Wirtinger's inequality for functions

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

In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today known as Wirtinger's inequality, all of which can be viewed as certain forms of the Poincaré inequality.

Theorem

There are several inequivalent versions of the Wirtinger inequality:

• Let Template:Mvar be a continuous and differentiable function on the interval Template:Closed-closed with average value zero and with y(0) = y(L). Then

$\underset{0}{\overset{L}{\int }}y(x{)}^2\mathrm{d}x\le \frac{L^2}{4{\pi }^2}\underset{0}{\overset{L}{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x,$

and equality holds if and only if y(x) = c sin Template:Sfrac for some numbers Template:Mvar and α.Template:Sfnm

• Let Template:Mvar be a continuous and differentiable function on the interval Template:Closed-closed with y(0) = y(L) = 0. Then

$\underset{0}{\overset{L}{\int }}y(x{)}^2\mathrm{d}x\le \frac{L^2}{{\pi }^2}\underset{0}{\overset{L}{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x,$

and equality holds if and only if y(x) = c sin Template:Sfrac for some number Template:Mvar.Template:Sfnm

• Let Template:Mvar be a continuous and differentiable function on the interval Template:Closed-closed with average value zero. Then

$\underset{0}{\overset{L}{\int }}y(x{)}^2\mathrm{d}x\le \frac{L^2}{{\pi }^2}\underset{0}{\overset{L}{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x.$

and equality holds if and only if y(x) = c cos Template:Sfrac for some number Template:Mvar.Template:Sfnm

Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of spectral geometry. They can also all be regarded as special cases of various forms of the Poincaré inequality, with the optimal Poincaré constant identified explicitly. The middle version is also a special case of the Friedrichs inequality, again with the optimal constant identified.

Proofs

The three versions of the Wirtinger inequality can all be proved by various means. This is illustrated in the following by a different kind of proof for each of the three Wirtinger inequalities given above. 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 Template:Mvar.

Fourier series

Consider the first Wirtinger inequality given above. Take Template:Mvar to be 2π. Since Dirichlet's conditions are met, we can write

$y(x)=\frac{1}{2}{a}_0+\sum _{n\ge 1}(a_n\frac{\sin nx}{\sqrt{\pi }}+b_n\frac{\cos nx}{\sqrt{\pi }}),$

and the fact that the average value of Template:Mvar is zero means that a₀ = 0. By Parseval's identity,

$\underset{0}{\overset{2\pi }{\int }}y(x{)}^2\mathrm{d}x=\sum _{n=1}^{\mathrm{\infty }}(a_n^2+b_n^2)$

and

$\underset{0}{\overset{2\pi }{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x=\sum _{n=1}^{\mathrm{\infty }}{n}^2(a_n^2+b_n^2)$

and since the summands are all nonnegative, the Wirtinger inequality is proved. Furthermore, it is seen that equality holds if and only if a_n = b_n = 0 for all n ≥ 2, which is to say that y(x) = a₁ sin x + b₁ cos x. This is equivalent to the stated condition by use of the trigonometric addition formulas.

Integration by parts

Consider the second Wirtinger inequality given above.Template:Sfnm Take Template:Mvar to be π. Any differentiable function y(x) satisfies the identity

$y(x{)}^2+(y^{\prime }(x)-y(x)\cot x{)}^2=y^{\prime }(x{)}^2-\frac{d}{dx}(y(x{)}^2\cot x).$

Integration using the fundamental theorem of calculus and the boundary conditions y(0) = y(π) = 0 then shows

$\underset{0}{\overset{\pi }{\int }}y(x{)}^2\mathrm{d}x+\underset{0}{\overset{\pi }{\int }}(y^{\prime }(x)-y(x)\cot x{)}^2\mathrm{d}x=\underset{0}{\overset{\pi }{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x.$

This proves the Wirtinger inequality, since the second integral is clearly nonnegative. 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 Template:Mvar.

