Chapter 3 - Extremal problems of Markov–Bernstein type in integral norms

Abstract

This chapter deals with extremal problems of Markov–Bernstein type for polynomials in integral norms. Orthogonal polynomials on the real line, especially those with respect to the classical weight functions, as one of the basic tools in the study of extremal problems of this type, are treated in Section 3.1. Special attention is paid to the recurrence relations and to formulas for differentiation of the classical orthogonal polynomials (Jacobi, generalized Laguerre, and Hermite polynomials). Besides the standard extremal problems of Markov's type in L2 norm for the classical weight functions, in this chapter we consider different modifications of the weighted L2 Markov–Bernstein extremal problems and the corresponding inequalities, different generalizations in Lr norm, and the extremal problems on some restricted classes of polynomials.