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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An $ l\sb{1}$ extremal problem for polynomials

Authors: E. Beller and D. J. Newman
Journal: Proc. Amer. Math. Soc. 29 (1971), 474-481
MSC: Primary 30.10; Secondary 42.00
MathSciNet review: 0280688
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Abstract: Let $ {\mathfrak{M}_n}$ be the maximum of the $ {l_1}$ norm, $ \mathop \sum \nolimits^n \vert{c_k}\vert$, of all nth degree polynomials satisfying $ \vert\mathop \sum \nolimits^n {c_k}{z^k}\vert \leqq 1$ for $ \vert z\vert = 1$. We prove that $ {\mathfrak{M}_n}$ is asymptotic to $ \surd n$, by exhibiting polynomials $ {P_n}$ (which are partial sums of certain Fourier series), whose $ {l_1}$ norm is asymptotic to $ \surd n$.

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Keywords: Close to constant polynomials, coefficients of close to constant modulus, extremal polynomials, $ {l_1}$ norm of polynomials, partial sums of Fourier series, upper bound for kth derivative
Article copyright: © Copyright 1971 American Mathematical Society

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