Analytic solutions of the Borel problem (Q1585645)

Let \(F\subset\mathbb C\) be a dense-in-itself set that has a nonempty connected interior and contains the origin. A complex-valued function \(f\) defined on \(F\) is said to be differentiable on \(F\) if for every point \(z_0\in F\) there exists a finite limit \(\lim_{z\to z_0, z\in F}\frac{f(z)-f(z_0)}{z-z_0}\). Let us denote by \(\mathcal C^\infty(F)\) the set of all infinitely differentiable functions on \(F\) in the sense of the above definition. The author considers the following Borel problem: for a given sequence \(\{d_n\}^\infty_{n=0}\) of complex numbers, find a function \(y\in\mathcal C^\infty(F)\) such that \(y^{(0)}(0)=d_n\) for \(n=0,1,\dots\). In particular, he shows that if \(F=\{z\in\mathbb C: |z-\alpha|\leq|\alpha|\}\), where \(\alpha\neq 0\), then the Borel problem has a solution in the form of a power series. This generalizes the classical result of Borel that corresponds to the case of \(\alpha=1\). Solutions of a multidimensional Borel problem are also considered in the paper.