Give an example of a simple curve $C=\{(x,y)\in \mathbb R^2: f(x,y)=0 \}$ ($f$ is an analytic function) that can not be implicitly characterized by a uniformly convergent series of functions as $$ \sum_{n=1}^\infty F_n(a_n x + b_n y) = 0, \tag{*} \label{sum} $$ where $a_n,b_n \in \mathbb R$ and $F_n:\mathbb R \to \mathbb R$ is a continuous function, $n\in \mathbb N$.
The implicit characterization $F_1(x) + F_2(y) = 0$ is very restrictive, for example a spiral can not be characterized that way (Can a spiral be represented as $F(x)+G(y)=0$), and what closed curves can be written that way is discussed in the post Closed curves of the form $F(x)+G(y)=0$.
Notice that the curve $xy=0$ can not be characterized as $F_1(x) + F_2(y) = 0$ (recall that the functions have to be continuous, so $y-1/x=0$ is not of the desired form). In contrast, we can write $$ xy = - x^2 - y^2 + \tfrac 1 2 (x+y)^2, $$ so the curve can be represented as $F_1(x) + F_2(y) + F_3(x+y) = 0$. A discussion of such functions can be found here: Can a spiral be represented implicitly as $F(x)+G(y)+H(x+y) = 0$?
Although I don't see a way the curve $x^2y=0$ could be represented by $F_1(x) + F_2(y) + F_3(x+y) = 0$, it can be represented by $F_1(x) + F_2(y) + F_3(x+y) + F_4(x+2y) = 0$ for example.
Can you find a curve can not be represented by \eqref{sum}?
Given an $n$, the set of functions $x^m y^{n-m}$ with $0 \le m \le n$ can be represented using a basis of $n+1$ functions of the form $(a_i x+ b_i y)^n$ for suitable sets of $a$ and $b$.
So any $f$ with a globally convergent Taylor series should be representable.
I think this logic can be extended to apply to all analytic $f$ -- since you could use add additional complexity to an $F_i$ to have it be zero outside of some neighborhood.