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 y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) = y(L). Then

${\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{4\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,}$

and equality holds if and only if y(x) = c sin 2π(x − α)/L for some numbers c and α.^[1]

• Let y be a continuous and differentiable function on the interval [0, L] with y(0) = y(L) = 0. Then

${\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,}$

and equality holds if and only if y(x) = c sin πx/L for some number c.^[1]

• Let y be a continuous and differentiable function on the interval [0, L] with average value zero. Then

${\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x.}$

and equality holds if and only if y(x) = c cos πx/L for some number c.^[2]

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 L.

Fourier series

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

${\displaystyle y(x)={\frac {1}{2}}a_{0}+\sum _{n\geq 1}\left(a_{n}{\frac {\sin nx}{\sqrt {\pi }}}+b_{n}{\frac {\cos nx}{\sqrt {\pi }}}\right),}$

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

${\displaystyle \int _{0}^{2\pi }y(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }(a_{n}^{2}+b_{n}^{2})}$

and

${\displaystyle \int _{0}^{2\pi }y'(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }n^{2}(a_{n}^{2}+b_{n}^{2})}$

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.^[1] Take L to be π. Any differentiable function y(x) satisfies the identity

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

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

${\displaystyle \int _{0}^{\pi }y(x)^{2}\,\mathrm {d} x+\int _{0}^{\pi }{\big (}y'(x)-y(x)\cot x{\big )}^{2}\,\mathrm {d} x=\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x.}$

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

${\displaystyle |y(x)|=\left|\int _{0}^{x}y'(x)\,\mathrm {d} x\right|\leq \int _{0}^{x}|y'(x)|\,\mathrm {d} x\leq {\sqrt {x}}\left(\int _{0}^{x}y'(x)^{2}\,\mathrm {d} x\right)^{1/2},}$

which shows that as long as

${\displaystyle {\int <!--\; \int \; -->}_0^{}}$Zdroj:https://en.wikipedia.org?pojem=Wirtinger's_inequality_for_functions
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