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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the $ L\sp 2$ inequalities involving trigonometric polynomials and their derivatives


Author: Weiyu Chen
Journal: Trans. Amer. Math. Soc. 347 (1995), 1753-1761
MSC: Primary 42A05; Secondary 41A17
DOI: https://doi.org/10.1090/S0002-9947-1995-1254834-7
MathSciNet review: 1254834
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Abstract: In this note we study the upper bound of the integral

$\displaystyle \int_0^\pi {{{({t^{(k)}}(x))}^2}w(x)} dx$

where $ t(x)$ is a trigonometric polynomial with real coefficients such that $ \left\Vert t \right\Vert\infty \leqslant 1$ and $ w(x)$ is a nonnegative function defined on $ [0,\pi ]$. When $ w(x) = \sin ^jx$, where $ j$ is a positive integer, we obtain the exact upper bound for the above integral.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1254834-7
Keywords: Bernstein-type inequalities, trigonometric polynomials
Article copyright: © Copyright 1995 American Mathematical Society