This always increases the rate of convergence of the series. On the inner-analytic side of the discussion, each logarithmic differentiation increases the degree of hardness of each singularity, while logarithmic integration decreases it. At the same time, these operations work in the opposite way on the degrees of softness. No point of singularity over the unit circle ever vanishes or appears as a result of these operations. Only the degrees of hardness and softness change.

We are now in a position to use these preliminary facts to analyze the question of the convergence of the series on the unit circle. While the derivations and integrations with respect to $\theta$ change the convergence status of the DP Fourier series on the unit circle, the related logarithmic derivations and integrations of the corresponding inner analytic function do not change the analyticity or the set of singular points of that function. The only thing that these logarithmic operations do change is the type of the singularities over the unit circle, as described by their degrees of hardness or softness. Hence, there must be a relation between the mode of convergence or lack of convergence of the DP Fourier series on the unit circle and the nature of the singularities of the inner analytic function on that circle. It is quite clear that the rate of convergence and the very existence of convergence of the DP Fourier series are tied up to the degree of hardness or softness of the singularities present. The less soft the singularities, the slower the convergence, leading eventually to hard singularities and to the total loss of convergence.

Note that the relation between the singularities of $w(z)$ on the unit circle and the convergence of the DP Fourier series is non-local, because processes of differentiation or integration will change the degree of hardness or softness only locally at the singular points, but will affect the speed of convergence to zero of the coefficients $a_{k}$, which has its effects on the rate of convergence of the DP Fourier series everywhere over the unit circle. Since the relation between the hardness or softness of the singularities and the convergence of the DP Fourier series is non-local, the rate of convergence or the lack of it will be ruled by the hardest or least soft singularity or set of singularities found anywhere over the whole unit circle. We will call these the dominant singularities. Therefore, from now on we will think in terms of the dominant singularity or set of singularities which exists on that circle.

The problem of establishing a general, complete and exact set of criteria determining the convergence or arbitrary DP Fourier series with basis on the set of dominant singularities of the corresponding inner analytic function is, so far as we can tell, an open one. We will, however, be able to classify and obtain the convergence criteria for a fairly large class of DP Fourier series. In order to establish this class of series, consider the set of all possible integral-differential chains involving the series $S_{z}$, $S_{v}$, $S_{\rm c}$ and $S_{\rm s}$, and the respective functions $w(z)$, $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$. Let us use $S_{v}$ to characterize the elements of these chains. Since the operations of logarithmic differentiation and logarithmic integration always produce definite and unique results, it is clear the each series $S_{v}$ belongs to only one of these chains. Let us now select from the set of all possible integral-differential chains those that satisfy the following two conditions.

• There is in the chain a series $S_{v}$ with coefficients $a_{k}$ that can be bounded in the following way: there exist positive real constants $A_{(-)}$ and $A_{(+)}$, a minimum value $k_{m}$ of $k$ and a positive real number $0<\varepsilon<1$ such that for $k>k_{m}$

  
  

  $$\frac{A_{(-)}}{k} \leq \vert a_{k}\vert \leq \frac{A_{(+)}}{k^{\varepsilon}}.$$
  
  

• The series $S_{v}$ qualified in the previous condition diverges to infinity on at least one point of the unit circle.

We will call the integral-differential chains that satisfy these conditions regular integral-differential chains. We will adopt as a shorthand for the first condition the statement that $\vert a_{k}\vert$ behaves as $1/k^{p}$ for large values of $k$, or $\vert a_{k}\vert\propto 1/k^{p}$, where $0<p\leq 1$. What the condition means is that, while the coefficients go to zero as $k\to\infty$, they do it sufficiently slowly to prevent the series $S_{v}$ from being absolutely and uniformly convergent. Therefore the series $S_{v}$ may still converge, but does not converge absolutely. Because the series satisfies the second condition we know, from the extended version of Abel's theorem [2], that the corresponding inner analytic function has a divergent limit going to infinity, on at least one point of the unit circle. Therefore, it has at least one hard singularity on that circle. We see then that the set of dominant singularities that the inner analytic function has on the unit circle is necessarily a set of hard singularities. We will denote the series that satisfies these conditions in any given regular integral-differential chain by $S_{v,h0}$, and the corresponding inner analytic function by $w_{h0}(z)$.

The next series in the chain, obtained from this one by logarithmic integration, which is the same as integration with respect to $\theta$, has coefficients that behave as $\vert a_{k}\vert\propto 1/k^{p}$ with $1<p\leq 2$, and is therefore absolutely and uniformly convergent everywhere. It follows therefore that all the singularities of the corresponding inner analytic function are soft. We will denote the series obtained from $S_{v,h0}$ in this way by $S_{v,s0}$, and the corresponding inner analytic function by $w_{s0}(z)$. The set of dominant singularities of $w_{s0}(z)$ is thus seen to be a set of soft singularities. It follows that the dominant singularities of $w_{h0}(z)$, which are hard and became soft by means of a single operation of logarithmic integration, constitute a set of borderline hard singularities. Since we may go back from $w_{s0}(z)$ to $w_{h0}(z)$ by a single operation of logarithmic differentiation, it also follows that the set of dominant singularities of $w_{s0}(z)$ is a set of borderline soft singularities.

