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Résolution d'équations aux dérivées partielles dans des espaces de distributions d'ordre de régularité variable

André Unterberger (1971)

Annales de l'institut Fourier

L’objet de cet article est de prouver des théorèmes du genre suivant : “Soient P un opérateur différentiel sur R n , ρ une fonction C à valeurs réelles, k un nombre réel et u une distribution à support compact : alors, si P u H ρ , u H ρ + k ” ; l’espace H ρ est ici l’espace de Sobolev “d’ordre variable” associé à ρ  ; bien entendu, il faut des hypothèses sur P , ρ et k . Les cas traités sont :1) certains opérateurs à coefficients variables déjà considérés dans le chapitre VIII du livre de L. Hörmander ;2) tous les opérateurs...

Results on Colombeau product of distributions

Blagovest Damyanov (1997)

Commentationes Mathematicae Universitatis Carolinae

The differential -algebra 𝒢 ( m ) of generalized functions of J.-F. Colombeau contains the space 𝒟 ' ( m ) of Schwartz distributions as a -vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in 𝒟 ' ( m ) . This is particularly useful for evaluation of certain products of distributions, as they are embedded in 𝒢 ( m ) , in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions x ± a and δ ( p ) ( x ) , with x in m ,...

Results on generalized models and singular products of distributions in the Colombeau algebra 𝒢 ( )

Blagovest Damyanov (2015)

Commentationes Mathematicae Universitatis Carolinae

Models of singularities given by discontinuous functions or distributions by means of generalized functions of Colombeau have proved useful in many problems posed by physical phenomena. In this paper, we introduce in a systematic way generalized functions that model singularities given by distributions with singular point support. Furthermore, we evaluate various products of such generalized models when the results admit associated distributions. The obtained results follow the idea of a well-known...

Ridgelet transform on tempered distributions

R. Roopkumar (2010)

Commentationes Mathematicae Universitatis Carolinae

We prove that ridgelet transform R : 𝒮 ( 2 ) 𝒮 ( 𝕐 ) and adjoint ridgelet transform R * : 𝒮 ( 𝕐 ) 𝒮 ( 2 ) are continuous, where 𝕐 = + × × [ 0 , 2 π ] . We also define the ridgelet transform on the space 𝒮 ' ( 2 ) of tempered distributions on 2 , adjoint ridgelet transform * on 𝒮 ' ( 𝕐 ) and establish that they are linear, continuous with respect to the weak * -topology, consistent with R , R * respectively, and they satisfy the identity ( * ) ( u ) = u , u 𝒮 ' ( 2 ) .

Right inverses for partial differential operators on Fourier hyperfunctions

Michael Langenbruch (2007)

Studia Mathematica

We characterize the partial differential operators P(D) admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual (Ω̅)-type estimate valid for the bounded holomorphic functions on the characteristic variety V P near d . The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén-Lindelöf-type condition.

Schwartz kernel theorem in algebras of generalized functions

Vincent Valmorin (2010)

Banach Center Publications

A new approach to the generalization of Schwartz’s kernel theorem to Colombeau algebras of generalized functions is given. It is based on linear maps from algebras of classical functions to algebras of generalized ones. In particular, this approach enables one to give a meaning to certain hypotheses in preceding similar work on this theorem. Results based on the properties of G -generalized functions class are given. A straightforward relationship between the classical and the generalized versions...

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