Polynomial approximation on algebraic variety

The talk is devoted to the discussion of the following problem: let $A\subset \mathbb C^N$, $\operatorname{dim}A=n$, be an algebraic variety, $f(z)\in C(K)$ be some continuous function on a compact set $K\subset A$. If the approximation rate
$$ \varlimsup_{m\to\infty}\rho_m^{1/m}(f,K)=\delta<1. $$
where $\rho_m(f,K)=\min\{\|f-p_m\|_K, \operatorname{deg}p_m\leq{m}\}$ is the minimal deviation of $f$ from polynomials $p_m$ of degree $\leq{m}$, then what can we say about analyticity of $f$ in a neighborhood of the compact set $K$ ?