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

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

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

The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of ${\displaystyle D_{n}(x)}$ with any function ${\displaystyle f}$ of period ${\displaystyle 2\pi }$ is the ${\displaystyle n}$th-degree Fourier series approximation to ${\displaystyle f}$, i.e., we have

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

where

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

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