Geodesic
On the Normalizing Multiplier of the Generalized Jackson Kernel
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 20-28.

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We consider the question of evaluating the normalizing multiplier $$ \gamma_{n,k} = \frac1 \pi \int_{-\pi}^\pi {\biggl(\frac{\sin\frac{n t}2}{\sin\frac t 2}\biggr)}^{2k}\,dt $$ for the generalized Jackson kernel $J_{n,k}(t)$. We obtain the explicit formula $$ \gamma_{n,k} = 2 \sum_{p=0}^{[k-\frac k n]} (-1)^p \binom{2k}p \binom{k(n+1) - np - 1}{k(n-1) - np} $$ and the representation $$ \gamma_{n,k} = \sqrt{\frac{24}{\pi}}\cdot\frac {(n-1)^{2k-1}}{\sqrt{2k-1}}\left[ 1 - \frac 1{8}\cdot\frac{1}{2k-1} + \omega(n,k)\right], $$ where $$ |{\omega(n,k)}|\frac{4}{(2k-1)\sqrt{\ln(2k-1)}}+ \sqrt{12\pi}\cdot\frac{k^\frac{3}{2}}{n-1}\left(1+ \frac{1}{n-1}\right)^{2k-2}. $$
Keywords: approximation theory, generalized Jackson kernel. 
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M. S. Viazovskaya; N. S. Pupashenko. On the Normalizing Multiplier of the Generalized Jackson Kernel. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 20-28. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a2/

[1] N. I. Akhiezer, Lektsii po teorii aproksimatsii, Nauka, M., 1965 | MR

[2] R. A. DeVore, G. G. Lorentz, Constructive approximation, Springer-Verlag, Berlin, 1993 | MR

[3] V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[4] N. P. Korneichuk, Ekstremalnye zadachi teorii priblizheniya, Nauka, M., 1976 | MR

[5] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[6] A. F. Timan, Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[7] I. A. Shevchuk, Priblizhenie mnogochlenami i sledy nepreryvnykh na otrezke funktsii, Naukova dumka, Kiev, 1992

[8] S. B. Stechkin, “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Izv. AN. SSSR. Ser. matem., 15:3 (1951), 219–242 | Zbl

[9] A. V. Skorokhod, N. P. Slobodyanyuk, Predelnye teoremy dlya sluchainykh bluzhdanii, Naukova dumka, Kiev, 1970 | MR | Zbl