Best Approximation-Preserving Operators over Hardy Space

TL;DR

This paper characterizes best approximation-preserving operators over Hardy spaces, providing necessary and sufficient conditions, and applies these results to establish bounds for approximation of bounded holomorphic functions.

Contribution

It offers a complete characterization of BAP operators over Hardy space $H^$, and derives exact bounds for polynomial approximation of bounded holomorphic functions.

Findings

Necessary and sufficient conditions for BAP operators over $H^$

Exact lower bounds for best approximation in Hardy spaces

Landau-type inequality with optimal constant

Abstract

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 $n<N$, holds for every $f\in H^\infty$ iff $c\le\frac{1}{2}$ and $N\ge 2n+1$.