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Simultaneous Zero-free Approximation And Universal Optimal Polynomial Approximants

Catherine Bénéteau, Oleg Ivrii, Myrto Manolaki, Daniel Seco
Published 2020 · Mathematics, Computer Science
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Let $E$ be a closed subset of the unit circle of measure zero. Recently, Beise and M\"uller showed the existence of a function in the Hardy space $H^2$ 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 $H^2$ instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to Dirichlet-type spaces $\mathcal{D}_\alpha$ for $\alpha \in [0,1]$.
This paper references
10.1090/S0002-9947-1984-0748841-0
Cyclic vectors in the Dirichlet space
Leon Carl Brown (1984)
10.1007/S00209-016-1694-X
Generic boundary behaviour of Taylor series in Hardy and Bergman spaces
Hans-Peter Beise (2015)
10.4153/CMB-2017-058-4
Remarks on Inner Functions and Optimal Approximants
Catherine Bénéteau (2017)
Theory of H[p] spaces
Peter Duren (1970)
10.1007/s11854-015-0017-1
Cyclicity in Dirichlet-type spaces and extremal polynomials
Catherine Bénéteau (2015)
10.1017/CBO9780511810817
Matrix analysis
Roger A. Horn (1985)
10.1016/s0079-8169(08)x6157-4
Theory of Hp Spaces
Peter L. Duren (2000)
10.1007/s00365-020-09508-z
Boundary behavior of optimal polynomial approximants
Catherine Bénéteau (2019)
Zero-free polynomial approximation on a chain of Jordan domains
Paul M. Gauthier (2012)
Universality and Potential Theory
S. J. Gardiner (2018)
10.1016/j.jat.2012.12.005
Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions
Johan Andersson (2013)
10.1017/CBO9781107239425
A primer on the Dirichlet Spaces
Omar El-Fallah (2013)
10.1016/j.jat.2013.01.003
Mergelyan's theorem for zero free functions
Sergey Khrushchev (2013)
10.1112/blms/bdw037
Boundary behaviour of Dirichlet series with applications to universal series
Stephen J. Gardiner (2016)
10.1007/BF01186621
Representations of continuous functions
Lennart Carleson (1956)
10.1007/978-1-4939-7543-3_14
Taylor Series, Universality and Potential Theory
Stephen J. Gardiner (2018)
10.2307/2321411
Bounded Analytic Functions
John B. Garnett (2006)
10.1017/CBO9780511546617.002
Harmonic Measure: Jordan Domains
John B. Garnett (2005)
10.1016/j.jat.2015.10.003
On the zero-free polynomial approximation problem
Arthur A. Danielyan (2016)
10.5802/aif.2849
Universal Taylor series, conformal mappings and boundary behaviour
Stephen J. Gardiner (2013)
10.1090/S0002-9939-1956-0081948-0
Boundary values of continuous analytic functions
Walter Rudin (1956)
10.1090/PROC/12764
A convergence theorem for harmonic measures with applications to Taylor series
Stephen J. Gardiner (2014)
10.1016/J.CRMA.2013.12.008
Boundary behaviour of universal Taylor series
Stephen J. Gardiner (2014)
10.1007/978-1-4612-0497-8_1
The Bergman Spaces
Haakan Per Jan Hedenmalm (2000)
10.4171/rmi/1064
Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems
Catherine Bénéteau (2016)
10.1090/S0002-9939-1985-0796443-9
Some examples of cyclic vectors in the Dirichlet space
Leon Carl Brown (1985)
10.1080/17476933.2013.837048
Mergelyan’s theorem with polynomials non-vanishing on unions of sets
Johan Andersson (2014)
10.1142/9789812832139_0001
Orthogonal Polynomials
Vilmos Totik (1992)
10.1007/S00365-018-9425-7
Generic Behavior of Classes of Taylor Series Outside the Unit Disk
G. Costakis (2019)
10.1112/jlms/jdw057
Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
Catherine Bénéteau (2016)
10.1080/17476933.2015.1036048
Universal Taylor series on convex subsets of
Nicholas J. Daras (2015)



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