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Let Pn be the class of algebraic polynomials of degree at most n and ‖P‖ = (∫ R |P (t)| 2 dσ(t) )1/2 , where dσ(t) is a measure corresponding to the classical orthogonal polynomials. We study extremal problems of Markov’s type Cn,m = sup P∈Pn ‖DmP‖ ‖Am/2P‖ , where A is given by (1.1), and the differential operator Dm is defined by (1.3). The best constants are found for Legendre, Laguerre, and Hermite measures on (−1, 1), (0,+∞), and (−∞,+∞), respectively.

Extremal Problems of Markov's Type for Some Differential Operators