Extremal polynomials for the Rogosinski--Szegő estimates of the third coefficient of nonnegative sine polynomials
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908561159749632 |
|---|---|
| author | Dmitrishin, Dmitriy Stokolos, Alexander Trebels, Walter |
| author_facet | Dmitrishin, Dmitriy Stokolos, Alexander Trebels, Walter |
| contents | In the class of normalized sine-polynomials $S(t),$ non-negative on $[0,π],$ W.Rogosinski and G.Szegő 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient $a_3.$ Their proof is based on the Lukács representation of non-negative algebraic polynomials. This method does not lead to the construction of polynomials attaining the extreme values.
We consider the corresponding problem in the framework of normalized typically real polynomials $P(z)$ on the unit disc in $\mathbb C.$ By L.Fejér's method with the additional use of the Chebyshev polynomials of the second kind and their derivatives, we regain the sharp upper and lower estimates for $a_3$ and identify the extremal polynomials. The corresponding statements for sine polynomials follow by the observation $S(t)=\text{Im}\{P(e^{it})\}$. For odd $N$ the extremizers are unique, for even $N$ there is a one-parameter family of extremizers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal polynomials for the Rogosinski--Szegő estimates of the third coefficient of nonnegative sine polynomials Dmitrishin, Dmitriy Stokolos, Alexander Trebels, Walter Complex Variables Analysis of PDEs 42A05 In the class of normalized sine-polynomials $S(t),$ non-negative on $[0,π],$ W.Rogosinski and G.Szegő 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient $a_3.$ Their proof is based on the Lukács representation of non-negative algebraic polynomials. This method does not lead to the construction of polynomials attaining the extreme values. We consider the corresponding problem in the framework of normalized typically real polynomials $P(z)$ on the unit disc in $\mathbb C.$ By L.Fejér's method with the additional use of the Chebyshev polynomials of the second kind and their derivatives, we regain the sharp upper and lower estimates for $a_3$ and identify the extremal polynomials. The corresponding statements for sine polynomials follow by the observation $S(t)=\text{Im}\{P(e^{it})\}$. For odd $N$ the extremizers are unique, for even $N$ there is a one-parameter family of extremizers. |
| title | Extremal polynomials for the Rogosinski--Szegő estimates of the third coefficient of nonnegative sine polynomials |
| topic | Complex Variables Analysis of PDEs 42A05 |
| url | https://arxiv.org/abs/2509.22238 |