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). This is resolved as follows. It follows from the Hölder inequality and y(0) = 0 that

$|y(x)|=|\underset{0}{\overset{x}{\int }}{y}^{\prime }(x)\mathrm{d}x|\le \underset{0}{\overset{x}{\int }}|y^{\prime }(x)|\cdot 1\mathrm{d}x\le {\left(\underset{0}{\overset{x}{\int }}{1}^2\mathrm{d}x\right)}^{1/2}{(\underset{0}{\overset{x}{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x)}^{1/2}=\sqrt{x}{(\underset{0}{\overset{x}{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x)}^{1/2},$

which shows that as long as

$\underset{0}{\overset{\pi }{\int }}{y}^{\prime }(x{)}^2\mathrm{d}x$

is finite, the limit of Template:Sfrac y(x)² as Template:Mvar converges to zero is zero. Since cot x < Template:Sfrac for small positive values of Template:Mvar, it follows from the squeeze theorem that y(x)² cot x converges to zero as Template:Mvar converges to zero. In exactly the same way, it can be proved that y(x)² cot x converges to zero as Template:Mvar converges to π.

Functional analysis

Consider the third Wirtinger inequality given above. Take Template:Mvar to be 1. Given a continuous function Template:Mvar on Template:Closed-closed of average value zero, let Tf denote the function Template:Mvar on Template:Closed-closed which is of average value zero, and with u′′ + f = 0 and u′(0) = u′(1) = 0. From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of Template:Mvar are (kπ)⁻² for nonzero integers Template:Mvar, the largest of which is then π⁻². Because Template:Mvar is a bounded and self-adjoint operator, it follows that

$\underset{0}{\overset{1}{\int }}Tf(x{)}^2\mathrm{d}x\le {\pi }^{-2}\underset{0}{\overset{1}{\int }}f(x)Tf(x)\mathrm{d}x=\frac{1}{{\pi }^2}\underset{0}{\overset{1}{\int }}(Tf{)}^{\prime }(x{)}^2\mathrm{d}x$

for all Template:Mvar of average value zero, where the equality is due to integration by parts. Finally, for any continuously differentiable function Template:Mvar on Template:Closed-closed of average value zero, let g_n be a sequence of compactly supported continuously differentiable functions on Template:Open-open which converge in L² to y′. Then define

$y_n(x)=\underset{0}{\overset{x}{\int }}{g}_n(z)\mathrm{d}z-\underset{0}{\overset{1}{\int }}\underset{0}{\overset{w}{\int }}{g}_n(z)\mathrm{d}z\mathrm{d}w.$

Then each y_n has average value zero with y_n′(0) = y_n′(1) = 0, which in turn implies that −y_n′′ has average value zero. So application of the above inequality to f = −y_n′′ is legitimate and shows that

$\underset{0}{\overset{1}{\int }}{y}_n(x{)}^2\mathrm{d}x\le \frac{1}{{\pi }^2}\underset{0}{\overset{1}{\int }}{y_n}^{\prime }(x{)}^2\mathrm{d}x.$

It is possible to replace y_n by Template:Mvar, and thereby prove the Wirtinger inequality, as soon as it is verified that y_n converges in L² to y. This is verified in a standard way, by writing

$y(x)-y_n(x)=\underset{0}{\overset{x}{\int }}({y_n}^{\prime }(z)-g_n(z))\mathrm{d}z-\underset{0}{\overset{1}{\int }}\underset{0}{\overset{w}{\int }}({y_n}^{\prime }(z)-g_n(z))\mathrm{d}z\mathrm{d}w$

and applying the Hölder or Jensen inequalities.

This proves the Wirtinger inequality. 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 Template:Mvar 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 Template:Mvar.

