论文标题
最佳的近似值运算符在Hardy空间上
Best Approximation-Preserving Operators over Hardy Space
论文作者
论文摘要
令$ t_n $为线性hadamard卷积运算符，在Hardy Space $ H^Q $，$ 1 \ le q \ le \ le \ infty $中作用。如果$ t_n（e_n）= e_n $，我们将$ t_n $称为最佳近似值操作员（bap运算符），其中$ e_n（z）：= z^n，$，如果$ \ | t_n（f）\ | | _q \ | _q \ e_n（f）for All $ f \ y y y y^q $ in $ f \ e__q $，则$ h^q $空间中最多$ n-1 $的代数多项式。 我们为$ t_n $提供了必要的条件，使其成为$ h^\ infty $的BAP运营商。我们将此结果应用于建立一个精确的下限，以最佳的界面全体形态函数的最佳近似值。 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} $和$ n \ ge 2n+1 $。
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 $N\ge 2n+1$.
