An estimate for coeffcients of polynomials in $L^2$ norm, II

G.V. Milovanovi\'c and L.Z. Ranci\'c

Abstract: 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 Milovanovi\'c 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.

Classification (MSC2000): 26C05, 26D05; 33C45, 41A44

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