How can I find the Fourier series of $f(x) = x^2$ which is a $2\pi - periodic$ function on the interval $[0,2\pi)$

Question

Find the Fourier series of $f(x) = x^2$ which is a $2\pi - periodic$ function on the interval $[0,2\pi)$

My question exactly:

what is the difference between the solution of the above question and this question:

Find the Fourier series of $f(x) = x^2$ which is a $2\pi - periodic$ function on the interval $[-\pi,\pi)$

Answer 1

Those are completely different functions. For instance, the original function given in the exercise has a supremum value of $4\pi^2$ (it doesn't even have a maximum), while your alternative function has a maximum value of $\pi^2$. The original function has positive derivative everywhere while your function has negative derivative half of the time.

Answer 2

If you make a plot of the two functions, over (let's say) three or four periods, you will have very clear what's the difference.

For getting the Fourier series in each case, it depends on what you know about "managing" the properties of the series.
Just as a hint you might consider that:

• $x^2$ is two times the integral of $x$, i.e. the Saw-Tooth (periodic ramp) function.

• otherwise, you can solve the integral of the Fourier series by parts.

Answer 3

The function $f(x):=x^2$ is neither a periodic function on the interval $(0,2\pi)$ nor a periodic function on the interval $(-\pi,\pi)$. It is a fine function on all of ${\mathbb R}$. But it is allowed to restrict $f$ to the $x$-interval $(0,2\pi)$ and then saying that we are considering the $2\pi$-periodic function $g:\>{\mathbb R}\to{\mathbb R}$ coinciding with $f$ on $(0,2\pi)$. This function $g$ has a Fourier series whose coefficients are encapsulated in a function $$\hat g:\>{\mathbb Z}\to{\mathbb C},\qquad n\mapsto c_n=\hat g(n)\ ,$$ or similar. In the same way, it is allowed to restrict $f$ to the $x$-interval $(-\pi,\pi)$ and then saying that we are considering the $2\pi$-periodic function $h:\>{\mathbb R}\to{\mathbb R}$ coinciding with $f$ on $(-\pi,\pi)$. This function $h$ has a Fourier series whose coefficients are encapsulated in a function $\hat h:\>{\mathbb Z}\to{\mathbb C}$, or similar.

The "difference" between $g$ and $h$ can be seen by making a single plot showing both functions. This difference is $2\pi$-periodic as well, and has a certain infinite Fourier expansion, which is not terribly interesting.