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Proceedings of the American Mathematical Society

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



An extremal problem
for trigonometric polynomials

Authors: J. Marshall Ash and Michael Ganzburg
Journal: Proc. Amer. Math. Soc. 127 (1999), 211-216
MSC (1991): Primary 42A05; Secondary 41A44
MathSciNet review: 1458861
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Abstract: Let $T_{n}(x)=\sum _{k=0}^{n}(a_{k}\cos kx+b_{k}\sin kx)$ be a trigonometric polynomial of degree $n.$ The problem of finding $C_{np},$ the largest value for $C$ in the inequality $\max \{\left| a_{0}\right| ,\left| a_{1}\right| ,...,\left| a_{n}\right| ,\left| b_{1}\right| ,...,\left| b_{n}\right| \}$ $\leq (1/C)\left\| T_{n}\right\| _{p}$ is studied. We find $C_{np}$ exactly provided $p$ is the conjugate of an even integer $2s$ and $n\geq 2s-1,s=1,2,....$ For general $p,1\leq p\leq \infty ,$we get an interval estimate for $C_{np},$ where the interval length tends to $0$ as $n\ $tends to $\infty .$

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Additional Information

J. Marshall Ash
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614

Michael Ganzburg
Affiliation: Department of Mathematics, Hampton University, Hampton, Virginia 23668

Keywords: Trigonometric polynomial, inequalities between different norms, best constants
Received by editor(s): January 9, 1997
Received by editor(s) in revised form: May 12, 1997
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society