PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 58(72), pp. 137142 (1995) 

An estimate for coeffcients of polynomials in $L^2$ norm, IIG.V. Milovanovi\'c and L.Z. Ranci\'cElektronski fakultet, Nis, YugoslaviaAbstract: Let ${\Cal P}_n$ be the class of algebraic polynomials $P(x)=\sum_{k=0}^na_kx^k$ of degree at most $n$ and $\P\_{d\sigma}= (\int_RP(x)^2d\sigma(x))^{1/2}$, 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 $C_{n,n1}(d\sigma)$ 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 Full text of the article:
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© 2001 Mathematical Institute of the Serbian Academy of Science and Arts
