Proof of Wirtinger inequality

Question

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ \int_a^b\Big|\frac{\mathrm{d}f}{\mathrm{d}t}\Big|^2\mathrm{d}t \ge\frac{\pi^2}{(b-a)^2} \int_a^b\left|\ f\right|^2\mathrm{d}t." $$ I already found a few questions about this topic in the site, but I couldn't actually grasp what's happening here. Also, I would very much like you to show me where I go wrong with my try, which is sketched below.

I know that, for $f\in \mathcal{L}^2([0,2\pi])\supset C^1([0,2\pi])$, I can expand: $$ f(t)=\sum_{n=-\infty}^{+\infty}c_n e^{int},\ \ \ c_n=\frac{1}{2\pi}\int_0^{2\pi}\mathrm{d}t\ e^{-int}f(t). $$ Rescaling $t \to \omega (t - a)$, where $\omega = 2\pi/(b-a)$, we can get a more general form for $f\in C([a,b])$: $$ f(t)=\sum_{n=-\infty}^{+\infty}c_n e^{in\omega t},\ \ \ c_n=\frac{1}{b-a}\int_a^{b}\mathrm{d}t\ e^{-in\omega t}f(t). $$ Now, having: $$ \frac{\mathrm{d}}{\mathrm{d}t}f(t) = \sum_{n=-\infty}^{+\infty}\tilde{c}_ne^{i\omega nt},\ \ \tilde{c}_n=\frac{1}{b-a}\int_a^b\mathrm{d}t\ e^{-i\omega nt}\frac{\mathrm{d}}{\mathrm{d}t}f(t). $$ Using the fact that $\,f(a)=f(b)=0$ we get: $$ \tilde{c}_0 = \frac{1}{b-a}\int_a^b\mathrm{d}t \frac{\mathrm{d}}{\mathrm{d}t} f(t) = \frac{f(b)-f(a)}{b-a} =0 \longrightarrow \frac{\mathrm{d}}{\mathrm{d}t}f(t) = \sum_{n\in\mathbb Z\setminus\{0\}}\tilde{c}_n e^{i\omega nt} $$ Now deriving the series expansion of $f$ yields: $$ \frac{\mathrm{d}}{\mathrm{d}t}f(t) = \frac{\mathrm{d}}{\mathrm{d}t} \sum_{n=-\infty}^{+\infty}c_n e^{in\omega t} = \sum_{n=-\infty}^{+\infty}(in\omega)c_n e^{in\omega t}=\sum_{n\in\mathbb Z\setminus\{0\}}(in\omega)c_n e^{in\omega t} $$ Comparing the two expressions we establish: $\tilde{c}_n = i\omega n c_n$, for $n\not= 0$. Parseval's Equality reads here for $\mathrm{d}f/\mathrm{d}t$: $$ \int_a^b\left|\frac{\mathrm{d}f}{\mathrm{d}t}\right|^2\mathrm{d}t= \sum_{n\in\mathbb Z\setminus\{0\}} |\tilde{c_n}|^2 = \omega^2\sum_{n\in\mathbb Z\setminus\{0\}} n^2 |c_n|^2 \ge \omega^2\sum_{n\in\mathbb Z\setminus\{0\}} |c_n|^2 = \omega^2 \left(\int_a^b\mathrm{d}t\left|f(t)\right|^2-|c_0|^2\right) $$ where in the last passage we used Parseval's Equality for $f$: $\int_a^b|f|^2\mathrm{d}t=\sum_{n=-\infty}^{+\infty}|c_n|^2$.

We are now left with finding a suitable way of estimating $|c_0|^2$: $$ |c_0|^2 = \left|\frac{1}{b-a}\int_a^b\mathrm{d}t\ f(t)\right|^2, $$ thus $$ \int_a^b\left|\frac{\mathrm{d}f}{\mathrm{d}t}\right|^2\mathrm{d}t \ge \frac{4\pi^2}{(b-a)^2}\left(\int_a^b\left|f\right|^2\mathrm{d}t - \frac{1}{(b-a)^2}\left|\int_a^bf\mathrm{d}t\right|^2\right). $$ Which is not exactly what I wanted.

Please lend me a hand!

Answer 1

The standard Wirtinger's inequality requires that $\int_a^b f\,dx=0$, which implies that $c_0=0$, and hence the difficulty you are encounter does not exist.

