According to Parseval Theorem (or Plancherel theorem), we have the following property. If $f(x)$ and $g(x)$ are two $L^2$ functions, and $P$ denotes the Plancherel transform,
$$ \int_{-\infty}^{\infty} f(x) \overline{g(x)} d x=\int_{-\infty}^{\infty}(\mathcal{P} f)(\xi) \overline{(\mathcal{P} g)(\xi)} d \xi, $$ where $$ (\mathcal{P} f)(\xi)=\widehat{f}(\xi)=\int_{-\infty}^{\infty} f(x) e^{-2 \pi i \xi x} d x. $$
My question is how it will be if we add more function into the integral? Assume all function are finite-valued and real. For example: $$ \int_{-\infty}^{\infty} |f(x)| d x=? $$ $$ \int_{-\infty}^{\infty} f(x) g(x) h(x) d x=? $$ $$ \int_{-\infty}^{\infty} f(x) g(x) h(x) i(x) d x=? $$
In my current work, I use Parseval Theorem to exchange the inner product of "two" function between spatial domain and spectral domain, but I cannot deal the other cases.