Éditeur(s) : HAL CCSD
Résumé : Submitted to the Journal of Mathematical Analysis and Applications
International audience
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{ infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.
J. Math. Anal. Appl.

Droits : info:eu-repo/semantics/OpenAccess
hal-00020280
https://hal.archives-ouvertes.fr/hal-00020280
https://hal.archives-ouvertes.fr/hal-00020280/document
https://hal.archives-ouvertes.fr/hal-00020280/file/ADelcroixGRegularJan27.pdf
ARXIV : math/0603183

Éditeur(s) : HAL CCSD
Résumé : Prelimary version - 13 pages.
International audience
A new approach to the algebra G_{τ} of temperate nonlinear generalized functions is proposed, in which G_{τ} is based on the space O_{M} endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of G_{τ}. This is illustrated by the natural introduction of a regularity theory in G_{τ}, of the Fourier transform, with the definition of G_{O_{C}′}, the space of rapidly generalized distributions which is the Fourier image of G_{τ}.
Publ. Inst.Math. (Beograd) (N.S.)

hal-00201458
https://hal.archives-ouvertes.fr/hal-00201458
https://hal.archives-ouvertes.fr/hal-00201458/document
https://hal.archives-ouvertes.fr/hal-00201458/file/NewGOMdec29WithAdditions.pdf
ARXIV : 0801.0074

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz’ result is contained in our main theorem.
ISSN: 0022-247X

hal-01530748
https://hal.univ-antilles.fr/hal-01530748
https://hal.univ-antilles.fr/hal-01530748/document
https://hal.univ-antilles.fr/hal-01530748/file/delcroixgskttjan21.pdf
DOI : 10.1016/j.jmaa.2005.01.006

Éditeur(s) : HAL CCSD Institute of Mathematics Polish Academy of Sciences
Résumé : International audience
In analogy to the classical isomorphism between L(D(Rn);D0(Rm)) and D0(Rm+n) (resp. L(S(Rn); S0(Rm)) and S0(Rm+n)), we show that a large class of moderate linear mappings acting between the space GC(Rn) of compactly supported generalized functions and G(Rn) of generalized functions (resp. the space GS(Rn) of Colombeau rapidly decreasing generalized functions and the space G (Rn) of temperate ones) admits generalized integral representations, with kernels belonging to specic regular subspaces of G(Rm+n) (resp. G (Rm+n)). The main novelty is to use accelerated -nets, which are unit elements for the convolution product in these regular subspaces, to construct the kernels. Finally, we establish a strong relationship between these results and the classical ones.
ISSN: 0137-6934

hal-01529541
https://hal.univ-antilles.fr/hal-01529541
https://hal.univ-antilles.fr/hal-01529541/document
https://hal.univ-antilles.fr/hal-01529541/file/delcroix.pdf
DOI : 10.4064/bc88-0-7

Éditeur(s) : HAL CCSD Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad
Résumé : International audience
We present some remarks about the embedding of spaces of Schwartz distributions into spaces of Colombeau generalized functions. Following ideas of M. Nedeljkov et alii, we recall how a good choice of compactly supported mollifiers allows to perform globally the embedding of D' () into G (). We show that this embedding is equal to the one obtained with local and sheaf arguments by M. Grosser et alii, this giving various equivalent technics to embed D'() into G ().
ISSN: 1450-5444

hal-01530751
https://hal.univ-antilles.fr/hal-01530751
https://hal.univ-antilles.fr/hal-01530751/document
https://hal.univ-antilles.fr/hal-01530751/file/NS04_25.PDF

Éditeur(s) : HAL CCSD
Résumé : 15 pages
International audience
In analogy to the classical isomorphism between $mathcal{L}left( mathcal{S}left( mathbb{R}^{n}right) ,mathcal{S}^{prime}left( mathbb{R}^{m}right) right) $ and $mathcal{S}^{prime}left( mathbb{R}^{n+m}right) $, we show that a large class of moderate linear mappings acting between the space $mathcal{G}_{mathcal{S}}left( mathbb{R}^{n}right) $ of Colombeau rapidly decreasing generalized functions and the space $mathcal{G}_{ au}left( mathbb{R}^{n}right) $ of temperate ones admits generalized integral representations, with kernels belonging to $mathcal{G}_{ au}left( mathbb{R}^{n+m}right) $. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.
Math. Proc. Camb. Philos. Soc.

