Categories: Mathematics, Physics.

Parseval’s theorem

Parseval’s theorem is a relation between the inner product of two functions \(f(x)\) and \(g(x)\), and the inner product of their Fourier transforms \(\tilde{f}(k)\) and \(\tilde{g}(k)\). There are two equivalent ways of stating it, where \(A\), \(B\), and \(s\) are constants from the FT’s definition:

\[\begin{aligned} \boxed{ \begin{aligned} \braket{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \braket*{\tilde{f}(k)}{\tilde{g}(k)} \\ \braket*{\tilde{f}(k)}{\tilde{g}(k)} &= \frac{2 \pi A^2}{|s|} \braket{f(x)}{g(x)} \end{aligned} } \end{aligned}\]

For this reason, physicists like to define the Fourier transform with \(A\!=\!B\!=\!1 / \sqrt{2\pi}\) and \(|s|\!=\!1\), because then it nicely conserves the functions’ normalization.

References

  1. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.

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