Regarding Wirtinger's inequality

[psie]

TL;DR

I have a doubt about about an assumption in Wirtinger's inequality.

Consider the following problem:

Let $f$ be a continuous real-valued function on $0<x<\pi$ such that $f(0)=f(\pi)=0$ and $f' \in L^2([0,\pi])$.
a) Prove that $$\int_0^{\pi}(f(x))^2 dx \leq \int_0^{\pi} (f'(x))^2 dx.$$
b) For what functions does equality hold?

What I struggle with in this exercise is why $f'$ is in $L^2([0,\pi])$. Does this make sense? The way I prove this inequality is that I extend $f$ to an odd function on $[-\pi,\pi]$. I find its Fourier series, namely $f(x)\sim\sum_{n=1}^\infty b_n\sin nx$. Then, since $f'$ will be even, its Fourier coefficients are $$a_n=\frac{2}{\pi}\int_0^\pi f'(x)\cos(nx)dx=\frac{2n}{\pi }\int_0^\pi f(x)\sin(nx)=nb_n,$$ since $f(0)=f(\pi)=0$ by partial integration. Then I use Parseval's formula $$\int_0^\pi f^2= \frac{\pi}{2}\sum_{n=1}^\infty b_n^2 \quad \text{and} \quad \int_0^\pi (f')^2= \frac{\pi}{2}\sum_{n=1}^\infty n^2b_n^2.$$ And the inequality follows from this. Equality holds for $a_n=0$ for $n\neq 1$, i.e. when $f(x)=C\sin x$.

In order for this solution to work, we need $f'$ to be integrable, right? Does $f'\in L^2([0,\pi])$ guarantee this?

 

[fresh_42]

In order for this solution to work, we need $f'$ to be integrable, right? Does $f'\in L^2([0,\pi])$ guarantee this?

We need square integrability of $f'$ to even phrase the statement and $f'\in L^2$ is just that.

 

[psie]

We need square integrability of $f'$ to even phrase the statement and $f'\in L^2$ is just that.

I oversaw something there In my solution, however, I also need $a_n=\frac2{\pi}\int_0^\pi f’(x)\cos(nt)dt$ to be well-defined. Does $f’\in L^2$ ensure this?

 

[martinbn]

I oversaw something there In my solution, however, I also need $a_n=\frac2{\pi}\int_0^\pi f’(x)\cos(nt)dt$ to be well-defined. Does $f’\in L^2$ ensure this?

Cauchy-Schwarz

 

[fresh_42]

I know a version with the condition $\int f=0,$ see the solution to the January 2020 challenge
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
that refers to the Dirichlet conditions to guarantee the Fourier series.