Peano differentiable extensions in o-minimal structures

Kurdyka

2009

Proc. Amer. Math. Soc.

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Section: Theorem 21 (Parusiński) Let U Be An Open Subset Of R N Andmentioning

confidence: 99%

Subanalytic blow-$\mathcal {C}^m$ functions

Kurdyka

2009

Proc. Amer. Math. Soc.

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“…[13], we perform studies of the Peano derivatives of functions which belong to an o-minimal expansion of a real closed field.…”

Section: Introductionmentioning

confidence: 99%

Singularities of o‐minimal Peano derivatives

Mathematische Nachrichten

2010

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The notion of Peano differentiability generalizes the differentiability in the usual sense to higher order. Peano differentiable functions have derivatives which are sometimes differentiable or continuous or not even locally bounded. We give a complete characterisation of the sets in which Peano differentiable functions which are definable in an o-minimal expansion of a real closed field are continuously differentiable. Thereby, we also distinguish between several kinds of discontinuities.

“…For definable functions of one variable, the notion of P ∞ is equipollent to that of C ∞ (cf. [10,Proposition 7.2]). For functions of several variables the notions of P ∞ and C ∞ differ, at least if the o-minimal structure is not polynomially bounded and we only consider such o-minimal structures.…”

mentioning

confidence: 99%

Zero-set property of o-minimal indefinitely Peano differentiable functions

2008

Ann. Polon. Math.

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Abstract. Given an o-minimal expansion M of a real closed field R which is not polynomially bounded. Let P ∞ denote the definable indefinitely Peano differentiable functions. If we further assume that M admits P ∞ cell decomposition, each definable closed subset A of R n is the zero-set of a P ∞ function f : R n → R. This implies P ∞ approximation of definable continuous functions and gluing of P ∞ functions defined on closed definable sets.1. Introduction. In the present paper we discuss the zero-set property of indefinitely Peano differentiable functions in connection with o-minimality. Let R be a fixed real closed field and M be an o-minimal expansion of R. In the following, "definable" always means "definable with parameters in M". We further assume the reader to be familiar with the basic properties of o-minimal structures (see for example [3] Let R n be endowed with the Euclidean R-norm · and the corresponding topology (an R-norm has the same definition as the norm just taking its values in R). Peano differentiability can be seen as the natural generalisation of Fréchet differentiability to higher-in our case infinite-order. Indefinitely Peano differentiable functions are functions which have an infinite Taylor series at every point of their domain. More precisely: