There are several inequivalent versions of the Wirtinger inequality: Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of spectral geometry.
The middle version is also a special case of the Friedrichs inequality, again with the optimal constant identified.
In each case, by a linear change of variables in the integrals involved, there is no loss of generality in only proving the theorem for one particular choice of L. Consider the first Wirtinger inequality given above.
Furthermore, equality in the Wirtinger inequality is seen to be equivalent to y′(x) = y(x) cot x, the general solution of which (as computed by separation of variables) is y(x) = c sin x for an arbitrary number c. There is a subtlety in the above application of the fundamental theorem of calculus, since it is not the case that y(x)2 cot x extends continuously to x = 0 and x = π for every function y(x).
From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of T are (kπ)−2 for nonzero integers k, the largest of which is then π−2.
In the case that y(x) is a function for which equality in the Wirtinger inequality holds, then a standard argument in the calculus of variations says that y must be a weak solution of the Euler–Lagrange equation y′′(x) + y(x) = 0 with y′(0) = y′(1) = 0, and the regularity theory of such equations, followed by the usual analysis of ordinary differential equations with constant coefficients, shows that y(x) = c cos πx for some number c. To make this argument fully formal and precise, it is necessary to be more careful about the function spaces in question.
[2] In the language of spectral geometry, the three versions of the Wirtinger inequality above can be rephrased as theorems about the first eigenvalue and corresponding eigenfunctions of the Laplace–Beltrami operator on various one-dimensional Riemannian manifolds:[3] These can also be extended to statements about higher-dimensional spaces.
For example, the Riemannian circle may be viewed as the one-dimensional version of either a sphere, real projective space, or torus (of arbitrary dimension).