Consider a function . Recall is periodic (with period ) iff , . For example, is periodic:
recalling that for . Assume is periodic and continuous. Define, for ,
The are called the Fourier coefficients of , while the Fourier series is
Let’s try an example. Take . Of course, this isn’t periodic. However implicity it is meant that , and is the periodic extension of this mapping, i.e. , , etc. For ,
Evaluating these (real) integrals individually, it is found that
Also . Hence the Fourier series of is
implicitly having the -th term to be , not . However, immediately this a bit difficult to deal with, say if pointwise convergence were in question!
To deal with this, we derive an equivalent representation of the Fourier series. Notice that in general,
Continuing, for , . Similarly,
The same work shows
It is then immediate that . Hence
recalling that , . Consequently the equivalent expression
for the partials sums is obtained, thereby obtaining the equivalent expression
for the Fourier series of . This is much easier to work with for calculations. For example the recently calculated Fourier series can now be written as
Since and are decreasing, for pointwise convergence it is sufficient to show that
are bounded. But this is true, and a common proof is due to Dirichlet (e.g. as explained in this post–easier to show is bounded and substitute ). But more than pointwise convergence is true: Uniform convergence of the Fourier series of is guaranteed, and in fact to . This by a theorem since is continuous, periodic, and piecewise continuously differentiable.