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 O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.