Abstract. We characterize stability under composition of ultradifferentiable classes defined by weight sequences M , by weight functions ω, and, more generally, by weight matrices M, and investigate continuity of composition (g, f ) → f • g. In addition, we represent the Beurling space E (ω) and the Roumieu space E {ω} as intersection and union of spaces E (M ) and E {M } for associated weight sequences, respectively.
For Denjoy-Carleman differentiable function classes C M where the weight sequence M = (M k ) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C M -curves to C M -curves. The category of C M -mappings is cartesian closed in the sense that C M (E, C M (F, G)) ∼ = C M (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C M -diffeomorphisms is a C M -Lie group but not better.
Abstract. We characterize stability under composition, inversion, and solution of ordinary differential equations for ultradifferentiable classes, and prove that all these stability properties are equivalent.
We revisit Whitney's extension theorem in the ultradifferentiable Roumieu
setting. Based on the description of ultradifferentiable classes by weight
matrices, we extend results on how growth constraints on Whitney jets on
arbitrary compact subsets in $\mathbb R^n$ are preserved by their extensions to
$\mathbb R^n$. More precisely, for any admissible class $\mathcal C$ of
ultradifferentiable functions on $\mathbb R^n$ we determine a class $\mathcal
C'$ such that all ultradifferentiable Whitney jets of class $\mathcal C'$ on
arbitrary compact subsets admit extensions in $\mathcal C$. The class $\mathcal
C'$ can be explicitly computed from $\mathcal C$.Comment: 21 pages; minor changes, some references added; accepted for
publication in Math. Nach
For quasianalytic Denjoy-Carleman differentiable function classes C Q where the weight sequence Q = (Q k ) is log-convex, stable under derivations, of moderate growth and also an L-intersection (see (1.6)), we prove the following: The category of C Q -mappings is cartesian closed in the sense that C Q (E, C Q (F, G)) ∼ = C Q (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C Q -diffeomorphisms is a regular C Q -Lie group but not better.Classes of Denjoy-Carleman differentiable functions are in general situated between real analytic functions and smooth functions. They are described by growth conditions on the derivatives. Quasianalytic classes are those where infinite Taylor expansion is an injective mapping.
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