Markov-Bernstein type inequalities for polynomials

Bull. Korean Math. Soc. 1999 Vol. 36, No. 1, 63-78

K. H. Kwon and D. W. Lee KAIST, Kyungpook National University

Abstract : Let $\mu(x)$ be an increasing function on the real line with finite moments of all orders. We show that for any linear operator $T$ on the space of polynomials and any integer $n\ge 0$, there is a constant $\gamma_n(T) \ge 0 $, independent of $p(x)$, such that $$ \| Tp \| \le \gamma_n(T) \, \| p \| , $$ for any polynomial $p(x)$ of degree $\le n$, where $$ \| p \| = \biggl \{ \int_{-\infty}^\infty | p (x) |^2 \, d \mu(x) \biggr \} ^{\frac 1 2} . $$ We find a formula for the best possible value $\Gamma_n(T)$ of $\gamma_n(T)$ and estimations for $\Gamma_n(T)$. We also give several illustrating examples when $T$ is a differentiation or a difference operator and $d\mu(x)$ is an orthogonalizing measure for classical or discrete classical orthogonal polynomials.

Keywords : Markov-Bernstein type inequality, orthonormal polynomials, linear operator