Limits from Within

We have established that every DP Fourier series that converges on one point together with its FC series corresponds to an inner analytic function. Whenever it is possible to take the limit to the unit circle of restrictions of this analytic function to circles of smaller radii, centered at the origin, it gives us back the real function that corresponds to the coefficients of the series, even if the Fourier series itself is not convergent. We may also interpret this type of limit as a collection of point-by-point limits taken in the radial direction to each point of the unit circle. Let us discuss now under what conditions we may take such limits and what we can learn from them.

Our ability to take the limits to the unit circle depends on whether or not the inner analytic function $w(z)$ is well-defined on the unit circle, and therefore on whether or not it has singularities on that circle, as well as on the nature of these singularities. If there are no singularities at all on the unit circle, then $w(z)$ is analytic over the whole unit circle and therefore continuous there. In this case it is always possible to take the limit, to all points of the circle, and they will always result in a pair of $C^{\infty}$ real functions on the unit circle. Also, in this case it is true that the Taylor series of $w(z)$ and the corresponding pair of FC series converge everywhere on the unit circle. In fact, as we will see shortly, they all converge absolutely and uniformly, and this is the situation which we characterize and refer to as that of strong convergence. The $C^{\infty}$ functions thus obtained on the unit circle are those that give the coefficients $a_{k}$.

Let us suppose now that there is a finite number of singularities on the unit circle, which are therefore all isolated singularities. On the open subsets of the unit circle between two adjacent singularities $w(z)$ is still analytic, and hence continuous. Therefore, within these open subsets the limits to the unit circle may still be taken, resulting in segments of $C^{\infty}$ real functions, and reproducing almost everywhere over the circle the real functions that originated the $a_{k}$ coefficients. Therefore we learn that any Fourier series that converges to a sectionally continuous and differentiable function, and is such that the corresponding inner analytic function has a finite number of singularities on the unit circle, in fact converges to a sectionally $C^{\infty}$ function, possibly with an increased number of sections.

At the points of singularity, if the Fourier series converge, then by Abel's theorem [6] they converge to the limits of $w(z)$ at these points, taken from within the open unit disk, so that in this case it is also possible to take the limits. Since the real functions that generate the coefficients by means of the real integrals are determined uniquely only almost everywhere, if the series diverge at the singular points we may still adopt the limits of $w(z)$ from within as the values of the corresponding functions at those points, so long as these limits exist and are finite. This will be the case if the singularities do not involve divergences to infinity, so that the inner analytic functions are still well-defined on their location, although they are not analytic there. We will call such singularities, at which we may still take the limits to the unit circle, soft singularities.

Usually one tends to think of singularities in analytic functions in terms only of hard singularities such as poles, which always involve divergences to infinity when one approaches them. The paradigmatic hard singularities are poles such as

$$\frac{1}{(z-z_{1})^{n}} = \frac{\,{\rm e}^{-\mbox{\boldmath$\imath$}n\alpha}}{R^{n}},$$

with $n\geq 1$, where $z-z_{1}=R\exp(\mbox{\boldmath$\imath$}\alpha)$ with real $R$ and $\alpha$, and where $z_{1}$ is a point on the unit circle. We will think of the integer $n$ as the degree of hardness of the singularity. Note that such singularities are not integrable along arbitrary lines passing through the singular point. A somewhat less hard singularity is the logarithmic one given by

$$\ln(z-z_{1}) = \ln(R)+\mbox{\boldmath$\imath$}\alpha,$$

which is also hard but much less so that the poles, going to infinity much slower. We will call this a borderline hard singularity, or a hard singularity of degree zero. Note that this singularity is integrable along arbitrary lines passing through the singular point. Soft singularities are those for which the limit to the singular point still exists, for example those such as

$$(z-z_{1})\ln(z-z_{1}),$$

which we will call a borderline soft singularity, or a soft singularity of degree zero. Progressively softer singularities can be obtained by increasing $n$ in the generic soft singularity

$$(z-z_{1})^{n+1}\ln(z-z_{1}),$$

where we should have $n\geq 1$. In this case we will think of the integer $n$ as the degree of softness of the singularity. To complete this classification of singularities, any essential singularity will be classified as an infinitely hard one. A more precise and complete definition of this classification of singularities will be given in the follow-up paper [1].

If one has a countable infinity of singularities on the unit circle, but they have only a finite number of accumulation points, then the situation still does not change too much. In the open subsets of the unit circle between every pair of consecutive singular points the limits still exist and give us segments of $C^{\infty}$ functions. If the singularities are all soft, then the limits exist everywhere over the unit circle. In any case the limits give us back the real functions which generated the coefficients, at least almost everywhere over the whole circle.

Even if we have a countably infinite set of singularities which are distributed densely on the unit circle, so long as the singularities are all soft the limits can still be taken everywhere. Of course, in this case it is not to be expected that the resulting real functions will be $C^{\infty}$ anywhere. If a given inner analytic function has the open unit disk as the maximum disk of convergence of its Taylor series, and has a dense set of hard singularities over the unit circle, then the limits do not exist anywhere, and therefore the real functions are in fact not defined at all by the limits to that circle. In this case, due to Abel's theorem [6], the Taylor series must diverge everywhere over the unit circle.

Since a borderline hard singularity such as $\ln(z-z_{1})$ can be obtained by the differentiation of a borderline soft singularity such as $(z-z_{1})\ln(z-z_{1})$, it is reasonable to presume that the transition by the differentiation of $w(z)$ from a densely distributed countable infinity of borderline soft singularities to a densely distributed countable infinity of borderline hard singularities corresponds to real functions that exist everywhere over the unit circle but that are not differentiable anywhere.

The questions related to the issue of convergence will be discussed in more detail in the follow-up paper [1], in which we will show that the degrees of hardness or softness of the singularities are related to the analytic character of the real functions, all the way from simple continuity through $n$-fold or $C^{n}$ differentiability to $C^{\infty}$ status. In particular, borderline soft singularities are related to everywhere continuous functions, so that the transition described above should be related to real functions that are everywhere continuous but nowhere differentiable. One such proposed function will be discussed briefly in Appendix A.