On the Markov extremal problem in the $L^2$-norm with the classical weight functions

Abstract

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a normed space, starting with original Markov's result in uniform norm on $X=C[-1,1]$ from the end of the 19th century. The central part is devoted to extremal problems on the space $X=L^2[(a,b);w]$ for the classical weights $w$ on $(-1,1)$, $(0,+\infty)$ and $(-\infty,+\infty)$. Beside a short account on basic properties of the (classical) orthogonal polynomials on the real line, the explicit formulas for expressing $k$-th derivative of the classical orthonormal polynomials in terms of the same polynomials are presented, which are important in our study of this kind of extremal problems, using methods of linear algebra. Several results for all cases of the classical weights, including algorithms for numerical computation of the best constants $M_{n,k}$, as well as their lower and upper bounds, asymptotic behaviour, etc., are also given. Finally, some results on Markov's extremal problems on certain restricted classes of polynomials are also mentioned.