论文标题
$ t_n^{（k）} $的最大临界值的估计值
Estimates for the largest critical value of $T_n^{(k)}$
论文作者
论文摘要
Here we study the quantity $$ τ_{n,k}:=\frac{|T_n^{(k)}(ω_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is the $n$-th Chebyshev polynomial of the first kind and $ω_{n,k}$ is the $ t_n^{（k+1）} $的最大零。由于$ t_n^{（k）} $的本地极值的绝对值单调地增加了$ [ - 1,1] $的终点，因此值$τ_{n，k} $的值是$ \ \ \ \ \，t_n^^{（k）} \，$相对至其全局的$ \ \ $ \ \ $ \，t__n^（1）的值$τ_{n，k} $显示了$ flys的最大关键值。这是最近的论文\ cite {nns2018}的延续，在其中，在$τ_{n，k} $上以$τ_{n，k} $的上限和渐近式形式是根据Alexei Shadrin的明确形式的Schaeffer- duffer-duffin Point Commainits的明确形式，该型号的主要价值不超过$ 1 $ 1,1 $ 1,11 $ 1. 我们利用了knut petras \ cite {kp1996}的结果，围绕着与超球权重函数相关的高斯正交配方的权重$w_λ（x）=（1-x^2）^{λ-1/2} $τ_{n，k}^2 $。这使我们能够证明一个下限，并为在\ cite {nns2018}中获得的$τ_{n，k} $改进上限。显式公式还承认，在\ cite {nns2018}中，对$τ_{n，k} $的新推导，以$ n \ to \ infty $。新方法更简单，而无需使用有关贝塞尔函数的求和的深刻结果，并允许更好地分析估计值的清晰度。
Here we study the quantity $$ τ_{n,k}:=\frac{|T_n^{(k)}(ω_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is the $n$-th Chebyshev polynomial of the first kind and $ω_{n,k}$ is the largest zero of $T_n^{(k+1)}$. Since the absolute values of the local extrema of $T_n^{(k)}$ increase monotonically towards the end-points of $[-1,1]$, the value $τ_{n,k}$ shows how small is the largest critical value of $\,T_n^{(k)}\,$ relative to its global maximum $\,T_n^{(k)}(1)$. This is a continuation of the recent paper \cite{NNS2018}, where upper bounds and asymptotic formuae for $τ_{n,k}$ have been obtained on the basis of Alexei Shadrin's explicit form of the Schaeffer--Duffin pointwise majorant for polynomials with absolute value not exceeding $1$ in $[-1,1]$. We exploit a result of Knut Petras \cite{KP1996} about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function $w_λ(x)=(1-x^2)^{λ-1/2}$ to find an explicit (modulo $ω_{n,k}$) formula for $τ_{n,k}^2$. This enables us to prove a lower bound and to refine the upper bounds for $τ_{n,k}$ obtained in \cite{NNS2018}. The explicit formula admits also a new derivation of the assymptotic formula in \cite{NNS2018} approximating $τ_{n,k}$ for $n\to\infty$. The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates.
