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# Wirtinger’s inequality

Theorem: 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.

Keywords:

parseval, isoperimetric

Synonym:

Wirtinger inequality

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

42B05*no label found*

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