Analytic continuation and superconvergence of series of homogeneous polynomials

Let $D$ be a domain in $\mathbb R^n$ ($n\ge1$) and $x^0\in D$. We prove that a necessary and sufficient condition for the existence of a semicontinuous regular method ${\operatorname{A}}$ such that the series expansion of any real-analytic function $f$ in $D$ in homogeneous polynomials around $x^0$ is uniformly summed by this method to $f(x)$ on compact subsets of $D$ is that $D$ be rectilinearly star-shaped with respect to $x^0$.