Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations

has a real analytic solution $f$ on $\omega\subseteq\mathbb{R} ^d$ for every right-hand side $(b_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}$ and give a complete characterization of such sequences of analytic functionals $(\mu_n)$. We also show that every open set $\omega\subseteq\mathbb{R} ^d$ has a complex neighbourhood $\Omega\subseteq\mathbb{C} ^d$ such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on $\Omega$.