Sorry for the not-perfect question. I am asking for a reference for the following relation:


$$\int f . g. h ...= \int\_{\xi\_1 +\xi\_2 +...=0} \hat{f}(\xi\_1) \hat{g}(\xi\_2)... d\xi\_1 d\xi\_2...$$


Could you please show me where to find details, statements and proofs about this relationship? Thanks in Advance.



Asking for reference about a relation related to Fourier transform
If $f\_1,\ldots,f\_n$ belong to the Schwarz space, Fourier inversion formula and Fourier-convolution properties yield
\begin{eqnarray\*}
\int\_{\xi\_1+\cdots+\xi\_n=0} \hat{f\_1}(\xi\_1) \cdots \hat{f\_n}(\xi\_n) d\xi\_1 \cdots d\xi\_n
&=& (\hat{f\_1} \* \cdots \* \hat{f\_n})(0) \\
&=& (2\pi)^{-1} \mathcal{F}(\overline{\mathcal{F}}(\hat{f\_1} \* \cdots \* \hat{f\_n}))(0) \\
&=& \mathcal{F}(f\_1 \cdots f\_n)(0) \\
&=& \int\_\mathbb{R} f\_1 \cdots f\_n(x) dx.
\end{eqnarray\*}


