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Simultaneous zero-free approximation and universal optimal polynomial approximants

Bénéteau, Catherine and Ivrii, Oleg and Manolaki, Myrto and Seco, Daniel (2020) Simultaneous zero-free approximation and universal optimal polynomial approximants. Journal of Approximation Theory, 256 . Art. No. 105389. ISSN 0021-9045. doi:10.1016/j.jat.2020.105389.

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Let E be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space H² for which the partial sums of its Taylor series approximate any continuous function on E. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in H² instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to the Dirichlet space D and are expected for more general Dirichlet-type spaces.

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Additional Information:© 2020 Elsevier Inc. Received 19 May 2019, Revised 8 February 2020, Accepted 13 February 2020, Available online 25 February 2020. We confirm that all authors have contributed equally in multiple roles for writing this paper and that there is no distinction among them. Myrto Manolaki thanks the Department of Mathematics and Statistics at the University of South Florida for support during work on this project. Daniel Seco acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through the “Severo Ochoa Programme for Centers of Excellence in R&D” (SEV-2015-0554) and through the grant MTM2016-77710-P.
Funding AgencyGrant Number
Ministerio de Economía, Industria y Competitividad (MINECO)SEV-2015-0554
Ministerio de Economía, Industria y Competitividad (MINECO)MTM2016-77710-P
Subject Keywords:Optimal polynomial approximants; Universality; Zero-free approximation; Hardy spaces
Classification Code:MSC: primary 30B30; secondary 30K99, 30H10
Record Number:CaltechAUTHORS:20200304-090034462
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Official Citation:Catherine Bénéteau, Oleg Ivrii, Myrto Manolaki, Daniel Seco, Simultaneous zero-free approximation and universal optimal polynomial approximants, Journal of Approximation Theory, Volume 256, 2020, 105389, ISSN 0021-9045, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101697
Deposited By: Tony Diaz
Deposited On:04 Mar 2020 18:40
Last Modified:16 Nov 2021 18:05

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