On a problem concerning quasianalytic local rings

Hassan Sfouli. "On a problem concerning quasianalytic local rings." Annales Polonici Mathematici 111.1 (2014): 13-20. <http://eudml.org/doc/280150>.

@article{HassanSfouli2014,
abstract = {Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f ∈ _\{m\}$ and f̂ be its Taylor series at $0 ∈ ℝ^\{m\}$. Split the set $ℕ^\{m\}$ of exponents into two disjoint subsets A and B, $ℕ^\{m\} = A ∪ B$, 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 $g,h ∈ _\{m\}$ 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.},
author = {Hassan Sfouli},
journal = {Annales Polonici Mathematici},
keywords = {quasianalytic functions; Carleman's theorem; o-minimal structure},
language = {eng},
number = {1},
pages = {13-20},
title = {On a problem concerning quasianalytic local rings},
url = {http://eudml.org/doc/280150},
volume = {111},
year = {2014},
}

TY - JOUR
AU - Hassan Sfouli
TI - On a problem concerning quasianalytic local rings
JO - Annales Polonici Mathematici
PY - 2014
VL - 111
IS - 1
SP - 13
EP - 20
AB - Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f ∈ _{m}$ and f̂ be its Taylor series at $0 ∈ ℝ^{m}$. Split the set $ℕ^{m}$ of exponents into two disjoint subsets A and B, $ℕ^{m} = A ∪ B$, 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 $g,h ∈ _{m}$ 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.
LA - eng
KW - quasianalytic functions; Carleman's theorem; o-minimal structure
UR - http://eudml.org/doc/280150
ER -