Dirichlet Kernel

The Dirichlet kernel $D_n^M$ is obtained by integrating the Character $e^{i\left\langle{\xi,x}\right\rangle{}}$ over the Ball $\vert\xi\vert\leq M$,

$$D_n^M=-{1\over 2\pi r} {d\over dr} D^M_{n-2}.$$

The Dirichlet kernel of a Delta Sequence is given by

$$\delta_n(x) \equiv {1\over 2\pi} {\sin[(n+{\textstyle{1\over 2}})x]\over \sin({\textstyle{1\over 2}}x)}.$$

The integral of this kernel is called the Dirichlet Integral $D[u]$.

See also Delta Sequence, Dirichlet Integrals, Dirichlet's Lemma