Consider a pair of functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and $p:\mathbb{R}^2 \rightarrow \mathbb{R}$ and the following integral:

$g(x_1,x_2,\dots,x_n) = \int_{\mathbb{R}^n} f(t_1,t_2,\dots,t_n) p(x_1,t_1) p(x_2,t_2) \dots p(x_n,t_n) \, d t_1 \, d t_2 \, \dots\, d t_n$.

One can think of this as application of the same operator $\int p$ to $f$ w.r.t. each coordinate of $f$, where the operator is an integral transform with the kernel $p$.

Suppose $n$ is very large. I need to approximate $g$ with some $\hat{g}$ that is "linear" in p (currently $g$ has a multilinear form in $p$). One possible choice is $\hat{g} = \frac{1}{n} \sum_{k=1}^n \int_{\mathbb{R}} f(x_1,\dots,x_{k-1},t_k,x_{k+1},\dots, x_n) p(x_k,t_k) \, d t_k$, i.e. averaging coordinate-wise application of $p$ to $f$. I just gave this particular choice of $\hat{g}$ for illustration purpose, and I am not claiming the suggested $\hat{g}$ provides the best approximation to $g$ (under the constraint of $\hat{g}$ being linear in $p$).

The question is what is the best choice of $\hat{g}$ that gives smallest approximation error (measured with whatever reasonable metric). One can assume $f$ is smooth if that helps.

I thought maybe the structure that defines $g$ (1. application of the "same" operator $p$ to all coordinates, and 2. $n$ is very large) could somehow simplify the limit expression of $g$ and one could leverage that to obtain good approximation of $g$ under the constraint of linearity in $p$.

Any advice would be greatly appreciated.



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