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See detailEnhanced Laplace transform and holomorphic Paley-Wiener-type theorems
Dubussy, Christophe ULiege

in Rendiconti del Seminario Matematico della Università di Padova (in press)

Starting from a remark about the computation of Kashiwara-Schapira's enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we explain how to obtain explicit holomorphic ... [more ▼]

Starting from a remark about the computation of Kashiwara-Schapira's enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we explain how to obtain explicit holomorphic Paley-Wiener-type theorems. As an example, we get back some classical theorems due to Polya and Méril as limits of tempered Laplace-isomorphisms. In particular, we show how contour integrations naturally appear in this framework. [less ▲]

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See detailHolomorphic Cohomological Convolution, Enhanced Laplace Transform and Applications
Dubussy, Christophe ULiege

Doctoral thesis (2019)

The Hadamard product of power series has been studied for more than one hundred years and has become a classical tool in complex analysis. Nonetheless, this product only concerns functions which are ... [more ▼]

The Hadamard product of power series has been studied for more than one hundred years and has become a classical tool in complex analysis. Nonetheless, this product only concerns functions which are holomorphic near the origin. In 2009, T. Pohlen studied an extension of this Hadamard product on functions defined on open subsets of the Riemann sphere, which do not necessarily contain the origin. Using ad-hoc and explicit constructions, he could define this product thanks to a contour integration formula. However, his construction is non-symmetric with respect to 0 and the infinity. The first part of this thesis consists in the study of a generalization of Pohlen's extended Hadamard product. Using singular homology theory, we introduce more symmetric cycles and define a generalized Hadamard product which is equivalent to Pohlen's product when the functions vanish at infinity. Then, we show that this generalized Hadamard product is a particular case of a more general phenomenon called "holomorphic cohomological convolution". We study this convolution in detail on the multiplicative complex Lie group C^* and provide a contour integration formula to compute it. The second part of the thesis is devoted to the study of holomorphic Paley-Wiener type theorems due to Polya (in the compact case) and to Méril (in the non-compact case). These theorems use a contour integration version of the Laplace transform. Thanks to the theory of enhanced subanalytic sheaves developed by A. D'Agnolo and M. Kashiwara as well as the enhanced Laplace transform introduced by M. Kashiwara and P. Schapira, we show that such theorems can be understood from a cohomological point of view. Under some convex subanalytic conditions, we are even able to provide stronger Laplace isomorphisms between spaces which are described by tempered growth conditions. It appears that these spaces can be linked to certain spaces of analytic functionals. In the non-compact case, we define a convolution product between analytic functionals and conjecture that it is compatible with the additive version of the previously studied holomorphic cohomological convolution. Thanks to our results on the enhanced Laplace transform, we prove the conjecture in the subanalytic case. [less ▲]

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See detailEnhanced Laplace transform and holomorphic Paley-Wiener-type theorems
Dubussy, Christophe ULiege

Conference (2019, May 03)

Starting from a remark about the computation of Kashiwara-Schapira’s enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we will explain how to obtain explicit holomorphic ... [more ▼]

Starting from a remark about the computation of Kashiwara-Schapira’s enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we will explain how to obtain explicit holomorphic PaleyWiener-type theorems. As an example, we will get back some classical theorems due to Polya and Méril. In particular, we will show how contour integrations naturally appear in this framework. [less ▲]

Detailed reference viewed: 18 (4 ULiège)
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See detailThe category of enhanced subanalytic sheaves as a tool for Fourier analysis
Dubussy, Christophe ULiege

Scientific conference (2019, February 04)

In this talk, we introduce the category of enhanced subanalytic sheaves on a complex bordered space by using the subanalytic site and the addition of an extra real variable. We also define Grothendieck ... [more ▼]

In this talk, we introduce the category of enhanced subanalytic sheaves on a complex bordered space by using the subanalytic site and the addition of an extra real variable. We also define Grothendieck operations and convolution functors which allow to see this category as a commutative tensor category. Then, we consider a bordered complex vector space and we define enhanced Fourier-Sato functors, which are equivalence of categories. We finally explain how this framework allows to obtain a "device" which produces explicit holomorphic Paley-Wiener-type theorems thanks to cohomological computations. [less ▲]

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See detailEnhanced Laplace transform and holomorphic Paley-Wiener theorems
Dubussy, Christophe ULiege

Conference (2018, June 27)

In this talk we explain how the Kashiwara-Schapira isomorphism for the enhanced Laplace transform can be seen as the usual Laplace transform for enhanced distributions. From this observation, we derive ... [more ▼]