Droits : info:eu-repo/semantics/OpenAccess
hal-00019916
https://hal.archives-ouvertes.fr/hal-00019916
https://hal.archives-ouvertes.fr/hal-00019916/document
https://hal.archives-ouvertes.fr/hal-00019916/file/KernelSADelcroixfeb7th.pdf
ARXIV : math.FA/0603035

Éditeur(s) : HAL CCSD
Résumé : We extend classical results from the Colombeau algebra, concerning point-value characterizations of generalized functions, to the more general case of multi-parameter (C,E,P)-algebras. Our investigations include considerations about different definitions of subspaces related to tempered generalized functions.
https://hal.univ-antilles.fr/hal-00761114

hal-00761114
https://hal.univ-antilles.fr/hal-00761114
https://hal.univ-antilles.fr/hal-00761114/document
https://hal.univ-antilles.fr/hal-00761114/file/DT2010-05_Hasler_Marti.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
generalized functions 1996,
Novi Sad, Serbia

hal-01535417
https://hal.univ-antilles.fr/hal-01535417

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We show that a new definition of the asymptotic structure of the (C,E,P)-algebras (which are a refinement of the Colombeau Generalized Algebras) allows to define a natural topology on them. Furthermore, this topology can be mixed with functorial aspects of the construction in order to give extension principles, which give powerful tools to solve singular differential problems in this framework.
ISSN: 0022-247X

hal-01529551
https://hal.univ-antilles.fr/hal-01529551
DOI : 10.1016/j.jmaa.2009.05.046

Éditeur(s) : HAL CCSD
Résumé : International audience
Given a topological R-module or algebra E and an asymptotic scale MR Λ, we define a natural M-extended topology on the sequence space E Λ, and the M-extension of E as the Hausdorff space associated with the subspace of sequences for which multiplication is continuous with respect to this topology. Colombeau's generalized functions and similar constructions are obtained as special cases, but this new approach also allows the iteration of the construction, which was not possible with previously existing theories. We use only the topology, i.e. neigbourhoods of zero, but not its explicit definition in terms of seminorms, inductive or projective limits etc., which is particularly useful in non-metrizable spaces. Many ideas commonly used in generalized function spaces (functoriality, association, sheaf structure, algebraic analysis, ...) can be applied to a large extent; but reasoning on a category-theoretic level allows the establishment of several results so far only known for particular cases, to the whole class of such spaces.
Integral Transforms and Special Functions
Bedlewo, Poland

hal-00761666
https://hal.univ-antilles.fr/hal-00761666
DOI : 10.1080/10652460802565040

Éditeur(s) : HAL CCSD Seminario Matematico
Résumé : International audience
An original method for the construction of algebras of generalized functions isgiven, which covers both Colombeau simplified algebras and Rosinger algebras cases, themain used such algebras. Examples are given showing how this method works for the mostknown algebras in this area.
ISSN: 0373-1243

hal-01678082
https://hal.archives-ouvertes.fr/hal-01678082

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
We review our recent formulation of Colombeau-type algebras as Hausdorff sequence spaces with ultranorms, defined by sequences of exponential weights. We extend previous results and give new perspectives related to echelon-type spaces, possible generalizations, asymptotic algebras, concepts of association, and applications thereof.
ISSN: 1065-2469

hal-00758570
https://hal.univ-antilles.fr/hal-00758570
https://hal.univ-antilles.fr/hal-00758570/document
https://hal.univ-antilles.fr/hal-00758570/file/0705.2739v1.pdf
DOI : 10.1080/10652460500437807

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The aim of this paper is to prove that the framework of generalized functions of Sobolev type is more suitable to pose and solve some PDEs problems with very irregular data, than the one introduced by J.-F. Colombeau, when ${mathcal{C}}^{infty}$ estimates are not available or out of reach. In such type of algebras, one shows the existence and some qualitative properties of solutions for problems appearing in the mathematical modelisation of oil activities.
Journal of Mathematical Analysis and applications

hal-00685042
https://hal.archives-ouvertes.fr/hal-00685042
DOI : 10.1016/j.jmaa.2011.07.063

Éditeur(s) : HAL CCSD Académie Serbe des Sciences et des Arts
Résumé : International audience
ISSN: 0561-7332

hal-01530760
https://hal.univ-antilles.fr/hal-01530760