Best Approximation-Preserving Operators over Hardy Space

Let \(T_n\) be the linear Hadamard convolution operator acting over Hardy space \(H^q\), \(1\le q\le\infty\). We call \(T_n\) a best approximation-preserving operator (BAP operator) if \(T_n(e_n)=e_n\), where \(e_n(z):=z^n,\) and if \(\|T_n(f)\|_q\le E_n(f)_q\) for all \(f\in H^q\), where \(E_n(f)_q...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Abdullayev, Fahreddin G, Savchuk, Viktor V, Savchuk, Maryna V
Format: Article
Language:English
Subjects:
Analytic functions
Approximation
Lower bounds
Mathematical analysis
Polynomials
Online Access:Get full text
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Summary:Let \(T_n\) be the linear Hadamard convolution operator acting over Hardy space \(H^q\), \(1\le q\le\infty\). We call \(T_n\) a best approximation-preserving operator (BAP operator) if \(T_n(e_n)=e_n\), where \(e_n(z):=z^n,\) and if \(\|T_n(f)\|_q\le E_n(f)_q\) for all \(f\in H^q\), where \(E_n(f)_q\) is the best approximation by algebraic polynomials of degree a most \(n-1\) in \(H^q\) space. We give necessary and sufficient conditions for \(T_n\) to be a BAP operator over \(H^\infty\). We apply this result to establish an exact lower bound for the best approximation of bounded holomorphic functions. In particular, we show that the Landau-type inequality \(\left|\widehat f_n\right|+c\left|\widehat f_N\right|\le E_n(f)_\infty\), where \(c>0\) and \(n
ISSN:2331-8422