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On a problem concerning quasianalytic local rings

100%
Sfouli H.
Annales Polonici Mathematici
|
2014
|
tom 111
|
nr 1
13-20
EN
Let (𝓒ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f ∈ 𝓒_{m}$ 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 $g,h ∈ 𝓒_{m}$ 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 (𝓒ₙ)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.
2
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Weierstrass division theorem in quasianalytic local rings

63%
Elkhadiri A., Sfouli H.
Studia Mathematica
|
2008
|
tom 185
|
nr 1
83-86
EN
The main result of this paper is the following: if the Weierstrass division theorem is valid in a quasianalytic differentiable system, then this system is contained in the system of analytic germs. This result has already been known for particular examples, such as the quasianalytic Denjoy-Carleman classes.
3
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Weierstrass division theorem in definable $C^{∞}$ germs in a polynomially bounded o-minimal structure

63%
Elkhadiri A., Sfouli H.
Annales Polonici Mathematici
|
2006
|
tom 89
|
nr 2
127-137
EN
We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{∞}$ functions.
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