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A difference between minimal and ordinary fine topology in function theory


Author: J. S. Hwang
Journal: Proc. Amer. Math. Soc. 85 (1982), 239-244
MSC: Primary 30D40; Secondary 30D50, 31A20
DOI: https://doi.org/10.1090/S0002-9939-1982-0652450-5
MathSciNet review: 652450
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Abstract: After Choquet (see Brelot and Doob [6, p. 404]), we recently have presented an alternative answer of Doob's problem [8] by showing that there is a Blaschke product $ B$ having the minimal fine cluster value 0 and the angular limit 1 at $ \infty $. In our construction, the zeros of $ B$ lie on both the first and fourth quadrant. Naturally, we may ask if a product having the same property, but having zeros which lie on only one quadrant, can be constructed. We show that the answer to this question is no for the minimal fine topology, but yes for the ordinary one. There is a significant difference between these two topologies in function theory.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0652450-5
Keywords: Fine topology, cluster value, Blaschke product
Article copyright: © Copyright 1982 American Mathematical Society

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