Dirichlet kernel

The Dirichlet kernel

named after Johann Peter Gustav Lejeune Dirichlet, is 2π times the nth-degree Fourier series approximation to a "function" with period 2π given by

where δ is the Dirac delta function, which is not really a function, in the sense of mapping one set into another, but is rather a "generalized function", also called a "distribution". In other words, the Fourier series representation of this "function" is

This "periodic delta function" is the identity element for the convolution defined on functions of period 2π by

In other words, we have

for every function f of period 2π. The convolution of D_n(x) with any function f of period 2π is the nth-degree Fourier series approximation to f, i.e., we have

where

is the kth Fourier coefficient of f.

The trigonometric identity displayed at the top of this article may be established as follows. First recall that the sum of a finite geometric series is

The first term is a; the common ratio by which each term is multiplied to get the next is r; the number of terms is n + 1. In particular, we have

The expression to the left of "=" should make us expect the sum to be a symmetric function of r and 1/r, but the expression to the right of "=" is perhaps less-than-obviously symmetric in those two quantities. The remedy is to multiply both the numerator and the denominator by r^-1/2, getting

In case r = e^ix we have

and then "-2i" cancels.