Asymptotically sharp inequalities for polynomials involving mixed Gegenbauer norms

Article type: Research Article

Authors: Langenau, Holger

Affiliations: Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany. E-mail: holger.langenau@mathematik.tu-chemnitz.de

Abstract: The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L2 on the interval (−1,1) with the weight (1−t2)α and (1−t2)β, respectively. Under the assumption that β−α is larger than the order of the derivative, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.

Keywords: Markov inequality, Gegenbauer weight, matrix norm

DOI: 10.3233/ASY-171425

Journal: Asymptotic Analysis, vol. 103, no. 4, pp. 221-233, 2017