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Plancherel's Theorem


\begin{displaymath}
\int_{-\infty}^\infty f(x)g^*(x)\,dx = \int_{-\infty}^\infty F(s)G^*(s)\,ds,
\end{displaymath}

where $F(s)\equiv {\mathcal F}[f(x)]$ and ${\mathcal F}$ denotes a Fourier Transform. If $f$ and $g$ are real

\begin{displaymath}
\int_{-\infty}^\infty f(x)g(-x)\,dx = \int_{-\infty}^\infty F(s)G(s)\,ds.
\end{displaymath}

See also Fourier Transform, Parseval's Theorem




© 1996-9 Eric W. Weisstein
1999-05-25