Fejer kernelFejerKernel2004-02-26 06:44:322007-06-05 06:56:07Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe Fejer kernel $F_n$ of order $n$ is defined as
$$F_n(t)=\frac{1}{n}\sum_{k=0}^{n-1}D_k(t),$$
where $D_n$ is the Dirichlet kernel of order $n$. The Fejer kernel can be written as
\begin{equation}\label{eq:rep}
F_n(t)=\frac{1}{n}\left(\frac{\sin\frac{nt}{2}}{\sin\frac{t}{2}}\right)^2.
\end{equation}
\textbf{Proof:} Since
$$D_n(t)=\frac{\sin\left(\left(n+\frac{1}{2}\right)t\right)}{\sin\frac{t}{2}}$$
we have
$$\sin\frac{t}{2}D_n(t)=\sin\left(\left(n+\frac{1}{2}\right)t\right).$$
Therefore
\begin{align*}
n\sin^2\frac{t}{2}F_n(t)& =\sum_{k=0}^{n-1}\sin\left(\left(k+\frac{1}{2}\right)t\right)\sin\frac{t}{2}\\
&=\frac{1}{2}\sum_{k=0}^{n-1}(\cos kt-\cos((k+1)t)\\
&=\frac{1}{2}(1-\cos nt)\\
&=\sin^2\frac{nt}{2}.
\end{align*}
From this follows equation (\ref{eq:rep}).
\begin{figure}[h]
\begin{centering}
\includegraphics[scale=0.5]{fejer2.ps}
\caption{Graphs of some Fejer kernels}\label{fig:fejer}
\end{centering}
\end{figure}