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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Polynomial extremal problems in $ L\sp{p}$


Author: E. Beller
Journal: Proc. Amer. Math. Soc. 30 (1971), 249-259
MSC: Primary 30.10
MathSciNet review: 0281884
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Abstract: For $ p > 2$, let $ {m_{p,n}}$ be the minimum of the $ {L^p}$ norm all nth degree polynomials $ {\sum ^n} {a_k} {e^{ikt}}$ which satisfy $ \vert{a_k}\vert = 1, k = 0, 1, \cdots, n$. We exhibit certain polynomials $ {P_n}$ whose $ {L^p}$ norm $ ( 2 < p < \infty)$ is asymptotic to $ \surd{n}$, thereby proving that $ {m_{p,n}}$ is itself asymptotic to $ \surd{n}$. We also show that the sup norm of (essentially) the same polynomials is asymptotic to $ (1.1716 \ldots) \times \surd{n}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0281884-9
PII: S 0002-9939(1971)0281884-9
Keywords: Close to constant polynomials, coefficients of constant modulus, extremal polynomials, $ {L^p}$ norms of polynomials, sup norm of polynomials, Fresnel integral, van der Corput's lemma
Article copyright: © Copyright 1971 American Mathematical Society