Éditeur(s) : HAL CCSD
Résumé : 21 pages Publié sous le titre: Spectral asymptotic analysis in algebras of generalized functions dans Asymptot. Anal. 59/1-2, 83-107, 2008.
International audience
We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.
Asymptot. Anal.

Droits : info:eu-repo/semantics/OpenAccess
hal-00140656
https://hal.archives-ouvertes.fr/hal-00140656
https://hal.archives-ouvertes.fr/hal-00140656/document
https://hal.archives-ouvertes.fr/hal-00140656/file/MicrolocalDelcroixMartiOberguggenbergerArkiv.pdf
ARXIV : 0704.1077

Éditeur(s) : HAL CCSD
Résumé : We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a "frequential microlocal analysis " and into a "microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations.
https://hal.archives-ouvertes.fr/hal-00190006

Droits : info:eu-repo/semantics/OpenAccess
hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006/document
https://hal.archives-ouvertes.fr/hal-00190006/file/Regularity-Local-Microlocal-Analysis5.pdf
ARXIV : 0711.3688

Éditeur(s) : HAL CCSD
Résumé : International audience
Generalized functions 2004
Novi Sad, Serbia

hal-01535462
https://hal.univ-antilles.fr/hal-01535462

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 : International audience
Based on a previously established framework of asymptotic extensions of topological modules and algebras, we use results concerning sheaf theoretic properties and functoriality of the construction, to develop some tools for microlocal algebraic analysis in this setting. In particular, we introduce the notion of singular spectrum of sections, and give results concerning its linear and non-linear properties.
Progress in Analysis and its Applications
London, United Kingdom

hal-00761660
https://hal.univ-antilles.fr/hal-00761660
DOI : 10.1142/9789814313179_0079

Éditeur(s) : HAL CCSD
Résumé : 18 pages
This article adresses questions on localization and microlocalization in the framework of the so called (C,E,P)-algebras of generalized functions. We give new results on nonlinear properties of the asymptotic singular spectrum in the multi-parameter case, a tool to describe the singularities of generalized solutions associated to problems which are classically ill-posed. Wa apply the theory to a transport equation with strongly irregular coefficients and initial data.
https://hal.univ-antilles.fr/hal-00761112

hal-00761112
https://hal.univ-antilles.fr/hal-00761112
https://hal.univ-antilles.fr/hal-00761112/document
https://hal.univ-antilles.fr/hal-00761112/file/DT2010-03_Devoue_al3.pdf

Éditeur(s) : HAL CCSD International Publishing Service/IPS
Résumé : International audience
We give a description of various algebras of generalized functions based on the introduction of pseudo-ultranorms on spaces of sequences in given locally convex function algebras. We study sheaf properties of these algebras, needed for microlocal analysis, and also consider regularity theory, functoriality and different concepts of association and weak equality in a unified setting. Using this approach, we also give new results on embeddings of ultradistribution and hyperfunction spaces into corresponding algebras of generalized functions.
ISSN: 0012-3862

hal-00758044
https://hal.univ-antilles.fr/hal-00758044
https://hal.univ-antilles.fr/hal-00758044/document
https://hal.univ-antilles.fr/hal-00758044/file/DHPV-26oct-8nov06.pdf
DOI : 10.4064/dm447-0-1

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