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.

翻译：本文用于 Markov 的外表问题 $M ⁇ n, k ⁇ sup ⁇ p\ p\ p\ p\ p\ p\ p\ p\ p\ p\ p\ (k) {} {{} {} $(1\le k\ le n) 美元, 其中$\ p_ n$ 是所有代数多位数度数的集合, 以美元计, 美元为标准空间 。 开始于19世纪末 $X=C[ 1, 1] 的原始标准, 以美元=C[ 1, 美元为统一标准 。 中心部分用于空间的 $X=L2 [( a, b); / / / {p\\\ xX} 美元 $( 1\ le k\ k\ le k\ n) 美元, 美元是所有代数位数多位数的多位数多位数度数 。 。 在真实行( 类数) 或多位多位数多位数多位数调多位数 中, 的直径解的直径分析中, 的结果也是直径的直径。 。 。 的直径分析中, 。 的直径分析的直径分析结果的直径的直径。