In this talk we explain how the Kashiwara-Schapira isomorphism for the enhanced Laplace transform can be seen as the usual Laplace transform for enhanced distributions. From this observation, we derive two holomorphic Paley-Wiener theorems which can be seen as tempered versions of the classical Polya's representation theorem and Méril's theorem. In particular, we show how the contour integrations naturally arise in this context. [less ▲]

Detailed reference viewed: 38 (17 ULiège)
See detailHolomorphic cohomological convolution and Hadamard product
Dubussy, Christophe ULiege

Conference (2018, February 06)

In his thesis, T. Pohlen succeeded in defining a Hadamard product between holo- morphic functions defined on star eligible open sets of the Riemann Sphere. We show how this theory is actually a particular ... [more ▼]

In his thesis, T. Pohlen succeeded in defining a Hadamard product between holo- morphic functions defined on star eligible open sets of the Riemann Sphere. We show how this theory is actually a particular case of the holomorphic cohomological convolution, defined in a general way thanks to the integration map on a complex Lie group. [less ▲]

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See detailConvolution of analytic functionals with convex carrier
Dubussy, Christophe ULiege

Scientific conference (2018, January 22)

In this talk we use the generalization of the Polya-Ehrenpreis-Martineau theorem proved by A. Méril to establish a convolution theorem for holomorphic functions with growth conditions defined on the ... [more ▼]

In this talk we use the generalization of the Polya-Ehrenpreis-Martineau theorem proved by A. Méril to establish a convolution theorem for holomorphic functions with growth conditions defined on the complementary of noncompact closed convex plane sets. We show that this convolution can be seen as an avatar of the holomorphic cohomological convolution that we developed in our previous work. [less ▲]

Detailed reference viewed: 60 (17 ULiège)
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See detailHolomorphic cohomological convolution
Dubussy, Christophe ULiege

Poster (2017, June 28)

In his thesis, T. Pohlen succeeded in defining a Hadamard product between holomorphic functions defined on star-eligible subsets of the Riemann sphere. We show how this theory is actually a particular ... [more ▼]

In his thesis, T. Pohlen succeeded in defining a Hadamard product between holomorphic functions defined on star-eligible subsets of the Riemann sphere. We show how this theory is actually a particular case of the holomorphic cohomological convolution, defined in a general way thanks to the integration map on complex Lie groups. [less ▲]

Detailed reference viewed: 70 (25 ULiège)
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See detailExercices d'analyse complexe
Dubussy, Christophe ULiege

Learning material (2017)

Ce livre d'exercices est destiné aux étudiants de second bloc en sciences mathématiques. Il vise à compléter le cours théorique "Analyse II, 2e partie" et à servir de base pour les séances de travaux ... [more ▼]

Ce livre d'exercices est destiné aux étudiants de second bloc en sciences mathématiques. Il vise à compléter le cours théorique "Analyse II, 2e partie" et à servir de base pour les séances de travaux pratiques. Chaque pan du cours est structuré en chapitre et illustré par de nombreux exercices de difficulté variable. [less ▲]

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See detailTopos de Grothendieck et fonctions holomorphes tempérées
Dubussy, Christophe ULiege

Scientific conference (2016, October 24)

Dans les années 60, Grothendieck révolutionna la géométrie algébrique en introduisant le concept de topos. Faisant face à des situations où la notion d'ouvert était trop restrictive pour comprendre les ... [more ▼]

Dans les années 60, Grothendieck révolutionna la géométrie algébrique en introduisant le concept de topos. Faisant face à des situations où la notion d'ouvert était trop restrictive pour comprendre les phénomènes géométriques sous-jacents, il définit une nouvelle notion de topologie dans le langage de la théorie des catégories. Très utilisée pour résoudre des problèmes de géométrie et d'analyse contemporains, cette notion a aussi de nombreuses applications surprenantes en logique (théorie intuitionniste, équivalence de Morita, etc ...). Dans ce séminaire je proposerai une application de la théorie des topos de Grothendieck pour étudier les fonctions et les distributions tempérées. En effet, ces objets ne forment un faisceau que sur une topologie de Grothendieck particulière (la topologie sous-analytique), qui n'est pas une topologie au sens usuel du terme. Je montrerai alors, de manière abstraite, comment définir un faisceau de fonctions holomorphes tempérées qui a joué un rôle crucial dans la démonstration de la correspondance de Riemann-Hilbert (généralisation du 21ème problème d'Hilbert). En particulier, ce nouveau faisceau fournira une nouvelle définition, presque purement algébrique, des distributions de Schwartz. [less ▲]

Detailed reference viewed: 69 (24 ULiège)