Infinite systems of linear equations for real analytic functions

Abstract:

We study the problem when an infinite system of linear functional equations \[ \mu _n(f)=b_n\quad \text {for\ }n\in \mathbb {N}\] 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 $\Omega$.