Simultaneous zero-free approximation and universal optimal polynomial approximants

Catherine Bénéteau, Oleg Ivrii, Myrto Manolaki, Daniel Seco

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number105389
JournalJournal of Approximation Theory
Volume256
DOIs
StatePublished - Aug 2020
Externally publishedYes

Keywords

  • Hardy spaces
  • Optimal polynomial approximants
  • Universality
  • Zero-free approximation

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