# An inequality for the coefficients of a cosine polynomial

• Volume: 36, Issue: 3, page 427-428
• ISSN: 0010-2628

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## Abstract

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We prove: If $\frac{1}{2}+\sum _{k=1}^{n}{a}_{k}\left(n\right)cos\left(kx\right)\ge 0\phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}x\in \left[0,2\pi \right),$ then $1-{a}_{k}\left(n\right)\ge \frac{1}{2}\frac{{k}^{2}}{{n}^{2}}\phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}k=1,\cdots ,n.$ The constant $1/2$ is the best possible.

## How to cite

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Alzer, Horst. "An inequality for the coefficients of a cosine polynomial." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 427-428. <http://eudml.org/doc/247706>.

@article{Alzer1995,
abstract = {We prove: If $\frac\{1\}\{2\}+\sum \_\{k=1\}^\{n\}a\_k(n)\cos (kx)\ge 0 \text\{ for all \} x\in [0,2\pi ),$ then $1-a\_k(n)\ge \frac\{1\}\{2\} \frac\{k^2\}\{n^2\} \text\{ for \} k=1,\dots ,n.$ The constant $1/2$ is the best possible.},
author = {Alzer, Horst},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cosine polynomials; inequalities; inequality; coefficients; cosine polynomial},
language = {eng},
number = {3},
pages = {427-428},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An inequality for the coefficients of a cosine polynomial},
url = {http://eudml.org/doc/247706},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Alzer, Horst
TI - An inequality for the coefficients of a cosine polynomial
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 3
SP - 427
EP - 428
AB - We prove: If $\frac{1}{2}+\sum _{k=1}^{n}a_k(n)\cos (kx)\ge 0 \text{ for all } x\in [0,2\pi ),$ then $1-a_k(n)\ge \frac{1}{2} \frac{k^2}{n^2} \text{ for } k=1,\dots ,n.$ The constant $1/2$ is the best possible.
LA - eng
KW - cosine polynomials; inequalities; inequality; coefficients; cosine polynomial
UR - http://eudml.org/doc/247706
ER -

## References

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1. DeVore R.A., Saturation of positive convolution operators, J. Approx. Th. 3 (1970), 410-429. (1970) Zbl0243.42024MR0271612
2. Stark E.L., Über trigonometrische singuläre Faltungsintegrale mit Kernen endlicher Oszillation, Dissertation, TH Aachen, 1970.
3. Stark E.L., Inequalities for trigonometric moments and for Fourier coefficients of positive cosine polynomials in approximation, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544-576 (1976), 63-76. (1976) MR0438017
4. Szegö G., Koeffizientenabschätzungen bei ebenen und räumlichen harmonischen Entwicklungen, Math. Annalen 96 (1926-27), 601-632. (1926-27)

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