If we go further along in either direction of the chain, in the integration direction all subsequent series $S_{v,sn}$ are also absolutely and uniformly convergent, to continuous functions that are everywhere $C^{n}$ and almost everywhere $C^{n+1}$, typically sectionally $C^{n+1}$, where $n$ is the degree of softness. The corresponding inner analytic functions $w_{sn}(z)$ have only soft singularities on the unit circle. In the other direction, the next series in the chain, denoted by $S_{v,h1}$, obtained from $S_{v,h0}$ by logarithmic differentiation, has coefficients that behave as $\vert a_{k}\vert\propto k^{p}$ with $0\leq p<1$, and is therefore divergent everywhere, since the coefficients do not go to zero as $k\to\infty$. The same can be said of all the subsequent elements $S_{v,hn}$ of the chain in this direction. The corresponding inner analytic functions $w_{hn}(z)$ have sets of dominant singularities consisting of hard singularities. We arrive therefore at the following scheme of convergence diagnostics, for series $S_{v}$ on regular integral-differential chains, based on the behavior of the dominant singularities of $w(z)$.

• If the dominant singularities of $w(z)$ are hard with degree of hardness $n\geq 1$, then the series $S_{v}$ is everywhere divergent.

• If the dominant singularities of $w(z)$ are borderline hard ones, then $S_{v}$ may still be convergent almost everywhere to a discontinuous function, but will diverge in one or more points. If the number of dominant singularities is finite, then the limiting function will be sectionally continuous and differentiable.

• If the dominant singularities of $w(z)$ are borderline soft ones, then $S_{v}$ is everywhere convergent to an everywhere continuous but not everywhere differentiable function. If the number of dominant singularities is finite, then the limiting function will be sectionally differentiable.

• If the dominant singularities of $w(z)$ are soft with degree of softness $n\geq 1$, then the series $S_{v}$ is everywhere convergent to a $C^{n}$ function, which is also $C^{n+1}$ almost everywhere. If the number of dominant singularities is finite, then the limiting function will be sectionally $C^{n+1}$.

Given the convergence status of $S_{v}$, corresponding conclusions can then be drawn for the FC pair of DP Fourier series $S_{\rm c}$ and $S_{\rm s}$. Observe that in this argument the facts about the convergence of the DP Fourier series are in fact feeding back into the question of the convergence of the power series $S_{z}$ at the rim of its maximum disk of convergence. It follows therefore that this classification can be understood as a set of statements purely in complex analysis, since it also states conditions for the convergence or divergence of the Taylor series $S_{z}$ over the unit circle, in the cases when that circle is the boundary of its maximum disk of convergence.

Note that in the case in which the dominant singularities are borderline hard ones there is as yet no certainty of convergence almost everywhere. Therefore in this respect this alternative must be left open here, and will be discussed in the next section for some classes of series. It is an interesting question whether or not there are series $S_{v}$ with coefficients that behave as $\vert a_{k}\vert\propto 1/k^{p}$ with $0<p\leq 1$ and that do not diverge to infinity anywhere. The common examples all seem to diverge as we have assumed here. It seems to be difficult to find an example that has the opposite behavior, but we can offer no proof one way of the other. Therefore, we must now leave this here as a mere speculation.

Observe that, since the inner analytic function can be integrated indefinitely, as many times as necessary, we have here another way, at least in principle, to get some information about the function that originated an arbitrarily given DP Fourier series, besides taking limits of $w(z)$ from within the open unit disk. We may construct the series $S_{z}$ from the coefficients and, if the corresponding inner analytic function $w(z)$ is in fact analytic on the open unit disk, with some upper bound for the hardness of the singularities on the unit circle, then we may logarithmically integrate it as many times as necessary to reduce all the singularities over the unit circle to soft singularities. We will then have corresponding series $S_{\rm c}$ and $S_{\rm s}$ that are absolutely and uniformly convergent over the unit circle, with continuous functions as their limits.

The original function is then a multiple derivative with respect to $\theta$ of one of these continuous functions. Typically we will not be able to actually take these derivatives at the unit circle, but we will always be able to take the corresponding derivatives of $w(z)$ within the open unit disk, arbitrarily close to the unit circle, so that at least we will have a chance to understand the origin of the problem, which might give us an insight into the structure of the application, and related circumstances, that generated the given DP Fourier series. This procedure will fail only if the inner analytic function turns out to have an infinite number of singularities on the unit circle, with degrees of hardness without an upper bound, or an infinitely hard singularity, such as an essential singularity, anywhere on the unit circle.