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Keywords:
cosine polynomials; inequalities
cosine polynomials; inequalities
Summary:
We prove: If $$ \frac 12+\sum_{k=1}^{n}a_k(n)\cos (kx)\geq 0 \text{ for all } x\in [0,2\pi ), $$ then $$ 1-a_k(n)\geq \frac 12 \frac{k^2}{n^2} \text{ for } k=1,\dots ,n. $$ The constant $1/2$ is the best possible.
References:
[1] DeVore R.A.: Saturation of positive convolution operators. J. Approx. Th. 3 (1970), 410-429. MR 0271612 | Zbl 0243.42024
[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. MR 0438017
[4] Szegö G.: Koeffizientenabschätzungen bei ebenen und räumlichen harmonischen Entwicklungen. Math. Annalen 96 (1926-27), 601-632.
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