Convergence of Fourier series

The convergence of Fourier series is the question of whether — and in what sense — the partial sums of a Fourier series approach the original function as more terms are included. The answer depends on both the regularity of the function and the mode of convergence. Dirichlet proved pointwise convergence for piecewise monotonic functions; Fejér proved uniform convergence of the Cesàro means for continuous functions; and Carleson's theorem (1966) established that Fourier series of L² functions converge almost everywhere. The problem is not merely technical: it shaped the development of measure theory, functional analysis, and the theory of distributions. The subtlety of convergence is encoded in the properties of the Dirichlet kernel and the Fejér kernel, which govern how partial sums approximate their target.