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


Authors: Gradimir V. Milovanović and Allal Guessab
Journal: Proc. Amer. Math. Soc. 120 (1994), 165-171
MSC: Primary 41A17
MathSciNet review: 1189749
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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 $ \vert\vert P\vert{\vert _{d\sigma }} = {({\smallint _\mathbb{R}}\vert P(x){\vert^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 $ \vert{a_\nu }\vert \leqslant {C_{n,\nu }}(d\sigma )\vert\vert P\vert{\vert _{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}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1189749-0
Keywords: Algebraic polynomials, coefficient estimates, orthogonal polynomials, classical orthogonal polynomials, nonnegative measure, norm, extremal polynomial, best constant
Article copyright: © Copyright 1994 American Mathematical Society