论文标题
自由多项式的最佳近似问题
An Optimal Approximation Problem For Free Polynomials
论文作者
论文摘要
通过在单元磁盘中多项式进行最佳近似的最佳近似工作的动机，我们考虑以下非共同近似问题：对于$ d $中的多项式$ f $ f $ in $ d $ in $ d $自由违规论点，找到一个免费的多项式$ p_n $，最多$ n $，以最大程度地减少$ c_n：= $ c_ = $ = p_nf $ c_ p_nf-1。 （这里的规范是系数上的$ \ ell^2 $ norm。）我们表明，$ c_n \ to to $ to $ to $ to $ i时，并且仅当$ f $在某个nc域（行球）中不介绍，并证明定量界限。作为应用程序，我们获得了$ d $ shift的多项式循环表征的新证明。
Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n := \|p_nf-1\|^2$. (Here the norm is the $\ell^2$ norm on coefficients.) We show that $c_n\to 0$ if and only if $f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift.
