Chemistry - Wirtinger's inequality

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

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

Since Dirichlet's conditions are met, we can write

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

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.