In mathematics, historically Wirtinger's inequality for real functions was an inequality used in Fourier analysis. It was named after Wilhelm Wirtinger. It was used in 1904 to prove the isoperimetric inequality. A variety of closely related results are today known as Wirtinger's inequality.
Contents
First version
Let
Then
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).
This version of the Wirtinger inequality is the one-dimensional Poincaré inequality, with optimal constant.
Second version
The following related inequality is also called Wirtinger's inequality (Dym & McKean 1985):
whenever f is a C1 function such that f(0) = f(a) = 0. In this form, Wirtinger's inequality is seen as the one-dimensional version of Friedrichs' inequality.
Proof
The proof of the two versions are similar. Here is a proof of the first version of the inequality. Since Dirichlet's conditions are met, we can write
and moreover a0 = 0 since the integral of f vanishes. By Parseval's identity,
and
and since the summands are all ≥ 0, we get the desired inequality, with equality if and only if an = bn = 0 for all n ≥ 2.