This is a reference request. It is motivated by understanding an online article here.
In evaluating an integral, it is written in the explanation that one can use the "Parseval's theorem for convolution" which is $$ \int_{0}^{\infty} f(x) g(x)\; d x=\frac{2}{\pi} \int_{0}^{\infty} F_{c}(\omega) G_{c}(\omega)\; d \omega\tag{1} $$ where $$ F_{c}(\omega)=\int_{0}^{\infty} f(x) \sin (\omega x) d x\;. $$
I see no "convolution" in (1) unless one write $g(x)=h(t-x)$ by defining $h(x):=g(t-x)$. I suspect that it is a misnomer. The form of (1) is nevertheless very close to that of the version in this Wikipedia article. Anyway, I am more interested in knowing when and why exactly (1) is true.
I have (very!) limited knowledge in Fourier analysis and I believe the Parseval theorem is a well-known result but different books treat it rather differently depending on how Fourier transform is introduced. I am looking for a book that has a theorem and its proof for (1), or a theorem stated in the form that has (1) as its easy consequence.