Wirtinger’s inequality

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a periodic function of period $2\pi$, which is continuous and has a continuous derivative throughout $\mathbb{R}$, and such that

 $\int_{0}^{2\pi}f(x)=0\;.$ (1)

Then

 $\int_{0}^{2\pi}f^{\prime 2}(x)dx\geq\int_{0}^{2\pi}f^{2}(x)dx$ (2)

with equality if and only if $f(x)=a\cos x+b\sin x$ for some $a$ and $b$ (or equivalently $f(x)=c\sin(x+d)$ for some $c$ and $d$).

Proof: Since Dirichlet’s conditions are met, we can write

 $f(x)=\frac{1}{2}a_{0}+\sum_{n\geq 1}(a_{n}\sin nx+b_{n}\cos nx)$

and moreover $a_{0}=0$ by (1). By Parseval’s identity,

 $\int_{0}^{2\pi}f^{2}(x)dx=\sum_{n=1}^{\infty}(a_{n}^{2}+b_{n}^{2})$

and

 $\int_{0}^{2\pi}f^{\prime 2}(x)dx=\sum_{n=1}^{\infty}n^{2}(a_{n}^{2}+b_{n}^{2})$

and since the summands are all $\geq 0$, we get (2), with equality if and only if $a_{n}=b_{n}=0$ for all $n\geq 2$.

Hurwitz used Wirtinger’s inequality in his tidy 1904 proof of the isoperimetric inequality.

Title Wirtinger’s inequality WirtingersInequality 2013-03-22 14:02:38 2013-03-22 14:02:38 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Theorem msc 42B05 Wirtinger inequality