Fourier Series - Approximation and Convergence of Fourier Series

Approximation and Convergence of Fourier Series

An important question for the theory as well as applications is that of convergence. In particular, it is often necessary in applications to replace the infinite series   by a finite one,

This is called a partial sum. We would like to know, in which sense does (S_N ƒ)(x) converge to ƒ(x) as N tends to infinity.