If instead we assume that $f(a)=f(b)=0$, then we exploit this by expanding $f$ in a sine series, i.e., for $a=0$ and $b=\pi$, $$ f(x)=\sum_{n=1}^\infty a_n\sin nx, $$ and hence $$ \int_0^\pi f^2= \frac{\pi}{2}\sum_{n=1}^\infty a_n^2 \qquad \text{while} \qquad \int_0^\pi (f')^2= \frac{\pi}{2}\sum_{n=1}^\infty n^2a_n^2, $$ and therefore $$ \int_0^\pi (f')^2\ge \int_0^\pi f^2, $$ with best constant $c=1$.

For arbitrary $b>a$, $$ f(x)=\sum_{n=1}^\infty a_n\sin \left(\frac{n\pi(x-a)}{b-a}\right), $$ and hence $$ \int_a^b\!\! f^2= \frac{b-a}{2}\sum_{n=1}^\infty a_n^2 \quad \text{while} \quad \int_a^b\!\! (f')^2= \frac{b-a}{2}\sum_{n=1}^\infty \left(\frac{\pi}{b-a}\right)^2n^2a_n^2, $$ and hence $$ \int_a^b\! (f')^2\ge \frac{\pi^2}{(b-a)^2}\int_0^\pi f^2. $$

Answer 2

Here is a proof not involving Fourier Series at all.

Since $f(a) = 0$ we can write, using the fundamental theorem of calculus: $$ f(t) = \int_a^t \frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\mathrm{d}\tau,\ \ \ t\in[a,b], $$ hence $$ \left|f(t)\right| \le \int_a^t \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|\mathrm{d}{\tau}. $$ Cauchy-Schwarz inequality ensures: $$ \int_a^t \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|\mathrm{d}{\tau} \le \left(\int_a^t \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|^2\mathrm{d}{\tau}\right)^{1/2}\cdot \left(\int_a^t\mathrm{d}\tau\right)^{1/2}\\ \hspace{2.7cm}\le \left(\int_a^b \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|^2\mathrm{d}{\tau}\right)^{1/2}\cdot \left(\int_a^b\mathrm{d}\tau\right)^{1/2} \\ \hspace{2cm} =(b-a)^{1/2}\left(\int_a^b \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|^2\mathrm{d}{\tau}\right)^{1/2}. $$ Thus $$ \left|f(t)\right|^2 \le (b-a)\int_a^b \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|^2\mathrm{d}{\tau} $$

$$ \int_a^b\left|f(t)\right|^2\mathrm{d}t \le (b-a)^2 \int_a^b \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|^2\mathrm{d}{\tau} $$ $$ \int_a^b \left|\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}\right|^2\mathrm{d}{\tau} \ge \frac{1}{(b-a)^2}\int_a^b \left|f(t)\right|^2\mathrm{d}t, $$ which is a slightly less strong result.

EDIT: The fact that $f(a)=0=f(b)$ can be incorporated as follows. Let us first assume for simplicity that $a=0$ and $b=1$ and denote $||\varphi||^2=\int_0^1 \varphi(t)^2 dt$. $$ \varphi(t)=\sum_{\mathbb Z} e^{i2\pi n t} c_n\,,\qquad c_n=\int_0^1 \varphi(t) e^{-i2\pi nt}dt\,, $$ then $$ ||\varphi||^2=\sum_{\mathbb Z} |c_n|^2= |c_0|^2 + \sum_{\mathbb Z\setminus \{0\}} |c_n|^2\,,\qquad ||\dot \varphi||^2=4\pi^2 \sum_{\mathbb Z\setminus\{0\}} n^2 |c_n|^2\ge 4\pi^2 \sum_{\mathbb Z\setminus \{0\}} |c_n|^2\,. $$ On the other hand $\varphi(0)=0=\varphi(1)$ requires $$ 0=\sum_{\mathbb Z} c_n = c_0 + \sum_{\mathbb Z\setminus\{0\}} c_n $$ and using this equation to get rid of $c_0$ we obtain $$ ||\varphi||^2 = \Big|\sum_{\mathbb Z\setminus\{0\}} c_n \Big|^2+\sum_{\mathbb Z\setminus \{0\}} |c_n|^2 \le 2 \sum_{\mathbb Z\setminus \{0\}} |c_n|^2\,. $$ Comparing the two inequalities $$ ||\dot\varphi||^2 \ge 2\pi^2 ||\varphi||^2\,. $$ More generally, reinstating the dependence on the interval, $$ \int_a^b \dot f(t)^2 dt \ge \frac{2\pi^2}{(b-a)^2}\int_a^b f(t)^2dt\,. $$ Curiously enough, this approach gives an extra factor of two, unless I am missing something.