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:Math with average value zero and with Template:Math. Then

<math>\int_0^L y(x)^2\,\mathrm{d}x\leq\frac{L^2}{4\pi^2}\int_0^L y'(x)^2\,\mathrm{d}x,</math>

and equality holds if and only if Template:Math for some numbers Template:Mvar and Template:Math.Template:Sfnm

• Let Template:Mvar be a continuous and differentiable function on the interval Template:Math with Template:Math. Then

<math>\int_0^L y(x)^2\,\mathrm{d}x\leq\frac{L^2}{\pi^2}\int_0^L y'(x)^2\,\mathrm{d}x,</math>

and equality holds if and only if Template:Math for some number Template:Mvar.Template:Sfnm

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

<math>\int_0^L y(x)^2\,\mathrm{d}x\leq \frac{L^2}{\pi^2}\int_0^L y'(x)^2\,\mathrm{d}x.</math>

and equality holds if and only if Template:Math 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 Template:Math. Since Dirichlet's conditions are met, we can write

<math>y(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),</math>

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

<math>\int_0^{2\pi}y(x)^2\,\mathrm{d}x=\sum_{n=1}^\infty(a_n^2+b_n^2)</math>

and

<math>\int_0^{2\pi}y'(x)^2 \,\mathrm{d}x = \sum_{n=1}^\infty n^2(a_n^2+b_n^2)</math>

and since the summands are all nonnegative, the Wirtinger inequality is proved. Furthermore, it is seen that equality holds if and only if Template:Math for all Template:Math, which is to say that Template:Math. 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 Template:Math. Any differentiable function Template:Math satisfies the identity

<math>y(x)^2+\big(y'(x)-y(x)\cot x\big)^2=y'(x)^2-\frac{d}{dx}\big(y(x)^2\cot x\big).</math>

Integration using the fundamental theorem of calculus and the boundary conditions Template:Math then shows

<math>\int_0^\pi y(x)^2\,\mathrm{d}x+\int_0^\pi\big(y'(x)-y(x)\cot x\big)^2\,\mathrm{d}x=\int_0^\pi y'(x)^2\,\mathrm{d}x.</math>

This proves the Wirtinger inequality, since the second integral is clearly nonnegative. Furthermore, equality in the Wirtinger inequality is seen to be equivalent to Template:Math, the general solution of which (as computed by separation of variables) is Template:Math 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 Template:Math extends continuously to Template:Math and Template:Math for every function Template:Math. This is resolved as follows. It follows from the Hölder inequality and Template:Math that

<math>|y(x)|=\left|\int_0^x y'(x)\,\mathrm{d}x\right|\leq\int_0^x |y'(x)|\,\mathrm{d}x\leq\sqrt{x}\left(\int_0^x y'(x)^2\,\mathrm{d}x\right)^{1/2},</math>

which shows that as long as

<math>\int_0^\pi y'(x)^2\,\mathrm{d}x</math>

is finite, the limit of Template:Math as Template:Mvar converges to zero is zero. Since Template:Math for small positive values of Template:Mvar, it follows from the squeeze theorem that Template:Math converges to zero as Template:Mvar converges to zero. In exactly the same way, it can be proved that Template:Math converges to zero as Template:Mvar converges to Template:Math.

Functional analysis

Consider the third Wirtinger inequality given above. Take Template:Mvar to be Template:Math. Given a continuous function Template:Mvar on Template:Math of average value zero, let Template:Math denote the function Template:Mvar on Template:Math which is of average value zero, and with Template:Math and Template:Math. From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of Template:Mvar are Template:Math for nonzero integers Template:Mvar, the largest of which is then Template:Math. Because Template:Mvar is a bounded and self-adjoint operator, it follows that

<math>\int_0^1 Tf(x)^2\,\mathrm{d}x\leq\pi^{-2}\int_0^1 f(x)Tf(x)\,\mathrm{d}x=\frac{1}{\pi^2}\int_0^1 (Tf)'(x)^2\,\mathrm{d}x</math>

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:Math of average value zero, let Template:Math be a sequence of compactly supported continuously differentiable functions on Template:Math which converge in Template:Math to Template:Math. Then define

<math>y_n(x)=\int_0^x g_n(z)\,\mathrm{d}z-\int_0^1 \int_0^w g_n(z)\,\mathrm{d}z\,\mathrm{d}w.</math>

Then each Template:Math has average value zero with Template:Math, which in turn implies that Template:Math has average value zero. So application of the above inequality to Template:Math is legitimate and shows that

<math>\int_0^1 y_n(x)^2\,\mathrm{d}x\leq\frac{1}{\pi^2}\int_0^1 y_n'(x)^2\,\mathrm{d}x.</math>

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

<math>y(x)-y_n(x)=\int_0^x \big(y_n'(z)-g_n(z)\big)\,\mathrm{d}z-\int_0^1\int_0^w (y_n'(z)-g_n(z)\big)\,\mathrm{d}z\,\mathrm{d}w</math>

and applying the Hölder or Jensen inequalities.

This proves the Wirtinger inequality. In the case that Template:Math 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 Template:Math with Template:Math, and the regularity theory of such equations, followed by the usual analysis of ordinary differential equations with constant coefficients, shows that Template:Math 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 Template:Math is Template:Math, 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:Math is Template:Math and the corresponding eigenfunctions are given by Template:Math for arbitrary nonzero numbers Template:Mvar.
• the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval Template:Math is Template:Math and the corresponding eigenfunctions are given by Template:Math 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 Template:Math 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 Template:Math 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 Template:Math, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on Template:Math 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 Template:Math with itself) is Template:Math, 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 Template:Math is the square of the smallest positive zero of the Bessel function of the first kind Template:Math.Template:Sfnm
• the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in Template:Math is the square of the smallest positive zero of the first derivative of the Bessel function of the first kind Template:Math.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 Template:Math be a differentiable embedding of the circle in the plane. Parametrizing the circle by Template:Math so that Template:Math has constant speed, the length Template:Math of the curve is given by

<math>\int_0^{2\pi}\sqrt{x'(t)^2+y'(t)^2}\,\mathrm{d}t</math>

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

<math>-\int_0^{2\pi}y(t)x'(t)\,\mathrm{d}t.</math>

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

<math>\frac{L^2}{2\pi}-2A=\int_0^{2\pi}\big(x'(t)^2+y'(t)^2+2y(t)x'(t)\big)\,\mathrm{d}t</math>

which can be rewritten as

<math>\int_0^{2\pi}\big(x'(t)+y(t)\big)^2\,\mathrm{d}t+\int_0^{2\pi}\big(y'(t)^2-y(t)^2\big)\,\mathrm{d}t.</math>

The first integral is clearly nonnegative. Without changing the area or length of the curve, Template:Math can be replaced by Template:Math 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

<math>\frac{L^2}{4\pi}\geq A,</math>

which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the Wirtinger inequality and also the equality Template:Math, which amounts to Template:Math and then Template:Math for arbitrary numbers Template:Math and Template:Math. These equations mean that the image of Template:Math is a round circle in the plane.

References

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