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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Some positive trigonometric sums

Authors: Richard Askey and John Steinig
Journal: Trans. Amer. Math. Soc. 187 (1974), 295-307
MSC: Primary 33A45
MathSciNet review: 0338481
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Abstract: Vietoris found an interesting generalization of the classical inequality $ \Sigma _{k = 1}^n(\sin \,k\theta /k) > 0,0 < \theta < \pi $. A simplified proof is given for his inequality and his similar inequality for cosine series. Various new results which follow from these inequalities include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums of ultraspherical polynomials which extend an old inequality of Fejér. Both of Vietoris' inequalities are special cases of a general problem for Jacobi polynomials, and a summary is given of known results on this problem.

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PII: S 0002-9947(1974)0338481-3
Article copyright: © Copyright 1974 American Mathematical Society