Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let and f̂ be its Taylor series at . Split the set of exponents into two disjoint subsets A and B, , and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some m ≥ 2, then the system (ₙ)ₙ is contained in the system of analytic germs. As an application of this result, we give a simple proof of Carleman’s theorem (on the non-surjectivity of the Borel map in the quasianalytic case), under the condition that the quasianalytic classes considered are closed under differentiation, for n ≥ 2.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap111-1-2,
author = {Hassan Sfouli},
title = {On a problem concerning quasianalytic local rings},
journal = {Annales Polonici Mathematici},
volume = {111},
year = {2014},
pages = {13-20},
zbl = {1298.26090},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap111-1-2}
}
Hassan Sfouli. On a problem concerning quasianalytic local rings. Annales Polonici Mathematici, Tome 111 (2014) pp. 13-20. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap111-1-2/