Dirichlet kernel

In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as

D_{n}(x)=\sum _{k=-n}^{n}e^{ikx}=\left(1+2\sum _{k=1}^{n}\cos(kx)\right)={\frac {\sin \left(\left(n+1/2\right)x\right)}{\sin(x/2)}},

where $n$ is any nonnegative integer. The kernel functions are periodic with period $2\pi$ .

The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of $D n (x)$ with any function $f$ of period 2 $π$ is the nth-degree Fourier series approximation to $f$ , i.e., we have

(D_{n}*f)(x)=\int _{-\pi }^{\pi }f(y)D_{n}(x-y)\,dy=2\pi \sum _{k=-n}^{n}{\hat {f}}(k)e^{ikx},

where

{\widehat {f}}(k)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-ikx}\,dx

is the $k$ th Fourier coefficient of $f$ . This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.

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