Estimates for the largest critical value of $T_n^{(k)}$

TL;DR

This paper refines bounds and provides explicit formulas for the largest critical value of derivatives of Chebyshev polynomials, enhancing understanding of their extremal properties with a simpler approach.

Contribution

It introduces an explicit formula for the critical value ratio using Gaussian quadrature weights, improving bounds and asymptotic analysis compared to prior work.

Findings

Refined upper and lower bounds for u_{n,k}

Explicit formula involving Gaussian quadrature weights

Simpler derivation of asymptotic behavior

Abstract

Here we study the quantity $$ \tau_{n,k}:=\frac{|T_n^{(k)}(\omega_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is the $n$-th Chebyshev polynomial of the first kind and $\omega_{n,k}$ 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 $\tau_{n,k}$ 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 $\tau_{n,k}$ 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 $w_{\lambda}(x)=(1-x^2)^{\lambda-1/2}$ to find an explicit (modulo $\omega_{n,k}$) formula for $\tau_{n,k}^2$. This enables us to prove a lower bound and to refine the upper bounds for $\tau_{n,k}$ obtained in \cite{NNS2018}. The explicit formula admits also a new derivation of the assymptotic formula in \cite{NNS2018} approximating $\tau_{n,k}$ 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.