Timeline for Behaviour of power series on their circle of convergence
Current License: CC BY-SA 3.0
17 events
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| Apr 19, 2021 at 19:41 | comment | added | Andrés E. Caicedo | @jaRedDRedmp Hello. I believe that measure-theoretic considerations will play a key role in any characterization. They certainly seem to appear prominently in "recent" results. Körner's survey mentions some of them and includes some speculations of his own along these lines. | |
| Apr 19, 2021 at 15:18 | comment | added | tassle | Sorry for coming so late, can you explain what do you mean by the "the question of what sets are sets of convergence is not purely topological"? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Aug 8, 2014 at 14:03 | comment | added | Benjamin Dickman | (Just a comment, assuming I haven't misunderstood.) To show that $\sum_{n} \frac{z^n}{n}$ converges for $C \ni z \neq 1$ (as you mention is shown in Rudin using trigonometric estimates) can be done very quickly with the Dirichlet Convergent Test, or -- even better! -- with the quick geometric proof provided here: mathoverflow.net/q/109582/22971 | |
| Jan 28, 2013 at 10:12 | comment | added | Martin | The question came up on SE again: math.stackexchange.com/q/288765 user mrf refers to Lukašenko S. Ju., Sets of divergence and nonsummability for trigonometric series, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1978, no. 2, 65–70 for the result that there is a $G_\delta$-set which is not a set of convergence. | |
| Jan 12, 2012 at 21:21 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
added 2347 characters in body
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| Dec 15, 2010 at 16:02 | comment | added | Andrés E. Caicedo | @Kimball : Yes. | |
| Dec 15, 2010 at 16:00 | comment | added | Kimball | I assume $F_\sigma$ means countable union of closed subsets, yes? | |
| Dec 15, 2010 at 10:48 | vote | accept | Piotr | ||
| Dec 15, 2010 at 2:00 | comment | added | J. M. isn't a mathematician | I'll just point out that the two papers Andres is referring to are both available at Project Euclid: dx.doi.org/10.1215/S0012-7094-49-01647-6 and dx.doi.org/10.1215/S0012-7094-53-02005-5 | |
| Dec 14, 2010 at 19:17 | comment | added | Theo Buehler | @Andres: I second Andrey's comment full-heartedly. This is indeed very interesting, thanks a lot! | |
| Dec 14, 2010 at 18:57 | comment | added | Andrés E. Caicedo | @Theo : Thanks. I fixed the typo and added some remarks. @Andrey: Many thanks! | |
| Dec 14, 2010 at 18:56 | history | edited | Andrés E. Caicedo | CC BY-SA 2.5 |
Added some remarks in light of comments
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| Dec 14, 2010 at 18:35 | comment | added | Andrey Rekalo | @Andres Caicedo: That's a very thorough answer! | |
| Dec 14, 2010 at 18:17 | comment | added | Theo Buehler | I think it's worth pointing out that the main effort of the second paper is to provide examples of schlicht (= injective and holomorphic in the open unit disk) Taylor series whose points of convergence not $F_{\sigma}$ on the unit circle. They also mention that the work of Féjer is sufficient to produce examples of non-$F_{\sigma}$-sets of divergence (I believe that this is tightly connected with Gerald Edgar's comment on Fourier series). | |
| Dec 14, 2010 at 17:25 | history | answered | Andrés E. Caicedo | CC BY-SA 2.5 |