# Uniform convergence of Fourier Series

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!

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.

• Perhaps you should say something regarding the lower index, since complex Fourier series starts from $i=-\infty$. – AD. Feb 1 '12 at 16:12
• But $\sum\limits_{j=1}^\infty |c_j|<\infty$ is not a given, though, is it? – Ryker Feb 15 '16 at 0:36
• @Ryker I interpreted the OP's question as "why would assuming $\sum|c_j|<\infty$ imply $f(x)$ is uniformly convergent?" – David Mitra Feb 15 '16 at 1:20
• @DavidMitra, I see. Yeah, taking another look at it I think you're right. – Ryker Feb 15 '16 at 1:24
• Isn't the last equality actually an inequality? I believe you were adapting the proof itself of the Weierstrass M-Test for this sequence of functions specifically. – DrHAL Sep 1 '17 at 14:40