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| : ''For other inequalities named after Wirtinger, see [[Wirtinger's inequality]].''
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| In [[mathematics]], historically '''Wirtinger's inequality''' for real functions was an [[inequality (mathematics)|inequality]] used in [[Fourier analysis]]. It was named after [[Wilhelm Wirtinger]]. It was used in 1904 to prove the [[isoperimetric inequality]]. A variety of closely related results are today known as Wirtinger's inequality.
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| ==Theorem==
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| ===First version===
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| Let <math>f : \mathbb{R} \to \mathbb{R}</math> be a [[periodic function]] of period 2π, which is continuous and has a continuous derivative throughout '''R''', and such that
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| :<math>\int_0^{2\pi}f(x) \, dx = 0.</math>
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| Then
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| :<math>\int_0^{2\pi}f'^2(x) \, dx \ge \int_0^{2\pi}f^2(x) \, dx</math>
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| with equality [[if and only if]] ''f''(''x'') = ''a'' sin(''x'') + ''b'' cos(''x'') for some ''a'' and ''b'' (or equivalently ''f''(''x'') = ''c'' sin (''x'' + ''d'') for some ''c'' and ''d'').
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| This version of the Wirtinger inequality is the one-dimensional [[Poincaré inequality]], with optimal constant.
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| ===Second version===
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| The following related inequality is also called Wirtinger's inequality {{harv|Dym|McKean|1985}}:
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| :<math>\pi^{2}\int_0^a |f|^2 \le a^2 \int_0^a|f'|^2</math>
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| whenever ''f'' is a C<sup>1</sup> function such that ''f''(0) = ''f''(''a'') = 0. In this form, Wirtinger's inequality is seen as the one-dimensional version of [[Friedrichs' inequality]].
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| ===Proof===
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| The proof of the two versions are similar. Here is a proof of the first version of the inequality. Since [[Dirichlet's conditions]] are met, we can write
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| :<math>f(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> | |
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| and moreover ''a''<sub>0</sub> = 0 since the integral of ''f'' vanishes. By [[Parseval's identity]],
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| :<math>\int_0^{2\pi}f^2(x)dx=\sum_{n=1}^\infty(a_n^2+b_n^2)</math>
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| and
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| :<math>\int_0^{2\pi}f'^2(x) \, dx = \sum_{n=1}^\infty n^2(a_n^2+b_n^2)</math>
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| and since the summands are all ≥ 0, we get the desired inequality, with equality if and only if ''a<sub>n</sub>'' = ''b<sub>n</sub>'' = 0 for all ''n'' ≥ 2.
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| ==References==
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| *{{citation|first1=H|last1=Dym|authorlink1=Harry Dym|first2=H|last2=McKean|title=Fourier series and integrals|publisher=Academic press|year=1985|isbn=978-0-12-226451-1}}
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| *[[Vadim Komkov|Komkov, Vadim]] (1983) Euler's buckling formula and Wirtinger's inequality. Internat. J. Math. Ed. Sci. Tech. 14, no. 6, 661—668.
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| {{PlanetMath attribution|id=5393|title=Wirtinger's inequality}}
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| [[Category:Fourier analysis]]
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| [[Category:Inequalities]]
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| [[Category:Theorems in analysis]]
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Irwin Butts is what my wife enjoys to contact me although I don't really like becoming known as like that. What I adore performing is doing ceramics but I haven't produced a dime with it. Years ago we moved to North Dakota and I love every working day residing right here. Hiring has been my occupation for some home std test time but I've already applied for an additional one.
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