An estimate for coefficients of polynomials in $L^ 2$-norm

Abstract:

Let ${\mathcal {P}_n}$ be the class of algebraic polynomials $P(x) = \sum \nolimits _{v = 0}^n {{a_\nu }{x^\nu }}$ of degree at most $n$ and $||P|{|_{d\sigma }} = {({\smallint _\mathbb {R}}|P(x){|^2}d\sigma (x))^{1/2}}$, where $d\sigma (x)$ is a nonnegative measure on $\mathbb {R}$. We determine the best constant in the inequality $|{a_\nu }| \leqslant {C_{n,\nu }}(d\sigma )||P|{|_{d\sigma }}$, for $\nu = n$ and $\nu = n - 1$, when $P \in {\mathcal {P}_n}$ and such that $P({\xi _k}) = 0,\;k = 1, \ldots ,m$. The case $d\sigma (t) = dt$ on $[ - 1,1]$ and $P(1) = 0$ was studied by Tariq. In particular, we consider the cases when the measure $d\sigma (x)$ corresponds to the classical orthogonal polynomials on the real line $\mathbb {R}$.