On the Largest Critical Value of $T_n^(k)$

Abstract

We study the quantity τ n,k := |TFinally, we derive the exact asymptotic formulae for the quantities τ * k := lim n→∞ τ n,k and τ * * m := lim n→∞ n m/2 τn,n−m , which show that our upper bounds for τ n,k and τn,n−m are asymptotically correct with respect to the exponential terms given above.

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

“…This quantity has found applications in studying some extremal problems such as Markov-type inequalities [4], [7], [11] and the Landau-Kolmogorov inequalities for intermediate derivatives [5], [12]. Some upper bounds for τ n,k have been obtained in the recent paper [6]. The main ingredient for the results in [6] is the pointwise majorant D n,k (x) for polynomials of degree at most n with absolute value less than or equal to one in [−1, 1].…”

“…Some upper bounds for τ n,k have been obtained in the recent paper [6]. The main ingredient for the results in [6] is the pointwise majorant D n,k (x) for polynomials of degree at most n with absolute value less than or equal to one in [−1, 1]. This majorant was used by Schaeffer and Duffin [13] to obtain another proof of V. Markov's inequality.…”

“…where Tn is the n-th Chebyshev polynomial of the first kind and ω n,k is the largest zero of T n (1). This is a continuation of the recent paper [6], 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 [9] 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 2 n,k . This enables us to prove a lower bound and to refine the upper bounds for τ n,k obtained in [6]. The explicit formula admits also a new derivation of the assymptotic formula in [6] approximating τ n,k for n → ∞.…”

“…This enables us to prove a lower bound and to refine the upper bounds for τ n,k obtained in [6]. The explicit formula admits also a new derivation of the assymptotic formula in [6] 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.…”