Wirtinger's inequality for functions

In mathematics, Wirtinger's inequality is an inequality used in Fourier analysis. It was used in 1904 to prove the isoperimetric inequality.

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

$\int _{0}^{2\pi }f(x)=0$. (1)

Then

$\int _{0}^{2\pi }f'^{2}(x)dx\geq \int _{0}^{2\pi }f^{2}(x)dx$ (2)

with equality iff f(x) = a sin(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

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

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 iff a_n = b_n = 0 for all n ≥ 2.

Wirtinger's inequality at PlanetMath.