where n is any nonnegative integer. The kernel functions are periodic with period .
The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(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. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.