2021/11/01 by Gradimir V. Milovanović, Milovanović, Gradimir V.
Mathematics · Computer Science · #Mathematical functions and polynomials #Polynomial and algebraic computation #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2111.01094
This paper is devoted to Markov's extremal problems of the form\nMn,k=\sup_p\∈ PPn\∖ 0 \‖p(k)\‖X/\‖p\‖X (1\≤\nk\≤ n), where PPn is the set of all algebraic polynomials of degree at\nmost n and X is a normed space, starting with original Markov's result in\nuniform norm on X=C[-1,1] from the end of the 19th century. The central part\nis devoted to extremal problems on the space X=L2[(a,b);w] for the classical\nweights w on (-1,1), (0,+\∞) and (-\∞,+\∞). Beside a short\naccount on basic properties of the (classical) orthogonal polynomials on the\nreal line, the explicit formulas for expressing k-th derivative of the\nclassical orthonormal polynomials in terms of the same polynomials are\npresented, which are important in our study of this kind of extremal problems,\nusing methods of linear algebra. Several results for all cases of the classical\nweights, including algorithms for numerical computation of the best constants\nMn,k, as well as their lower and upper bounds, asymptotic behaviour, etc.,\nare also given. Finally, some results on Markov's extremal problems on certain\nrestricted classes of polynomials are also mentioned.\n