On the exact constant in the L2 Markov inequality

In this note we consider the classical extremal problem of estimating the L-2-norm of the derivative of an algebraic polynomial when its norm is given. For the supremum norm the corresponding extremal problem was solved by A.A. Markov, but finding the exact Markov-type inequality in the L-2-case turned out to be much more difficult. Some descriptions of the exact L-2-Markov constant based on spectral analysis can be found in Hille et al. [On some generalizations of a theorem of A. Markoff, Duke. Math. J. 3 (1937) 729-739] and Rahman and Schmeisser [Analytic Theory of Polynomials, London Mathematical Society Monographs, Clarendon Press, Oxford, 2002]. In this short note we present a simple new elementary method for treating the L-2-Markov problem which leads to a new representation of the extremal polynomials and also yields the same equation for the best Markov constant which was found in Hille et al. [On some generalizations of a theorem of A. Markoff, Duke. Math. J. 3 (1937) 729-739] using matrix analysis. (c) 2007 Elsevier Inc. All rights reserved.