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[parent] Wirtinger's inequality (Theorem)

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

$\displaystyle \int_0^{2\pi}f(x)=0\;.$ (1)

Then
$\displaystyle \int_0^{2\pi}f'^2(x)dx\ge\int_0^{2\pi}f^2(x)dx$ (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

$\displaystyle f(x)=\frac{1}{2}a_0+\sum_{n\ge 1}(a_n\sin nx+b_n\cos nx)$
and moreover $ a_0=0$ by (1). By Parseval's identity,
$\displaystyle \int_0^{2\pi}f^2(x)dx=\sum_{n=1}^\infty(a_n^2+b_n^2)$
and
$\displaystyle \int_0^{2\pi}f'^2(x)dx=\sum_{n=1}^\infty n^2(a_n^2+b_n^2)$
and since the summands are all $ \ge 0$, we get (2), with equality if and only if $ a_n=b_n=0$ for all $ n\ge 2$.

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



"Wirtinger's inequality" is owned by rspuzio. [ full author list (3) | owner history (4) ]
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Other names:  Wirtinger inequality
Keywords:  parseval, isoperimetric

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Cross-references: isoperimetric inequality, Parseval's identity, Dirichlet's conditions, equality, derivative, continuous, period, periodic function
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This is version 6 of Wirtinger's inequality, born on 2003-10-16, modified 2006-11-26.
Object id is 5393, canonical name is WirtingersInequality.
Accessed 5141 times total.

Classification:
AMS MSC: 42B05 (Fourier analysis :: Fourier analysis in several variables :: Fourier series and coefficients)
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