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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: 2024-10-04
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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