Estimates for the Largest Critical Value of Tn(k)

For T n ( x ) = cos n arccos x , x ∈ [ - 1 , 1 ] , the n -th Chebyshev polynomial of the first kind, we study the quantity τ n , k : = | T n ( k ) ( ω n , k ) | T n ( k ) ( 1 ) , 1 ≤ k ≤ n - 2 , where T n ( k ) is the k -th derivative of T n 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 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 τ 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 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 previously obtained upper bounds for τ n , k . The explicit formula admits also a new derivation of the asymptotic formula approximating τ n , k for n → ∞ . 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.