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}.