Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight

Abstract

The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd algorithm for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov–Bernstein constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203–243, 1999) and Draux (Numer Algor 24:31–58, 2000).