Markov Inequalities for Weight Functions of Chebyshev Type

✍ Scribed by D.K. Dimitrov

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 159 KB
• Volume: 83
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1115

✦ Synopsis

Denote by (\eta_{i}=\cos (i \pi / n), i=0, \ldots, n) the extreme points of the Chebyshev polynomial (T_{n}(x)=\cos (n \operatorname{arc} \cos x)). Let (\pi_{n}) be the set of real algebraic polynomials of degree not exceeding (n), and let (B_{n}) be the unit ball in the space (\pi_{n}) equipped with the discrete norm (|p|{n . x}:=\max {0 \leqslant i \leqslant n}\left|p\left(\eta{i}\right)\right|). We prove that the unique solutions of the extremal problems (\max {p \in B{n}} \int{-1}^{1}\left[p^{\prime k+1)}(x)\right]^{2}\left(1-x^{2}\right)^{k-1 / 2} d x, k=0, \ldots, n-1), and (\max {p \in B{n}} \int_{-1}^{1}\left[p^{1 k+21}(x)\right]^{2}\left(1-x^{2}\right)^{k-1 / 2} d x, k=0, \ldots, n-2), are (p(x)= \pm T_{n}(x)), and we obtain the extremal values in an explicit form. 1995 Academic Press, Inc