Find the Fourier series of the function $ \ f \ $ with period $ \ 2 \pi \ $ given by $ \ f(x)=|x| $

Question

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.

Answer 1

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.

Answer 2

There are few ways to show the Fourier series converge:

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1

$f(x)$ 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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2

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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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

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