scholarly journals Schwartz kernel theorem in algebras of generalized functions

2010 ◽  
Author(s):  
Vincent Valmorin
Keyword(s):  
Generalized Functions ◽  
Schwartz Kernel ◽  
Algebras Of Generalized Functions ◽  
Kernel Theorem

scholarly journals Regular nonlinear generalized functions and applications

2006 ◽  
Vol 133 (31) ◽  
pp. 163-174 ◽  
Author(s):  
A. Delcroix
Keyword(s):  
Fourier Transform ◽  
Integral Operators ◽  
Generalized Functions ◽  
Regular Growth ◽  
Simplified Model ◽  
Local Analysis ◽  
Schwartz Kernel ◽  
Linear Maps ◽  
Nonlinear Generalized Functions ◽  
Kernel Theorem

We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of G8-regularity introduced by M. Oberguggenberger. As a first application we show that these new spaces are useful in a problem of representation of linear maps by integral operators, giving an analogon to Schwartz kernel theorem in the framework of nonlinear generalized functions. Secondly, we remark that these new regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to micro local analysis of singularities of generalized functions, with respect to these regularities. AMS Mathematics Subject Classification (2000): 35A18, 35A27, 42B10, 46E10, 46F30.

scholarly journals Algebras of generalized functions with smooth parameter dependence

2012 ◽  
Vol 55 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Annegret Burtscher ◽  
Michael Kunzinger
Keyword(s):  
Systematic Study ◽  
Generalized Functions ◽  
Parameter Dependence ◽  
Order Structure ◽  
Colombeau Generalized Functions ◽  
Algebras Of Generalized Functions ◽  
Algebraic Properties

AbstractWe show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\tilde{\mathbb{K}}_\mathrm{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\tilde{\mathbb{K}}_\mathrm{sm}$ and establish some properties of its ideals.

FOUNDATION OF THE FRACTIONAL CALCULUS IN GENERALIZED FUNCTION ALGEBRAS

2012 ◽  
Vol 10 (04) ◽  
pp. 439-467 ◽  
Author(s):  
MIRJANA STOJANOVIĆ
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Generalized Functions ◽  
Generalized Function ◽  
Colombeau Algebra ◽  
Nonlinear Odes ◽  
Function Algebras ◽  
Algebras Of Generalized Functions ◽  
Special Algebra

We introduce an approach to fractional derivatives involving singularities based on the theory of algebras of generalized functions in the sense of Colombeau. We are interested in solving fractional nonlinear ODEs and PDEs with singularities with a lack of solutions in the space of classical functions or distributions. For these purposes, we embed different forms of fractional derivatives into space of Colombeau special algebra of generalized functions using appropriate techniques such as the regularization with delta sequences and multiplication with different cut-off functions. Finally, we present an example for application of the ideas presented in paper to confirm the reason of introducing fractional derivatives into Colombeau algebra of generalized functions.

scholarly journals A New Approach to Diffeomorphism Invariant Algebras of Generalized Functions

2015 ◽  
Vol 58 (3) ◽  
pp. 717-738
Author(s):  
E. A. Nigsch
Keyword(s):  
Generalized Functions ◽  
New Approach ◽  
Local Setting ◽  
Algebras Of Generalized Functions ◽  
Nonlinear Generalized Functions

AbstractWe develop the diffeomorphism invariant Colombeau-type algebra of nonlinear generalized functions in a modern and compact way. Using a unifying formalism for the local setting and on manifolds, the construction becomes simpler and more accessible than before.

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