Estimates for the Largest Critical Value of $$T_n^{(k)}$$

Abstract

For $$T_n(x)=\cos n\arccos x$$ , $$x\in [-1,1]$$ , the n-th Chebyshev polynomial of the first kind, we study the quantity $$\begin{aligned} \tau _{n,k}:=\frac{|T_n^{(k)}(\omega _{n,k})|}{T_n^{(k)}(1)},\quad 1\le k\le n-2, \end{aligned}$$ where $$T_n^{(k)}$$ is the k-th derivative of $$T_n$$ 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 our (joint with Alexei Shadrin) paper “On the largest critical value of $$T_n^{(k)}$$ ”, SIAM J. Math. Anal. 50(3), 2018, 2389–2408, where upper bounds and asymptotic formuae for $$\tau _{n,k}$$ have been obtained on the basis of the Schaeffer–Duffin pointwise upper bound for polynomials with absolute value not exceeding 1 in $$[-1,1]$$ . We exploit a 1996 result of Knut Petras 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 previously obtained upper bounds for $$\tau _{n,k}$$ . The explicit formula admits also a new derivation of the asymptotic formula approximating $$\tau _{n,k}$$ for $$n\rightarrow \infty $$ . 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.