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I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at every $2^n$-th root of unity for every $n$, I feel I'm missing some intuition into what exactly is going on.

Specifically, there is certainly the intuition that the faster a power series' coefficients decrease, the larger the radius of convergence will be - say, comparing the geometric series with the exponential power series. When contrasted with lacunary series, this seems to fail: the coefficients seem to be increasingly "smaller", at least in an average sense, but the function becomes terribly ill-behaved. (One could try and argue that in the Cesàro sense the coefficients do tend to zero: if $\sum_{n=0}^\infty z^{2^n}=\sum_{k=0}^\infty a_k z^k$, then $\frac{1}{n}\sum_{k=0}^n a_k\approx\frac{\lfloor\log_2(n)\rfloor}{n}\rightarrow0$ as $n\rightarrow\infty$. On the other hand, the power series $\sum_{k=0}^\infty \frac{z^k}{k}$, while having the same radius of convergence, can easily, if non-uniquely, be analytically extended to the whole complex plane; I'd expect the same of any series of the form $\sum_{k=0}^\infty \frac{\log(k)}{k}z^k$.)

Can anyone share some insight?

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I blame the parents. – Will Jagy Mar 5 '12 at 1:26
up vote 8 down vote accepted

You can read more about this in the excellent survey

J.-P. Kahane: A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. 29(1997), 257-279

In particular you can learn from this survey that the phenomenon you mentioned is rather typical. It's definitely worth having a look at it.

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Maybe your question is backwards. Natural boundary at the radius of convergence is the usual thing, and analytic continuation outside the circle of convergence is the fluke. Only VERY SPECIAL series have continuations.

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Can this veryspecialness be more precise? – Mariano Suárez-Alvarez Mar 5 '12 at 23:49
@Mariano: Yes. A nice place to see how (historically) this came to be is the paper by Kahane mentioned in Liviu Nicolaescu's answer. Around the time of his dissertation, Borel came to the realization that Taylor series are, in general, not continuable. But he didn't have a way of making his intuition precise as randomness and probability where not yet formally developed. This was addressed in 1929 by Steinhaus. – Andrés E. Caicedo Mar 7 '12 at 4:14

The mentioned gap theorem was generalized by Fabry (Acta Math. 1899, pp. 65-87): if the power series $f(z)=\sum_n a_n z^{\lambda_n}$ has radius of convergence $1$, and the exponents $\lambda_n\in\mathbb{N}$ satisfy $\lambda_n/n\to\infty$, then the unit circle is a natural boundary for $f(z)$.

Turán (Acta Math. Hung. 1947, pp. 21-29) gave a simple proof which might provide some insight into the phenomenon. His main inequality, from which he deduces the result, reads as follows:

$$ \max_{0\leq x\leq 2\pi}\ \left| \sum_{n=1}^N a_n e^{i\lambda_n x} \right| \leq \left(\frac{48\pi}{\delta}\right)^N \max_{a\leq x\leq a+\delta}\ \left| \sum_{n=1}^N a_n e^{i\lambda_n x} \right| $$

In other words, the key feature seams to be that on every arc of the unit circle, the partial sums are considerably bounded away from zero. For more details I would recommend to study Turán's paper.

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I love the phrasing «considerably bounded away from zero.» :) – Mariano Suárez-Alvarez Mar 4 '12 at 21:03
@Mariano: Well, what can we do... I should add that this approach to Fabry's gap theorem made quite a revolution in complex analysis and analytic number theory. Two closely related papers of Turán are Acta Sci. Szeged 1952, pp. 209-218; Rev. Math. Pures Appl. 1956, pp. 27-32 – GH from MO Mar 4 '12 at 21:37
I gave the specific sources (author, journal, year, page numbers). I can give the titles and volumes as well, but you can find these by going to MathSciNet or a library. I tried to save some time. – GH from MO Mar 6 '12 at 15:01
@GH, it was a compliment, really! Math writing, its wordings and style are, very often, like an uneventful tetris game, in which pieces more or less fall in their natural places—yet sometimes, just as in tetris, one finds little gems of phrasing like yours. – Mariano Suárez-Alvarez Mar 6 '12 at 22:13
@Mariano: Thank you! I should have put a smiley after "Well, what can we do..." :-) – GH from MO Mar 6 '12 at 23:35

"Objection, the question assumes facts not in evidence!"

Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. Maybe the point is just that a lacunary form makes it easier to construct badly behaved series, which is slightly different. An example: we know that a real entire function $f$, say with real coefficients, may grow as fast as any given increasing function on $g:\mathbb{R}\to\mathbb{R}$, and building an example is easy by means of lacunary series. But $f(z+1)$ grows even faster, although the translation destroys the lacunary form.

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