Dirichlet kernel

The Dirichlet $D_{n}$ of order $n$ is defined as

$D_{n}(t)=\sum_{k=-n}^{n}e^{ikt}.$

It can be represented as

$D_{n}(t)=\frac{\sin\left(n+\frac{1}{2}\right)t}{\sin\frac{t}{2}}.$

Proof: It is

	$\displaystyle\sum_{k=-n}^{n}e^{ikt}$	$\displaystyle=e^{-int}\frac{1-e^{i(2n+1)t}}{1-e^{it}}$
		$\displaystyle=\frac{e^{i\left(n+\frac{1}{2}\right)t}-e^{-i\left(n+\frac{1}{2}% \right)t}}{e^{i\frac{t}{2}}-e^{-i\frac{t}{2}}}$
		$\displaystyle=\frac{\sin\left(n+\frac{1}{2}\right)t}{\sin\frac{t}{2}}.\qquad\qquad\Box$

The Dirichlet kernel arises in the analysis of periodic functions because for any function $f$ of period $2\pi$ , the convolution of $D_{N}$ and $f$ results in the Fourier-series approximation of order $n$ :

$(D_{N}*f)(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)D_{n}(x-y)dy=\sum_{k=-n}^{n}% \hat{f}(k)e^{ikx}.$