Say $F$ is the class of separable functions, $$\Big\{(x,y) \to \sum_{k=1}^Nu_k(x)v_k(y)\Big\}.$$ Under what conditions are such functions positive definite?
By "positive definite function" I mean functions where the matrix $ A $, defined as $A_{ij} = f(x_i-x_j,y_i-y_j)$, is positive semidefinite, for some set of points $\{(x_i,y_i)\}$.
By working with two points, $\{(x_1,y_1),(x_2,y_2)\}$, I found, using $\det{A}$, that: $$f(0,0)^2\ge f(a,b)f(-a,-b) $$ but I am unsure if this is the general requirement for the main question, and if so, how to prove it for the general case of $n$ points $\{(x_i, y_i)\}$.