| Title:
|
An inequality for the coefficients of a cosine polynomial (English) |
| Author:
|
Alzer, Horst |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
36 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
427-428 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
cosine polynomials |
| Keyword:
|
inequalities |
| MSC:
|
26D05 |
| MSC:
|
42A05 |
| idZBL:
|
Zbl 0833.26012 |
| idMR:
|
MR1364482 |
| . |
| Date available:
|
2009-01-08T18:19:01Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118770 |
| . |
| Reference:
|
[1] DeVore R.A.: Saturation of positive convolution operators.J. Approx. Th. 3 (1970), 410-429. Zbl 0243.42024, MR 0271612 |
| Reference:
|
[2] Stark E.L.: Über trigonometrische singuläre Faltungsintegrale mit Kernen endlicher Oszillation.Dissertation, TH Aachen, 1970. |
| Reference:
|
[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 |
| Reference:
|
[4] Szegö G.: Koeffizientenabschätzungen bei ebenen und räumlichen harmonischen Entwicklungen.Math. Annalen 96 (1926-27), 601-632. |
| . |
