Article
| 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 |
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| Category: | math |
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| 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 |
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| Date available: | 2022-08-22T08:15:42Z |
| Last updated: | 2022-12-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150607 |
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| 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 |
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