Introduction to Partial Differential Equations pp 285-312 | Cite as

# Convergence of Fourier Series

Chapter

## Abstract

Let
. converge to the function

*f*be a piecewise continuous function defined on [-1, 1] with a full Fourier series given by$$
\frac{{a_0 }}
{2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos \left( {k\pi x} \right) + b_k \sin \left( {k\pi x} \right)} \right).}
$$

*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.## Keywords

Fourier Series Triangle Inequality Uniform Convergence Pointwise Convergence Uniform Limit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York, Inc. 1998