1,551 research outputs found
Characterization of ultradifferentiable test functions defined by weight matrices in terms of their Fourier transform
We prove that functions with compact support in non-quasianalytic classes of
Roumieu-type and of Beurling-type defined by a weight matrix with some mild
regularity conditions can be characterized by the decay properties of their
Fourier transform. For this we introduce the abstract technique of constructing
from the original matrix multi-index matrices and associated function spaces.
We study the behaviour of this construction in detail and characterize its
stability. Moreover non-quasianalyticity of the classes is characterized.Comment: 25 pages This version is accepted for publication in Note di
Matematic
On the Borel mapping in the quasianalytic setting
The Borel mapping takes germs at of smooth functions to the sequence of
iterated partial derivatives at . We prove that the Borel mapping restricted
to the germs of any quasianalytic ultradifferentiable class strictly larger
than the real analytic class is never onto the corresponding sequence space.Comment: 14 pages; minor changes, accepted for publication in Math. Scand.;
typos corrected and numbering of equations changed in order to be in
accordance with the published articl
Composition in ultradifferentiable classes
We characterize stability under composition of ultradifferentiable classes
defined by weight sequences , by weight functions , and, more
generally, by weight matrices , and investigate continuity of
composition . In addition, we represent the Beurling
space and the Roumieu space
as intersection and union of spaces and
for associated weight sequences, respectively.Comment: 28 pages, mistake in Lemma 2.9 and ramifications corrected, Theorem
6.3 improved; to appear in Studia Mat
Indices of O-regular variation for weight functions and weight sequences
A plethora of spaces in Functional Analysis (Braun-Meise-Taylor and Carleman
ultradifferentiable and ultraholomorphic classes; Orlicz, Besov, Lipschitz,
Lebesque spaces, to cite the main ones) are defined by means of a weighted
structure, obtained from a weight function or sequence subject to standard
conditions entailing desirable properties (algebraic closure, stability under
operators, interpolation, etc.) for the corresponding spaces. The aim of this
paper is to stress or reveal the true nature of these diverse conditions
imposed on weights, appearing in a scattered and disconnected way in the
literature: they turn out to fall into the framework of O-regular variation,
and many of them are equivalent formulations of one and the same feature.
Moreover, we study several indices of regularity/growth for both functions and
sequences, which allow for the rephrasing of qualitative properties in terms of
quantitative statements.Comment: 37 page
A Phragm\'en-Lindel\"of theorem via proximate orders, and the propagation of asymptotics
We prove that, for asymptotically bounded holomorphic functions in a sector
in , an asymptotic expansion in a single direction towards the
vertex with constraints in terms of a logarithmically convex sequence admitting
a nonzero proximate order entails asymptotic expansion in the whole sector with
control in terms of the same sequence. This generalizes a result by A. Fruchard
and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on
a suitably refined version of the classical Phragm\'en-Lindel\"of theorem, here
obtained for functions whose growth in a sector is specified by a nonzero
proximate order in the sense of E. Lindel\"of and G. Valiron.Comment: 20 page
The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces
We consider r-ramification ultradifferentiable classes, introduced by J.
Schmets and M. Valdivia in order to study the surjectivity of the Borel map,
and later on also exploited by the authors in the ultraholomorphic context. We
characterize quasianalyticity in such classes, extend the results of Schmets
and Valdivia about the image of the Borel map in a mixed ultradifferentiable
setting, and obtain a version of the Whitney extension theorem in this
framework.Comment: 31 pages; this version has been accepted for publication in Monatsh.
Mat
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