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

Olivier Le Gal (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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 exist incompatible...

A note on Bierstone-Milman-Pawłucki's paper "Composite differentiable functions"

Krzysztof Jan Nowak (2011)

Annales Polonici Mathematici

We demonstrate that the composite function theorems of Bierstone-Milman-Pawłucki and of Glaeser carry over to any polynomially bounded, o-minimal structure which admits smooth cell decomposition. Moreover, the assumptions of the o-minimal versions can be considerably relaxed compared with the classical analytic ones.

Analytic functions are -density continuous

Krzysztof Ciesielski, Lee Larson (1994)

Commentationes Mathematicae Universitatis Carolinae

A real function is -density continuous if it is continuous with the -density topology on both the domain and the range. If f is analytic, then f is -density continuous. There exists a function which is both C and convex which is not -density continuous.

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

In rings Γ M of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence M = ( M l ) l 0 (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in Γ M such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to Γ M , provided Γ M is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...

Caractérisation des anneaux noethériens de séries formelles à croissance controlée. Application à la synthèse spectrale.

Jacques Chaumat, Anne-Marie Chollet (1997)

Publicacions Matemàtiques

Given a subring of the ring of formal power series defined by the growth of the coefficients, we prove a necessary and sufficient condition for it to be a noetherian ring. As a particular case, we show that the ring of Gevrey power series is a noetherian ring. Then, we get a spectral synthesis theorem for some classes of ultradifferentiable functions.

Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives

Paweł Domański (2004)

Banach Center Publications

This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces and the theory of the functor Proj¹ are applied to questions like solvability of linear partial differential equations, existence of a solution depending...

Closed universal subspaces of spaces of infinitely differentiable functions

Stéphane Charpentier, Quentin Menet, Augustin Mouze (2014)

Annales de l’institut Fourier

We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace...

Composition in ultradifferentiable classes

Armin Rainer, Gerhard Schindl (2014)

Studia Mathematica

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ( ω ) and the Roumieu space ω as intersection and union of spaces ( M ) and M for associated weight sequences, respectively.

Decomposition into special cubes and its applications to quasi-subanalytic geometry

Krzysztof Jan Nowak (2009)

Annales Polonici Mathematici

The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.

Examples of functions -extendable for each finite, but not -extendable

Wiesław Pawłucki (1998)

Banach Center Publications

In Example 1, we describe a subset X of the plane and a function on X which has a k -extension to the whole 2 for each finite, but has no -extension to 2 . In Example 2, we construct a similar example of a subanalytic subset of 5 ; much more sophisticated than the first one. The dimensions given here are smallest possible.

Explicit extension maps in intersections of non-quasi-analytic classes

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ( ) ( [ - 1 , 1 ] r ) ; (b) there is no continuous linear extension map from Λ ( ) ( r ) into ( ) ( r ) ; (c) under some additional assumption on , there is an explicit extension map from ( ) ( [ - 1 , 1 ] r ) into ( ) ( [ - 2 , 2 ] r ) by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

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