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Wirtinger's Inequality

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

\begin{displaymath}
\int_0^{2\pi} y\,dx=0,
\end{displaymath}

then

\begin{displaymath}
\int_0^{2\pi} y^2\,dx<\int_0^{2\pi} y'^2\,dx
\end{displaymath}

unless

\begin{displaymath}
y=A\cos x+B\sin x.
\end{displaymath}


References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184-187, 1988.




© 1996-9 Eric W. Weisstein
1999-05-26