Wirtinger's inequality

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In mathematics, historically Wirtinger's inequality was an inequality used in Fourier analysis. It was named after Wilhelm Wirtinger. It was used in 1904 to prove the isoperimetric inequality. (There is a second Wirtinger's inquality (complex analysis).)

[edit] Theorem

Let f : R → R be a periodic function of period 2π, which is continuous and has a continuous derivative throughout R, and such that

\int_0^{2\pi}f(x)=0.

Then

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

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

[edit] Proof

Since Dirichlet's conditions are met, we can write

f(x)=\frac{1}{2}a_0+\sum_{n\ge 1}(a_n\sin nx+b_n\cos nx)

and moreover a0 = 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'^2(x)dx=\sum_{n=1}^\infty n^2(a_n^2+b_n^2)

and since the summands are all ≥ 0, we get (2), with equality if and only if an = bn = 0 for all n ≥ 2.

[edit] Modern usage

During XX century many similar inequalities were proved in different functional spaces by different authors. Most of them are also called Wirtinger's inequality.

This article incorporates material from Wirtinger's inequality on PlanetMath, which is licensed under the GFDL.

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