关于带约束的$L_{\infty}$ -极值多项式的几个经典问题
On some classical problems concerning $L_{\infty}$-extremal polynomials with constraints
摘要：
首先，我们考虑以下问题可以追溯到切比雪夫，Zolotarev和Achieser：在所有具有给定首项系数的三角多项式中，$a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$找到了一个在$[0,2 π].$上具有最小最大范数的多项式。我们证明了极小多项式在$[0,2 π]$上渐近等于一个Blaschke乘积乘以一个常数，其中常数是与$τ_j = a_j + i b_j.$相关的Hankel矩阵的最大奇异值。代数多项式的情况对应语句如下。最后确定了三角多项式空间上某些线性泛函的极小范数。作为一个猜想的后果克伦肖从六十年代的行为的截断傅立叶级数的比率和最小偏差被证明。
英文摘要：
First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 π].$ We show that the minimal polynomial is on $[0,2 π]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $τ_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linear functionals on the space of trigonometric polynomials is determined. As a consequence a conjecture by Clenshaw from the sixties on the behavior of the ratio of the truncated Fourier series and the minimum deviation is proved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
