| Title:
|
Generic power series on subsets of the unit disk (English) |
| Author:
|
Maga, Balázs |
| Author:
|
Maga, Péter |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
637-652 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda $ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a nonreal boundary point on the unit circle, $f(U)$ is a dense set of $\mathbb {C}$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given. (English) |
| Keyword:
|
complex power series |
| Keyword:
|
boundary behaviour |
| Keyword:
|
Baire category |
| MSC:
|
28A05 |
| MSC:
|
30B30 |
| MSC:
|
54H05 |
| idZBL:
|
Zbl 07584092 |
| idMR:
|
MR4467932 |
| DOI:
|
10.21136/CMJ.2022.0021-21 |
| . |
| Date available:
|
2022-08-22T08:15:42Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150607 |
| . |
| Reference:
|
[1] : Bolyai János Mathematical Society: Miklós Schweitzer Memorial Competition 2020, Problems and Solutions.Available at \brokenlink{https://www.bolyai.hu/files/Schweitzer_{2020_megoldasok.pdf}} Hungarian. MR 1162554 |
| Reference:
|
[2] Breuer, J., Simon, B.: Natural boundaries and spectral theory.Adv. Math. 226 (2011), 4902-4920. Zbl 1219.30001, MR 2775889, 10.1016/j.aim.2010.12.019 |
| Reference:
|
[3] Kahane, J.-P.: Some Random Series of Functions.Cambridge Studies in Advanced Mathematics 5. Cambridge University Press, Cambridge (1985). Zbl 0571.60002, MR 0833073 |
| Reference:
|
[4] Kahane, J.-P.: Baire's category theorem and trigonometric series.J. Anal. Math. 80 (2000), 143-182. Zbl 0961.42001, MR 1771526, 10.1007/BF02791536 |
| Reference:
|
[5] Kierst, S., Szpilrajn, E.: Sur certaines singularités desfonctions analytiques uniformes.Fundam. Math. French 21 (1933), 276-294. Zbl 0008.07401, 10.4064/FM-21-1-276-294 |
| Reference:
|
[6] Kuratowski, K.: Topology. Vol. 1.Academic Press, New York (1966). Zbl 0158.40802, MR 0217751, 10.1016/C2013-0-11022-7 |
| Reference:
|
[7] Maga, B., Maga, P.: Random power series near the endpoint of the convergence interval.Publ. Math. 93 (2018), 413-424. Zbl 1424.60048, MR 3875344, 10.5486/PMD.2018.8130 |
| . |
