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Find the Fourier series of the function $ \ f \ $ with period $ \ 2 \pi \ $ given by $ \ f(x)=|x| , \ \ x \in [-\pi,\pi] \ $.

Does the Fourier series converges?

Answer:

I have found the Fourier series to $ \ f(x) \sim \large \frac{\pi^2}{2}+\sum_{k=2n-1}^{\infty} \frac{-4}{\pi k^2} \ \cos (kx) \ $

Apparently , I can see that the series is convergent by Comparison test with convergent series $ \ \sum_{n=1}^{\infty} \frac{1}{n^2} \ $

But how to conclude that the obtained Fourier series is convergent from the point of view of Fourier convergence?

help me out.

MAS
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2 Answers2

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Here is one anywhere for pointwise convergence.

Carleson's theorem states that if $f $ is an $L^p$ periodic function with $p \in (1,\infty)$ then the (symmetric) partial sums of the Fourier series converge pointwise to $f$ for ae. point.

Since both $f$ and the Fourier sum are continuous everywhere it follows that the sum converges pointwise everywhere.

copper.hat
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    We can say even more! The Fourier sum is converges absolutely, so the series converges not only pointwise, but also uniformly – ℋolo Apr 15 '18 at 04:17
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    @Holo: There are many stronger convergence results but I think Carleson's a useful one that doesn't get much press when Fourier methods are taught. – copper.hat Apr 15 '18 at 04:18
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    Indeed, since $f$ is Lipschitz, the series converges uniformly. – copper.hat Apr 15 '18 at 04:20
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    For stronger results, look at the Dini and Dini Lipschitz convergence results. – copper.hat Apr 15 '18 at 04:21
  • I agree with that that Carlson's theorem can be useful but in this case I think that it is not the most useful, I always compare how useful the method to how hard to achieve it, and as OP already proved its convergence (and, although OP didn't add it, its absolute convergence) I think that the most natural way to go is that, but this is, of course, my own opinion. Also, $|x|$ is hölder class at this domain so Dini Lipschitz comes automatically – ℋolo Apr 15 '18 at 04:35
  • @Holo: There are a few results that are not immediate (Brouwer, Carleson, etc.) but are worth knowing early on. (Besides, the Dini Lipschitz criterion is not entirely trivial!) Feel free to add your own answer as an illustration of other approaches. – copper.hat Apr 15 '18 at 04:41
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There are few ways to show the Fourier series converge:

1

$f(x)$ is continuous$\implies$ converges almost everywhere. @copper.hat talked about this and explained that this case it is continuous pointwise everywhere.

2

From $1$ we can get stronger result, the Fourier coefficient are absolutely converge. This implies that the Fourier series converge uniformly.

3

$f(x)$ is Hölder continuous.

Dini-Lipschitz test gives us that any Hölder continuous function with $\alpha>0$ has uniformly converges Fourier series. Even more, $f$ is Lipschitz

There are even more ways to prove this, I suggest that if you interest in this read about the above theorems and others

ℋolo
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