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An extension of the Fejér-Jackson inequality

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

Gavin Brown  and
Kun-Yang Wang
Gavin Brown
Affiliation:
Department of Mathematics The University of AdelaideAdelaide SA 5005, Australia
Kun-Yang Wang
Affiliation:
Department of Mathematics Beijing Normal UniversityBeijing 100875, China
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Abstract

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Best-possible results are established for positivity of the partial sums of Σ sin k θ (k + α)−1. In fact odd sums are positive for −1 ≤ α ≤ α0 = 2.1 …, while 2k terms are positive on the subinterval ]0, π − 2μ0π(4k +1)−1 [, μ0 = 0.8128 …. This is analagous to the Gasper extension of the Szegö-Rogosinski-Young inequality for cosine sums.

MSC classification

Secondary: 42A05: Trigonometric polynomials, inequalities, extremal problems 42A10: Trigonometric approximation 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 1 , February 1997 , pp. 1 - 12
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Askey, R. and Gasper, G., ‘Inequalities for Polynomials’, in: The Bieberbach conjecture, Math. Surveys Monographs 21 (Amer. Math. Soc., Providence, 1986) pp. 7–32.Google Scholar
[2]Brown, G. and Wilson, D. C., ‘A class of positive trigonometric sums II’, Math. Ann. 285 (1989), 57–74.CrossRefGoogle Scholar
[3]Fejér, L., ‘Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Funktion beziehen’, Monats. Math. Phys. 35 (1928), 305–344.CrossRefGoogle Scholar
[4]Gasper, G., ‘Nonnegative sums of cosine, ultraspherical and Jacobi polynomials’, J. Math. Anal. Appl. 26 (1969), 60–68.CrossRefGoogle Scholar
[5]Rogosinski, W. and Szegö, G., ‘Über die Abschnitte von Potenzreihen, die in einem Kreise beschränkt bleiben’, Math. Z. 28 (1928), 73–94.CrossRefGoogle Scholar
[6]Young, W. H., ‘On a certain series of Fourier’, Proc. London Math. Soc. 11 (1912), 357–366.Google Scholar