Suppose $F: \mathbb{R}_{\ge 0}\times \mathbb{R}_{\ge 0}\rightarrow \mathbb{R}$ is symmetric i.e. $F(x_i, x_j) = F(x_j, x_i)$ and I know the $n(n+1)/2$ unique functional evaluations $F(x_i, x_j)$ for $x_i, x_j \in \{x_1, \cdots, x_n\}$. Without loss of generality, let's assume $x_1 < \cdots < x_n$. I'm trying to find a function $G: \mathbb{R}_{\ge 0}\rightarrow \mathbb{R}$ that best approximates $F$ in the sense \begin{align*} F(x, y) \approx G(x)G(y) \end{align*} on an interval $I$ that is not much larger than $[x_1, x_n]$. To make the problem more well-defined, $G$ may be restricted to the following scenarios:
$G$ be a piecewise cubic polynomial on each interval $[x_k, x_{k+1}]$ (cubic spline interpolation)
$G$ may be approximated locally in a linear fashion, similar to local regression.
I'm totally adept to solve the problem if I were just given outcomes $(x_i, f(x_i))$ and need to approximate $f$ with G, but now we are given $(x_i, x_j, F(x_i, x_j))$ and the product condition above, and it's not entirely clear to me how I should proceed from there. Any help would be appreciated!