Timeline for Maximal domain of holomorphy of a series
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| Aug 11, 2011 at 1:32 | comment | added | fedja | Look up Sibilev, R. V. A uniqueness theorem for Wolff-Denjoy series. (Russian. Russian summary) Algebra i Analiz 7 (1995), no. 1, 170--199; translation in St. Petersburg Math. J. 7 (1996), no. 1, 145–168 30B50 (30B40 47A15 47B38) He was Nikolski's student and this paper of his seems to be most relevant to your question. | |
| Aug 10, 2011 at 22:32 | comment | added | Gro-Tsen | @fedja → Right, I should have been more careful about the possibility of converging to $0$ (I thought a mild condition like $c_n = o(n^-\alpha)$ for $\alpha>3/2$ would automatically preclude this, but it seems my reasoning was faulty, so I don't know); I'd love to see an example. If I don't get an answer here, I'll try bothering Nikolai as you suggest. @Igor Rivin → I guess I wasn't too clear on quantifiers. I would be asking: (given some meaning of “rapidly”) does there exist $(c_n)$ decreasing rapidly and does there exist some enumeration $(a_n)$ such that $f$ can be exended? | |
| Aug 10, 2011 at 21:46 | comment | added | Igor Rivin | Do you think the answer depends on which enumeration you choose? | |
| Aug 10, 2011 at 17:39 | comment | added | fedja | It is perfectly possible for the series to converge to $0$ uniformly in the closed unit disk without having zero coefficients. On the other hand, if the decay of the coefficients is really fast (like $e^{-n}$, say), then, if I remember it right, the function has the unit circle an the natural boundary. My memory is somewhat shaky but, since you are in France, you probably know Nikolay Nikolskii at the University of Bordeaux. He would know for sure :). | |
| Aug 10, 2011 at 15:02 | history | asked | Gro-Tsen | CC BY-SA 3.0 |