Wirtinger's inequality for functions

For other inequalities named after Wirtinger, see Wirtinger's inequality.

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.

Theorem

First version

Let $f:{\mathbb {R}}\to {\mathbb {R}}$ 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) \, dx = 0.

Then

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

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):

\pi^{2}\int_0^a |f|^2 \le a^2 \int_0^a|f'|^2

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

f(x)=\frac{1}{2}a_0+\sum_{n\ge 1}\left(a_n\frac{\sin nx}{\sqrt{\pi}}+b_n\frac{\cos nx}{\sqrt{\pi}}\right),

and moreover a₀ = 0 since the integral of f vanishes. 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 the desired inequality, with equality if and only if a_n = b_n = 0 for all n ≥ 2.

References

Dym, H; McKean, H (1985), Fourier series and integrals, Academic press, ISBN 978-0-12-226451-1
Komkov, Vadim (1983) Euler's buckling formula and Wirtinger's inequality. Internat. J. Math. Ed. Sci. Tech. 14, no. 6, 661—668.

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