# What functions satisfy $\int\limits_{-\pi}^\pi f(x)^2\,dx = \int\limits_{-\pi}^\pi f'(x)^2\,dx$

If $f$ is a real values function that is $2\pi$-periodic and of class $C^1$.

What functions satisfy:

$$\int_{-\pi}^\pi f(x)\,dx=0$$ and

$$\int_{-\pi}^\pi f(x)^2\,dx = \int_{-\pi}^\pi f'(x)^2\,dx$$

My thought is functions of sines and cosines that are $2\pi$-periodic, are class $C^1$, and are symmetric about the $y$-axis. Am I correct and are there functions that I haven't thought of?

• If $f(x)=\sum\limits_{n\in\mathbb{Z}}\,a_n\,\exp(\text{i}nx)$ with $a_n\in\mathbb{C}$ for each $n\in\mathbb{Z}$, then the given condition is equivalent to requiring that $a_0=0$ and $\sum\limits_{n\in\mathbb{Z}_{>0}}\,a_n\,a_{-n}=\sum\limits_{n\in\mathbb{Z}_{>0}}\,n^2\,a_n\,a_{-n}$. There are an uncountable number of such functions, and I don't know how else to characterize all of them. Commented Apr 16, 2016 at 17:46

Since $f$ is real-valued, we can use Parseval's identity. $$\sum_{n=-\infty}^\infty|a_n|^2=\dfrac{1}{2\pi}\int_{-\pi}^\pi f(x)^2\,dx = \dfrac{1}{2\pi}\int_{-\pi}^\pi f'(x)^2\,dx=\sum_{n=-\infty}^\infty|na_n|^2.$$ So, it must be $a_n=0$, except from $n=1,-1$ ($a_0=0$).
Thus, $f(x)=a cos(x)+b sin(x), a,b\in \mathbb{R}.$ Also, we do not need $f$ to be of class $C^1$, just $f'$ to be of class $L^2[-\pi,\pi]$.