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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.

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2 Answers 2

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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.

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    $\begingroup$ We can say even more! The Fourier sum is converges absolutely, so the series converges not only pointwise, but also uniformly $\endgroup$
    – Holo
    Commented Apr 15, 2018 at 4:17
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    $\begingroup$ @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. $\endgroup$
    – copper.hat
    Commented Apr 15, 2018 at 4:18
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    $\begingroup$ Indeed, since $f$ is Lipschitz, the series converges uniformly. $\endgroup$
    – copper.hat
    Commented Apr 15, 2018 at 4:20
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    $\begingroup$ For stronger results, look at the Dini and Dini Lipschitz convergence results. $\endgroup$
    – copper.hat
    Commented Apr 15, 2018 at 4:21
  • $\begingroup$ 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 $\endgroup$
    – Holo
    Commented Apr 15, 2018 at 4:35
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There are few ways to show the Fourier series converge:

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

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

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$f(x)$ 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

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