Dirichlet kernel

In mathematical analysis, the Dirichlet kernel, is the collection of periodic functions defined as

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

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

Plot restricted to one period $[-L,L],~L=\pi ,~$ of the first few Dirichlet kernels showing their convergence to one of the Dirac delta distributions of the Dirac comb.

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 $2\pi$ is the $n$th-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

$${\widehat {f}}(k)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)e^{-iky}\,dy$$

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.

The Dirichlet kernel was named for Peter Gustav Lejeune Dirichlet.