Continuation of functions beyond natural boundaries

Question

The article Continuation of functions beyond natural boundaries by John L. Gammel states

I am particularly interested in the convergence of the $[N/N+1]$ Padé approximants beyond the natural boundary, since, as is well known, Borel [2] has shown that there exists a kind of analytic continuation which differs from the usual kind (the theory of the usual kind is due to Weierstrass), and Borel made use of examples such as the ones studied here in showing that in some cases it is possible to continue functions beyond what Weierstrass called natural boundaries. I am interested in these examples because they seem to me suggestive of the direction in which comprehensive theorems about the domains in which Padé approximants converge and theorems about to what they converge are to be sought.

where [2] references

E. Borel, Lecons sur les fonctions monogènes d'une variable complexe, Gauthier-Villars, Paris, 1917.

Unfortunately, it is in French and inaccessible to me. Does anyone know what technique of Borel Gammel is referring to? How is it possible to continue a function beyond its natural boundary?

Answer 1

There are two phenomena where a given holomorphic function can be continued in some sense beyond its natural boundary, namely overconvergence and generalized analytic continuation: Gammel most likely refers to the last one, i.e. to generalized analytic continuation, which was precisely studied by Borel and his followers. A modern and very readable reference is [1], and particularly relevant to our question is chapter 3, pp. 21-35. And quoting verbatim the Authors of [1], (p. xi of the preface)

"Generalized analytic continuation (...) studies ways in which the component functions $f|_{\Omega_j}$ - where $f$ is a meromorphic function on disconnected open set $\Omega$ in the complex plane $\Bbb C$ and $\{\Omega_j\}$ are the connected components of $\Omega$ - may possibly be related to each other in certain cases where the Weierstrassian notion of analytic continuation says there is a 'natural boundary'."

The 'way' this happens is that an appropriate rational approximation (i.e. one satisfying appropriate conditions) to a part $f|_{\Omega_j}$ of a function $f$ characterized as above, determines also other parts.

An example (the Poincaré example) ([1] chapter 3, p. 11)

A typical holomorphic function which exibit this "superconvergence" behavior is $$ f(z)=\sum_{n=1}^{\infty}\frac{c_n}{z-e^{i\theta_n}}\label{1}\tag{1} $$ where

• $\{c_n\}_{n\in\Bbb N}$ is an absolutely summable sequence of complex numbers $\big(\sum_{n=0}^{\infty}|c_n|<\infty\big)$.
• $\{\theta_n\}_{n\in\Bbb N}$ is a sequence of real numbers which is dense on the boundary of the unit disk $\partial \Bbb D$

It can be proved that, for $f|_{\Bbb D}$ and $f|_{\Bbb{C\setminus D}}$, the boundary $\partial \Bbb D$ is a natural boundary, thus standard analytic continuation cannot happen across any of its arcs, yet there exists a rational approximation \eqref{1} (as we can see, de facto the whole $f$ is defined by a series of rational functions, thus any finite truncation of it will do the job) such that, while univocally determining one of its parts, it does simultaneously the same for the other.

Notes

• According to [1] (pp. xii-xiii of the preface), generalized analytic continuation is a field of research which is wide open, and there are also several types of analytic continuation depending on the conditions assumed on the approximation used to define it.
• A similar phenomenon is overconvergence: it was discovered by Milton Brocket Porter and further studied by Jentzsch and Ostrowski.

Reference

[1] Ross, William T.; Shapiro, Harold S., Generalized analytic continuation, University Lecture Series. 25. Providence, RI: American Mathematical Society (AMS). xiii, 149 p. (2002), ISBN: 0-8218-3175-5, MR1895624, ZBL1009.30002.