Éditeur(s) : HAL CCSD Cambridge University Press (CUP)
Résumé : International audience
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.
ISSN: 0305-0041

hal-01530755
https://hal.univ-antilles.fr/hal-01530755

Éditeur(s) : HAL CCSD
Résumé : International audience
Colloque en l'honneur de Robert Lutz.
Mulhouse, France

hal-01535440
https://hal.univ-antilles.fr/hal-01535440

Éditeur(s) : HAL CCSD
Résumé : International audience
Anwendungen des infinitesimalmathematik
Oberwolfach, Germany

hal-01535427
https://hal.univ-antilles.fr/hal-01535427

Éditeur(s) : HAL CCSD
Résumé : International audience
In this paper we investigate solutions to the semi-linear wave equation in canonical form with non Lipschitz non-linearity and distributions or other generalized functions as data. To give a meaning to the Goursat problem with irregular data, we replace it by a biparametric family of problems. The first parameter turns the problem into a family of Lipschitz problems, the second one regularizes the data. Finally, the problem is solved in an appropriate algebra. We show that the solution is equal to the non-regularized one. In the examples, we take advantage of our results to give a new approach of the blow-up problem.
Differential Equations & Applications

Droits : info:eu-repo/semantics/OpenAccess
hal-00345090
https://hal.archives-ouvertes.fr/hal-00345090
https://hal.archives-ouvertes.fr/hal-00345090v2/document
https://hal.archives-ouvertes.fr/hal-00345090/file/Devoue-Goursat-wellposed-12-08.pdf

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 : 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 : International audience
Rencontres européennes d'Analyse Non Standard,
Luminy - Marseille, France

hal-01535423
https://hal.univ-antilles.fr/hal-01535423

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 Springer Verlag
Résumé : International audience
We recall the construction of the Reed-Muller hermitian codes and some results about hermitians forms and exponential sums. With the help of these results, we give the weight distribution of the Reed-Muller hermitian codes and give some examples for which experimental computations have been made.
ISSN: 0938-1279

hal-00771918
https://hal.univ-antilles.fr/hal-00771918

Éditeur(s) : HAL CCSD
Résumé : Note, 5 pages
We give a new definition of the so-called overgenerated rings, which are the usual tool used to define the asymptotic structure of a (C,E,P)-algebra, written as a factor space M_{(A,E,P)}/N_{(I_{A},E,P)}. With this new definition and in the particular case of E=C^{∞}, we show that a moderate element i.e. in M_{(A,E,P)} is negligible if and only if it satisfies the C⁰-order estimate for the ideal N_{(I_{A},E,P)}.
https://hal.archives-ouvertes.fr/hal-00326671

Droits : info:eu-repo/semantics/OpenAccess
hal-00326671
https://hal.archives-ouvertes.fr/hal-00326671
https://hal.archives-ouvertes.fr/hal-00326671/document
https://hal.archives-ouvertes.fr/hal-00326671/file/AustrianLemmaVer1.pdf
ARXIV : 0810.0831

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 Elsevier Masson
Résumé : International audience
ISSN: 1631-073X

hal-01683263
https://hal.univ-antilles.fr/hal-01683263
DOI : 10.1016/j.crma.2004.04.002

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We discuss various methods of regularization of singular differential problems. Their common point is that we use the flexibility of the theories of nonlinear generalized functions for adapting the regularization to the singularity of the problem. We particularly underline the relationship between the generalized solutions and those classical or distribution, when they exist, giving a general result for the case of the regularization of data.
Journal of Mathematical Analysis and applications

hal-01529550
https://hal.univ-antilles.fr/hal-01529550
DOI : 10.1016/j.jmaa.2008.11.077

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 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
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 Cambridge Scientific Publishers Ltd
Résumé : International audience
https://hal.univ-antilles.fr/hal-01529538
Pointe-à-Pitre, Guadeloupe

ISBN : 1-904868-31-2
hal-01529538
https://hal.univ-antilles.fr/hal-01529538

Éditeur(s) : HAL CCSD
Résumé : 25 pages, Equipe AANL
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.
https://hal.archives-ouvertes.fr/hal-00004888

hal-00004888
https://hal.archives-ouvertes.fr/hal-00004888
https://hal.archives-ouvertes.fr/hal-00004888/document
https://hal.archives-ouvertes.fr/hal-00004888/file/Article5.pdf
ARXIV : math/0505126

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
We show that the so-called low-regret (or least-regret) control by J. L. Lions [8] fits on the control of ill-posed problems. At each time, we give the characterization of the so-called no-regret control by means of singular optimality systems. For the backward heat ill-posed problem, no Slater hypothesis is assumed on the admissible set of controls ${{cal U}_{mbox{ iny ad}}}$. This work is two pieces, and two methods are considered : the regularization method and the null-controllability method. For the first method, a zero order corrector is used, while for the second method, the passage to the limit is easy. The results presented here generalize the works in [2,3] to the no-regret control.
ISSN: 1270-900X

hal-00720814
https://hal.univ-antilles.fr/hal-00720814
https://hal.univ-antilles.fr/hal-00720814/document
https://hal.univ-antilles.fr/hal-00720814/file/esaimpr1.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
NSS'88
Amherst, United States

hal-01535426
https://hal.univ-antilles.fr/hal-01535426