Suppose we have differentiable functions $F$, $f\_1, \dots, f\_n$, and $g\_1, \dots, g\_n$ satisfy the following relation
$$ F(x+y) = \sum\_{i=1}^n f\_i(x) g\_i(y).$$
What are the possible forms of $F$?



Functions with special separability
Differentiate with respect to $x$ and then with respect to $y$, you get
$$F'(x+y)=\sum\_i f\_i^\prime(x)g\_i(y)=\sum\_i f\_i(x)g\_i^\prime(y).$$
Substitute to the last equality some values $y\_j$ then
$$\sum\_i f\_i^\prime(x)g\_i(y\_j)=\sum\_i f\_i(x)g\_i^\prime(y\_j),\quad j=1,\ldots,n.$$
Assuming that $\det g\_i(y\_j)\neq 0$, solve this, and obtain
$$f\_i^\prime(x)=\sum\_{j} c\_{i,j}f\_j(x).$$
This is a system of linear equation with constant coefficients from which it follows
that $f\_i$ are (generalized) exponential sums. For similar reasons, $g\_i$ are also generalized exponential sums. The rest is algebra.


