Markov type polynomial inequality for some generalized Hermite weight

In: Tatra Mountains Mathematical Publications, vol. 49, no. 2
Branislav Ftorek - Mariana Marčoková

Details:

Year, pages: 2011, 111 - 118
Keywords:
Markov type inequality, weight function, generalized Hermite polynomials
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 $$W(x)=|x|^{2\alpha } { e}^{- x^2},\qquad \alpha > -{1\over 2}.$$ The classical Hermite polynomials $H_n (x)$ present the special case for $\alpha =0$.