Absolutely convergent Fourier series

Summary

A Fourier series converges absolutely when the sum of the absolute values of its coefficients is finite. Then the series converges uniformly to a continuous function, which agrees with the original function almost everywhere. If the function is continuous, it equals the sum of its Fourier series at every point. The functions with this property form the Wiener algebra. Criteria such as Bernstein's theorem tell which smoothness forces absolute convergence.