To make this argument fully formal and precise, it is necessary to be more careful about the function spaces in question.Template:Sfnm

Spectral geometry

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:Template:Sfnm

• the first eigenvalue of the Laplace–Beltrami operator on the Riemannian circle of length L is Template:Sfrac, and the corresponding eigenfunctions are the linear combinations of the two coordinate functions.
• the first Dirichlet eigenvalue of the Laplace–Beltrami operator on the interval Template:Closed-closed is Template:Sfrac and the corresponding eigenfunctions are given by c sin Template:Sfrac for arbitrary nonzero numbers Template:Mvar.
• the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval Template:Closed-closed is Template:Sfrac and the corresponding eigenfunctions are given by c cos Template:Sfrac for arbitrary nonzero numbers Template:Mvar.

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). The Wirtinger inequality, in the first version given here, can then be seen as the n = 1 case of any of the following:

• the first eigenvalue of the Laplace–Beltrami operator on the unit-radius Template:Mvar-dimensional sphere is Template:Mvar, and the corresponding eigenfunctions are the linear combinations of the n + 1 coordinate functions.Template:Sfnm
• the first eigenvalue of the Laplace–Beltrami operator on the Template:Mvar-dimensional real projective space (with normalization given by the covering map from the unit-radius sphere) is 2n + 2, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on R^{n + 1} to the unit sphere (and then to the real projective space).Template:Sfnm
• the first eigenvalue of the Laplace–Beltrami operator on the Template:Mvar-dimensional torus (given as the Template:Mvar-fold product of the circle of length 2π with itself) is 1, and the corresponding eigenfunctions are arbitrary linear combinations of Template:Mvar-fold products of the eigenfunctions on the circles.Template:Sfnm

The second and third versions of the Wirtinger inequality can be extended to statements about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on metric balls in Euclidean space:

• the first Dirichlet eigenvalue of the Laplace−Beltrami operator on the unit ball in Rⁿ is the square of the smallest positive zero of the Bessel function of the first kind J_{(n − 2)/2}.Template:Sfnm
• the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in Rⁿ is the square of the smallest positive zero of the first derivative of the Bessel function of the first kind J_n/2.Template:Sfnm

Application to the isoperimetric inequality

In the first form given above, the Wirtinger inequality can be used to prove the isoperimetric inequality for curves in the plane, as found by Adolf Hurwitz in 1901.Template:Sfnm Let (x, y) be a differentiable embedding of the circle in the plane. Parametrizing the circle by Template:Closed-closed so that (x, y) has constant speed, the length L of the curve is given by

$\underset{0}{\overset{2\pi }{\int }}\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}\mathrm{d}t$

and the area Template:Mvar enclosed by the curve is given (due to Stokes theorem) by

$-\underset{0}{\overset{2\pi }{\int }}y(t)x^{\prime }(t)\mathrm{d}t.$

Since the integrand of the integral defining Template:Mvar is assumed constant, there is

$\frac{L^2}{2\pi }-2A=\underset{0}{\overset{2\pi }{\int }}(x^{\prime }(t{)}^2+y^{\prime }(t{)}^2+2y(t)x^{\prime }(t))\mathrm{d}t$

which can be rewritten as

$\underset{0}{\overset{2\pi }{\int }}(x^{\prime }(t)+y(t){)}^2\mathrm{d}t+\underset{0}{\overset{2\pi }{\int }}(y^{\prime }(t{)}^2-y(t{)}^2)\mathrm{d}t.$

The first integral is clearly nonnegative. Without changing the area or length of the curve, (x, y) can be replaced by (x, y + z) for some number Template:Mvar, so as to make Template:Mvar have average value zero. Then the Wirtinger inequality can be applied to see that the second integral is also nonnegative, and therefore

$\frac{L^2}{4\pi }\ge A,$

which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the Wirtinger inequality and also the equality x′(t) + y(t) = 0, which amounts to y(t) = c₁ sin(t – α) and then x(t) = c₁ cos(t – α) + c₂ for arbitrary numbers c₁ and c₂. These equations mean that the image of (x, y) is a round circle in the plane.