I have a function of $2n$ variables:
$u(x_1,\ldots,x_n,y_1,\ldots,y_n)$
I am interested in the existence of approximations of such a function as a sum of products of paired variables. Let me write for brevity $x=(x_1,\ldots,x_n)$ and $y=(y_1,\ldots,y_n)$. The approximations I am interested in have the form:
$u(x,y)=\sum_i f(x_i)g(y_i)$
Additionally, I can assume that $u(x,y)$ is monotonically increasing in all of the $y_i$ variables and I want the same to hold of all of the $g(y_i)$ functions. Otherwise, the $f$ and $g$ can be arbitrary. I can assume strong regularity conditions on $u$.
To be clear, I am not interested necessarily in constructing the approximation (although that would be neat), but in its existence and error bounds. I have found a similar question with approximating $u(x,y)$ as $f(x)g(y)$, but the only answer required discretizing the function and using single-value decomposition (low-rank approximation). This may be acceptable, but I am having a hard time generalizing to my case of $n$ variables. Also, it would be OK if the sum included more than $n$ terms, i.e., more than 1 term to approximate the contribution of each variable pair.
Thanks!
For a theoretical survey see Light, Cheney, Approximation Theory in Tensor Product Spaces which concerns "the approximation of multivariate functions by combinations of univariate ones"
Practically, multivariate functions or data are often approximated using tensor product polynomials or splines.