This article is cited in 5 papers

On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary

M. A. Komarov

Vladimir State University

Abstract: We study uniform approximation in the open unit disc $D=\{z\colon |z|<1\}$ by logarithmic derivatives of $C$-polynomials, that is, polynomials whose zeros lie on the unit circle $C=\{z\colon |z|\,{=}\,1\}$. We find bounds for the rate of approximation for functions in Hardy class $H^1(D)$ and certain subclasses. We prove bounds for the rate of uniform approximation (either in $D$ or its closure) by $h$-sums $\sum_k \lambda_k h(\lambda_k z)$ with parameters $\lambda_k\in C$.

Keywords: $C$-polynomial, logarithmic derivative, simple partial fraction, uniform approximation, $h$-sum.

UDC: 517.538.5

MSC: 41A20

Received: 29.01.2019
Revised: 29.04.2019

DOI: 10.4213/im8901

 English version: Izvestiya: Mathematics, 2020, 84:3, 437–448

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