Dirichlet kernel

Summary

The Dirichlet kernel $D_N(x) = \sum_{n=-N}^{N} e^{inx}$ is the trigonometric polynomial whose convolution with a periodic function gives the $N$-th partial sum of its Fourier series. In closed form $D_N(x) = \frac{\sin((N + \frac{1}{2}) x)}{\sin(x/2)}$. Its integral over a period is constant, but the integral of its absolute value grows like $\log N$. This growth is the source of most convergence problems of Fourier series, including the existence of continuous functions with divergent Fourier series.