Constructive Borel–Ritt interpolation results for functions of several variables

Abstract

We give constructive proofs for two Borel–Ritt interpolation results, stating the existence of holomorphic functions on polysectors admitting an arbitrarily prescribed asymptotic expansion, whether in the sense of Gérard–Sibuya or in that of Majima. As a consequence, a new proof is obtained for the classical Borel's theorem on the existence of 𝒞^∞ functions on $\mathbb{R}^{n}$ with given derivatives at 0; in fact, our construction provides functions analytic in $(\mathbb{R}-\{0\})^{n}$ .