An estimate for coeffcients of polynomials in L² norm, ii

G.V. Milovanović   ; L.Z. Rančić

Publications de l'Institut Mathématique, _N_S_58 (1995) no. 72, p. 137

Résumé

Let ${\Cal P}_n$ be the class of algebraic polynomials $P(x)=\sum_{k=0}^na_kx^k$ of degree at most $n$ and $\|P\|_{d\sigma}= (\int_R|P(x)|^2d\sigma(x))^{1/2}$, where $d\sigma(x)$ is a nonnegative measure on $R$. We determine the best constant in the inequality $|a_k|\le C_{n,k} (d\sigma)\|P\|_{d\sigma}$, for $k=0,1,\dots,n$, when $P\in {\Cal P}_n$ and such that $P(\xi_k)=0$, $k=1,\dots,m$. The cases $C_{n,n}(d\sigma)$ and $C_{n,n-1}(d\sigma)$ were studed by Milovanović and Guessab [6]. In particular, we consider the case when the measure $d\sigma(x)$ corresponds to generalized Laguerre orthogonal polynomials on the real line.

Détail

Classification : 26C05 26D05 33C45 41A44
Keywords: polynomial