Abstract:
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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