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Displaying similar documents to “Extending Tamm's theorem”

A generic condition implying o-minimality for restricted C ∞ -functions

Olivier Le Gal (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove that the expansion of the real field by a restricted C ∞ -function is generically o-minimal. Such a result was announced by A. Grigoriev, and proved in a different way. Here, we deduce quasi-analyticity from a transcendence condition on Taylor expansions. This then implies o-minimality. The transcendance condition is shown to be generic. As a corollary, we recover in a simple way that there exist o-minimal structures that doesn’t admit analytic cell decomposition, and that there...

On the analyticity of generalized eigenfunctions (case of real variables)

Eberhard Gerlach (1968)

Annales de l'institut Fourier

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On démontre que, dans les espaces fonctionnels propres de Hilbert (avec un noyau reproduisant), formés de fonctions analytiques de n variables dans un domaine G , pour tout opérateur auto-adjoint, les fonctions propres généralisées sont des fonctions réelles-analytiques dans G .

Extending analyticK-subanalytic functions

Artur Piękosz (2004)

Open Mathematics

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Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.

Approximation of C ∞ -functions without changing their zero-set

F. Broglia, A. Tognoli (1989)

Annales de l'institut Fourier

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For a C ∞ function ϕ : M → ℝ (where M is a real algebraic manifold) the following problem is studied. If ϕ - 1 ( 0 ) is an algebraic subvariety of M , can ϕ be approximated by rational regular functions f such that f - 1 ( 0 ) = ϕ - 1 ( 0 ) ? We find that this is possible if and only if there exists a rational regular function g : M → ℝ such that g - 1 ( 0 ) = ϕ - 1 ( 0 ) and g(x) · ϕ ( x ) ≥ 0 for any x in ℝ n . Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus...

Topological invariants of analytic sets associated with Noetherian families

Aleksandra Nowel (2005)

Annales de l’institut Fourier

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Let Ω ⊂ ℝ n be a compact semianalytic set and let ℱ be a collection of real analytic functions defined in some neighbourhood of Ω . Let Y ω be the germ at ω of the set ⋂ f ∈ ℱ f - 1 ( 0 ) . Then there exist analytic functions v 1 , v 2 , ... , v s defined in a neighbourhood of Ω such that 1 2 χ ( lk ( ω , Y ω ) ) = ∑ i = 1 s sgn v i ( ω ) , for all ω ∈ Ω .