Extremal problems of Markov's type for some differential operators (Q1890797)

The authors study Markov's type extremal problems \(c_{n, m}= \sup_{P\in {\mathcal P}_n} {|D_m P|\over |A^{m/2} P|}\) in the class \({\mathcal P}_n\) of algebraic polynomials of degree at most \(n\) endowed with \(L^2(d\sigma)\)-norm on \((a, b)\) and find the best constants \(c_{n, m}\) in the three following cases: the Legendre measure \(d\sigma(t)= dt\) on \([-1, 1]\), the Laguerre measure \(d\sigma(t)= e^{- t} dt\) on \([0, +\infty)\) and the Hermite measure \(d\sigma(t)= e^{- t^2}dt\) on \((- \infty, +\infty)\). Here \(D_m P= {d^m\over dt^m} (A^m P)\) and \(A(t)\equiv 1- t^2\) or \(t\), or else 1, respectively.