Analytic continuation and superconvergence of series of homogeneous polynomials (Q1387391)

Let \(A=(a_k(t))_{k=0}^{\infty }\) be a sequence of complex-valued functions defined for \(0<t<\infty \), and let \(\sum_{k=0}^{\infty }u_k\) be some (not necessarily convergent) series of complex numbers. It is said that \(\sum_{k=0}^{\infty }u_k\) is summed to a number \(u(t)\in \mathbb{C}\) by the regular semicontinuous method \(A\) if for each \(t\) the series \(\sum_{k=0}^{\infty }a_k(t)u_k\) converges to a number \(u(t)\in \mathbb{C}\) and \(\lim_{t\to 0}u(t)=u\). (Moreover, it is assumed that if \(\sum_{k=0}^{\infty }u_k\) converges to a number \(s\in \mathbb{C}\), then necessarily \(u=s\).) In this paper the following theorem is proved: let \(D\) be a domain in \(\mathbb{R}^n\) and let \(x^0 \in D;\) a necessary and sufficient condition for the existence of a semicontinuous regular method \(A\) 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 \(x^0\).