Convergence of Fourier Series

Abstract

Let f be a piecewise continuous function defined on [-1, 1] with a full Fourier series given by

$$ \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos \left( {k\pi x} \right) + b_k \sin \left( {k\pi x} \right)} \right).} $$

. converge to the function f ?” If we here refer to convergence in the mean square sense, then a partial answer to this question is already established by Theorem 8.2. At least we have seen that we have convergence if and only if the corresponding Parseval’s identity holds. However, we like to establish convergence under assumptions which are easier to check. Also, frequently we are interested in notions of convergence other than convergence in the mean.