Colombeau's generalized functions: Topological structures, microlocal properties - a simplified point of view - part II

Scarpalézos, Dimitris

doi:10.2298/pim0476111s

Cited by 60 publications

(105 citation statements)

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“…In this paper, we continue the investigations in the field of generalized integral operators initiated by [18] (recently republished in [19,20]) and carried on by [2,5,9,10,23]. Let us recall that those operators generalize, in the Colombeau framework, the operators with distributional kernels in the space of Schwartz distributions [2].…”

Section: Introductionmentioning

confidence: 80%

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Kernel theorems in spaces of tempered generalized functions

Delcroix¹

2007

Math. Proc. Camb. Phil. Soc.

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In analogy to the classical isomorphism between L (S (R n ) , S ′ (R m )) and S ′ R n+m , we show that a large class of moderate linear mappings acting between the space GS (R n ) of Colombeau rapidly decreasing generalized functions and the space Gτ (R n ) of temperate ones admits generalized integral representations, with kernels belonging to Gτ R n+m . Furthermore, this result contains the classical one in the sense of the generalized distribution equality. (2000) Mathematics Subject Classification IntroductionDuring the three last decades, theories of nonlinear generalized functions have been developed by many authors (see [1,11,14,15],...), mainly based on the ideas of J.-F. Colombeau [3,4], which we are going to follow in the sequel. Those theories appear to be a natural continuation of the distributions' one [12,21,22], specially efficient to pose and solve differential or integral problems with irregular data.In this paper, we continue the investigations in the field of generalized integral operators initiated by [18] (recently republished in [19,20]) and carried on by [2,5,9,10,23]. Let us recall that those operators generalize, in the Colombeau framework, the operators with distributional kernels in the space of Schwartz distributions [2]. More specifically, in [5], we proved that any moderate net of linear maps (, that is satisfying some growth properties with respect to the parameter ε, gives rise to a linear map L :and G C (R n ) denote respectively the space of generalized functions and the space of compactly supported ones.) The main result is that L can be represented as a generalized integral operator in the spirit of Schwartz Kernel Theorem.Going further in this direction, we study here the generalization of the classical isomorphism between S ′ (R n+m ) and the space of continuous linear mappings acting between S (R n ) and S ′ (R m ) [22]. Thus, the spaces of generalized functions considered in this paper are the space G S (R n ) of rapidly decreasing generalized functions [7,10,17] and the space G τ (R n ) of tempered generalized functions [4,11,18]: G S (R n ) plays here, roughly speaking, the role of S (R n ) (resp. G C (R n )) in the classical case (resp. in [5]), whereas G τ (R n ) plays the role of S ′ (R m ) (resp. G (R m )) in the classical case (resp. in [5]). 1The main results are the following. First, any kernel H ∈ G τ (R m+n ) gives rise to a new type of linear generalized integral operator acting between G S (R n ) and G τ (R m ) and defined bywhere (H ε ) ε (resp. (f ε ) ε ) is any representative of H (resp. f ) and [·] τ , the class of an element in G τ (R m ). Moreover, the linear map H → H from G τ (R m+n ) to the space of linear maps Proposition 14). This gives the first part of the expected result. Then, new regular subspaces of G S (R n ) and G τ (R m ) are introduced in the spirit of [6]. They are used to define the moderate nets of linear maps, which can be extended to act between G S (R n ) and G τ (R m ) (Proposition 16). Finally, our main result states that ...

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Section: Introductionmentioning

confidence: 80%

“…Nets of maps (L ε ) ε between two topological algebras, having some good growth properties with respect to the parameter ε, can be extended to act between the corresponding Colombeau algebras, as it is shown in [5,11,18] for example. We are going to introduce here new notions adapted to our framework.…”

Section: Extension Of Linear Mapsmentioning

confidence: 99%

Kernel theorems in spaces of tempered generalized functions

Delcroix¹

2007

Math. Proc. Camb. Phil. Soc.

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“…In particular, a thorough study of algebraic properties was carried out (e.g., [1,29]) and topological and functional analytic structures on Colombeau spaces were developed and refined to a high degree (e.g., [25,26,9,10]). …”

Section: Introductionmentioning

confidence: 99%

Recent progress in special Colombeau algebras: geometry, topology, and algebra

Kunzinger

2010

Linear and Non-Linear Theory of Generalized Functions and Its Applications

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Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of modules over the ring of generalized numbers, and algebraic aspects of Colombeau theory. Some open problems are given and directions of further research are outlined. (2000): Primary 46F30; Secondary 46T30, 46A20, 16D25, 53B20. Mathematics Subject Classification

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“…Following a first approach done by D. Scarpalezos in [18], we introduce a natural concept of generalized integral kernel operators in this setting. In addition, we show that these operators are characterized by their kernel.…”

Section: Introductionmentioning

confidence: 99%

Generalized Integral Operators and Applications

Bernard¹,

Colombeau²,

Delcroix³

2006

Math. Proc. Camb. Phil. Soc.

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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.

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Colombeau's generalized functions: Topological structures, microlocal properties - a simplified point of view - part II

Abstract: This paper is the second part of [S-1]. Here we consider convolution products, microlocalization and pseudodifferential operators in the frame of Colombeau generalized functions.

Cited by 60 publications

References 3 publications

Kernel theorems in spaces of tempered generalized functions

Kernel theorems in spaces of tempered generalized functions

Recent progress in special Colombeau algebras: geometry, topology, and algebra

Generalized Integral Operators and Applications

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