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Published August 2020 | Submitted
Journal Article Open

Simultaneous zero-free approximation and universal optimal polynomial approximants

Abstract

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.

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.

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