I'm looking for the properties of integrals in the following form:
$$G(x_1, x_2, \ldots, x_n)=\int f(t) g(t-x_1, t-x_2, \ldots, t-x_n) dt $$
For $n=1$ the above integral is simply the convolution of $f$ and $g$, i.e. $G(x_1) = f * g$. Thus, I think the above integral should be some sort of generalized convolution.
Is there any reference that studies these type of integrals?
Is it possible to express the above integral as an ordinary convolution of two functions (perhaps using some auxiliary variables)?
I've figured out the answer! The given integral can be expressed in terms of an $n$-dimensional convolution.
We may write $$G(x_1, x_2, \ldots, x_n)=\int f(t) g(t-x_1, t-x_2, \ldots, t-x_n) dt \\ =\iiiint f(t)\delta(t-t_2)\delta(t-t_3)\ldots\delta(t-t_n) g(t-x_1, t_2-x_2, \ldots, t_n-x_n) dtdt_2dt_3\ldots dt_n$$ Where $\delta(t)$ is the Dirac delta function. Hence, we may write:
$$G(x_1, x_2, \ldots, x_n) = \hat{f}(x_1,\ldots, x_n) * g(x_1, x_2, \ldots, x_n)$$ Where $\hat{f}(x_1,\ldots, x_n) = f(x_1)\delta(x_1-x_2)\delta(x_1-x_3)\ldots\delta(x_1-x_n)$.