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\begin{center}
\bf The Coefficients of Generalized Jackson Kernel
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The generalized Jackson kernels are of the form
$$
\mathcal{K}_{m,2r}(\theta)  \equiv \frac{k_{m,2r}}{m^{2r-1}} \left[
\frac{\sin(\frac{m \theta}{2})}{\sin(\frac{\theta}{2})}\right]^{2r}, \quad r = 1, 2, \ldots.
$$
When $r = 1$, we have
$$
\mathcal{K}_{m,2}(\theta) = \sum_{j=-m+1}^{m-1} b_j^{(m,2)} e^{\ii j \theta} =
b_0^{(m,2)} + 2 \sum_{j=1}^{m-1} b_j^{(m,2)} \cos (j \theta),
$$
where
$$
b_j^{(m,2)} = \frac{m-|j|}{m}, \quad j = 0, \pm 1, \pm 2, \ldots, \pm (m-1).
$$
For general $r$, we have
\begin{eqnarray}
\mathcal{K}_{m,2r}(\theta) & = & \sum_{j=-r(m-1)}^{r(m-1)} b_j^{(m,2r)} e^{\ii j \theta} \label{eq1} \\
& = & b_0^{(m,2r)} + 2 \sum_{j=1}^{r(m-1)} b_j^{(m,2r)} \cos (j \theta). \label{eq2}
\end{eqnarray}

Here we present three algorithms to compute the coefficients $b_j^{(m,2r)}$, $j = 0, 1, \ldots,r(m-1)$.
\begin{enumerate}
\item The code convol.m computes the coefficients by Convoluting the vector
$$
(b_{-m+1}^{(m,2)}, b_{-m+2}^{(m,2)}, \ldots, b_{0}^{(m,2)}, b_{1}^{(m,2)}, \ldots,b_{m-1}^{(m,2)})
$$
with itself for $r-1$ times, as has been presented in the book.

\item The code jackson$\_$fft.m, which was written by Szymon Modzelewski, computes the coefficients
by interpolating $\mathcal{K}_{m,2r}(\theta)$ (in form (\ref{eq1}))
at the grid points $\frac{2j\pi}{k_1}$ ($j=0,1,\ldots,k_1-1$),
where $k_1 = 2^l$ with $ 2^{l-1} < 2r(m-1)+1 \leq 2^l$.

\item The code jackson$\_$dct.m, which was
written by Furong Lin, computes the coefficients
by interpolating $\mathcal{K}_{m,2r}(\theta)$ (in form (\ref{eq2}))
at the grid points $\frac{(2j+1)\pi}{2 k_2}$ ($j=0,1,\ldots,k_2-1$). Here $k_2 = 2^l$
with $ 2^{l-1} < r(m-1)+1 \leq 2^l$. Note that $k_2 = k_1/2$.
\end{enumerate}

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