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# Wirtinger’s inequality

Let be a periodic function which is continuous and has a continuous derivative throughout such that . Prove that:

Solution

We are basing the whole fact on Fourier series. Since Dirichlet’s conditions are met then can be expanded into a Fourier series. Therefore we can write:

However, since the integral of vanishes we have that . Using Parseval’s identity we get that

as well as

Since all summands are positive we get the desired inequality. Equality holds if and only if .