Asymptotics of the best constant in a certain Markov-type inequality
Let \(\gamma_n(\alpha)\) be the best possible constant in the Markov-type inequality: \(| | p^\prime | | \leq c | | p | | , p \in \Pi_n \)( the space of complex polynomials of degree not exceeding \(n\)), where \(| | . | | \) be the weighted \(L^2\)-norm with Laguerre weight : \(w(t) = t^\alpha e^{-t}, \alpha > -1.\) \textit{P. Turán} [Mathematica, Cluj 2(25), 373--378 (1960; Zbl 0101.04702)] found the exact value of \(\gamma_{n(0)}\) which in particular implied that (*): \( lim \frac{\gamma_{n(0)}}{n} = \frac{2}{\pi}\) as \(n\) tends to infinity. However for arbitrary \(\alpha > -1\), very little is known [see P. Dörfler, Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 200, No. 1--10, 13--20 (1991; Zbl 0748.41009)]. The author obtains here the value of \( lim \frac{\gamma_n(\alpha)}{ n}\) as \(n\) tends to infinity, in terms of the first positive zero of a certain Bessel function of the first kind. In particular when \(\alpha = 0.\) this result gives the above mentioned result (*). It is also oberved that in general it is very difficult to obtain the value of \(\gamma_n(\alpha)\).
scientific article; zbMATH DE number 1763610
scientific article