MARKOV TYPE POLYNOMIAL INEQUALITY FOR SOME GENERALIZED HERMITE WEIGHT

Abstract

 In this paper we study some weighted polynomial inequalities of Markov type in $L^2$-norm.
We use the properties of
the system of generalized Hermite
polynomials $\{H^{(\alpha)} _n (x)\}_{n=0}^{\infty} $. The polynomials
$H^{(\alpha)} _n (x) $ are orthogonal in
$\mathbb{R}=(-\infty,\infty )$ with respect to the weight function $$
W(x)=|x|^{2\alpha } { e}^{- x^2},\ \ \alpha > -{1\over 2}. $$
The classical Hermite polynomials $H_n (x)$ present the special case
for $\alpha =0$.