Document Type

Article

Publication Date

2023

DOI

10.2140/pjm.2023.327.267

Publication Title

Pacific Journal of Mathematics

Volume

327

Issue

2

Pages

267-295

Abstract

We study optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, Hp (1 < p < ∞). For fixed f ∈ Hp and n ∈ N, the OPA of degree n associated to f is the polynomial which minimizes the quantity ∥qf −1∥p over all complex polynomials q of degree less than or equal to n. We begin with some examples which illustrate, when p ≠ 2, how the Banach space geometry makes the above minimization problem interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions of certain fixed-point problems. Finally, using duality arguments, we provide several bounds concerning the error incurred in the OPA approximation.

Rights

© 2023 MSP (Mathematical Sciences Publishers).

Distributed under the Creative Commons Attribution 4.0 International (CC BY 4.0) License.

Original Publication Citation

Cheng, R., & Felder, C. (2023). More properties of optimal polynomial approximants in Hardy spaces. Pacific Journal of Mathematics, 327(2), 267-295. https://doi.org/10.2140/pjm.2023.327.267

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