Simultaneous zero-free approximation and universal optimal polynomial approximants Articles uri icon

publication date

  • August 2020

start page

  • 1

end page

  • 13

issue

  • 105389

volume

  • 256

International Standard Serial Number (ISSN)

  • 0021-9045

Electronic International Standard Serial Number (EISSN)

  • 1096-0430

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.

subjects

  • Mathematics

keywords

  • optimal polynomial approximants; universality; zero-free approximation; hardy spaces