Given above is a theorem about Fourier series converging pointwise to a periodic
function or more generally to the mid point of the jump of the function. Notice
that some sort of smoothness of the function approximated was required, the Dini
condition. It can be shown that if this sort of thing is not present, the Fourier series of a
continuous periodic function may fail to converge to it in a very spectacular manner. In
fact, Fourier series don’t do very well at converging pointwise. However, there is
another way of converging at which Fourier series cannot be beat. It is mean square
convergence.

Definition 11.6.1Let f be a function defined on an interval,

[a,b]

. Then asequence,

{gn}

of functions is said to converge uniformly to f on