Application of Plancherel/Parseval

Question

Assuming $u,v\in L^1\cap L^2$, then how do you show that $$\int uv=\int \hat{u}\hat{v}$$ I tried using Plancherel, but didnt give any nice result. Any ideas/hints?

Thanks

Answer 1

Hint:

$$uv = \frac{1}{4} ((u+v)^2-(u-v)^2).$$

Apply Plancherel's theorem (twice) and conclude.

Note: In your question, it should read $\int \hat{u} \bar{\hat{v}}$ instead of $\int \hat{u} \hat{v}$.

Answer 2

Plancherel's Theorem gives $$ \int u\overline{v}dx = \int \hat{u}\overline{\hat{v}}ds. $$ So that gives you $$ \int uv dx = \hat{u}\overline{\hat{\overline{v}}}ds. $$ And, $$ \overline{\hat{\overline{v}}}(s)=\overline{\frac{1}{\sqrt{2\pi}}\int e^{-isx}\overline{v(x)}dx} =\frac{1}{\sqrt{2\pi}}\int e^{isx}v(x)dx, $$ which is the inverse Fourier transform, and that means your equality does not hold in general.