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# Dirichlet kernel

The *Dirichlet kernel* $D_{n}$ 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 $n$:

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

Related:

ExampleOfTelescopingSum

Type of Math Object:

Definition

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Reference

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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

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new image: plot W(t) = P(waiting time <= t) by robert_dodier