Wirtinger’s inequality

Theorem: Let f: be a periodic function of period 2π, which is continuousMathworldPlanetmathPlanetmath and has a continuous derivativePlanetmathPlanetmath throughout , and such that

02πf(x)=0. (1)


02πf2(x)𝑑x02πf2(x)𝑑x (2)

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

Proof: Since Dirichlet’s conditions are met, we can write


and moreover a0=0 by (1). By Parseval’s identity,




and since the summands are all 0, we get (2), with equality if and only if an=bn=0 for all n2.

Hurwitz used Wirtinger’s inequalityMathworldPlanetmath in his tidy 1904 proof of the isoperimetric inequalityMathworldPlanetmath.

Title Wirtinger’s inequality
Canonical name WirtingersInequality
Date of creation 2013-03-22 14:02:38
Last modified on 2013-03-22 14:02:38
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Theorem
Classification msc 42B05
Synonym Wirtinger inequality