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Wirtinger's Inequality


If y has period 2pi, y^' is L^2, and

 int_0^(2pi)ydx=0,
(1)

then

 int_0^(2pi)y^2dx<int_0^(2pi)y^('2)dx
(2)

unless

 y=Acosx+Bsinx
(3)

(Hardy et al. 1988).

Another inequality attributed to Wirtinger involves the Kähler form, which in C^n can be written

 omega=-1/2isumdz_k ^ dz^__k.
(4)

Given 2k vectors X_1,...,X_(2k) in R^(2n)=C^n, let X=X_1 ^ ... ^ X_(2k) denote the oriented k-dimensional parallelepiped and |X| its k-dimensional volume. Then

 omega^k(X)<=k!|X|,
(5)

with equality iff the vectors span a k-dimensional complex subspace of C^n, and they are positively oriented. Here, omega^k is the kth exterior power for 1<=k<=n, and the orientation of a complex subspace is determined by its complex structure.


See also

Kähler Form

Portions of this entry contributed by Todd Rowland

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References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, 1988.

Referenced on Wolfram|Alpha

Wirtinger's Inequality

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Wirtinger's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WirtingersInequality.html

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