I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand:
Given that we know the series
$f(x) = \sum c_k e^{ikx}$
converges pointwise (where the $c_k$ are the Fourier coefficients and $f$ is a periodic function), to show uniform convergence it is enough to show that
$
\sum |c_k| < \infty
$
I tried to find the result regarding uniform convergence that this comment refers to but so far I wasn't successful.
Could somebody help me and give a hint as to why this is true ? Many thanks!
Best Answer
This is just an application of the Weierstrass M-test.
It's a rather simple result, and it is edifying to prove the uniform convergence of your series directly:
Let $m> n$, then $$ \biggl| \sum_{j=1}^m c_j e^{ijx} - \sum_{j=1}^n c_j e^{ijx} \biggr| =\biggl| \sum_{j=n+1}^m c_j e^{ijx} \biggr| \le \sum_{j=n+1}^m |c_j e^{ijx} |= \sum_{j=n+1}^m |c_j |. $$ Since $\sum\limits_{j=1}^\infty |c_j|<\infty$, we can make the right hand side of the above as small as we wish provided $n$ is sufficiently large. Thus, we can make the left hand side as small as we wish, independently of $x$, as long as $n$ is sufficiently large.
It follows that $ \sum\limits_{j=1}^\infty c_j e^{ijx} $ is uniformly Cauchy, and, thus, uniformly convergent.