Multiplication of Schwartz Distributions and Colombeau Generalized Functions

1999 ◽  
Vol 5 (2) ◽  
Author(s):  
B. P. Damyanov
Keyword(s):  
Generalized Functions ◽  
Colombeau Generalized Functions ◽  
Schwartz Distributions

scholarly journals An embedding of Schwartz distributions in the algebra of asymptotic functions

1998 ◽  
Vol 21 (3) ◽  
pp. 417-428 ◽  
Author(s):  
Michael Oberguggenberger ◽  
Todor Todorov
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions ◽  
Asymptotic Functions ◽  
Space Of Distributions ◽  
Asymptotic Function

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper “asymptotic function,” similar to but different from J. F Colombeau's algebras of new generalized functions.

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.

scholarly journals On Products of Distributions in Colombeau Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Marija Miteva ◽  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Schwartz Distributions ◽  
Products Of Distributions ◽  
Renormalization Theory

Results on products of distributionsx+-kandδ(p)(x)are derived. They are obtained in Colombeau differential algebra𝒢(R)of generalized functions that contains the space𝒟'(R)of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.

scholarly journals The classical theory of calculus of variations for generalized functions

2017 ◽  
Vol 8 (1) ◽  
pp. 779-808 ◽  
Author(s):  
Alexander Lecke ◽  
Lorenzo Luperi Baglini ◽  
Paolo Giordano
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Conjugate Points ◽  
Lagrange Equations ◽  
Smooth Functions ◽  
Nonlinear Properties ◽  
Schwartz Distributions ◽  
Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

scholarly journals Hypoelliptic differential operators with generalized constant coefficients

1998 ◽  
Vol 41 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović
Keyword(s):  
Small Parameter ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Order Estimate ◽  
Colombeau Generalized Functions ◽  
Constant Coefficients ◽  
Necessary And Sufficient

The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

scholarly journals On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jaeyoung Chung ◽  
Jeongwook Chang
Keyword(s):  
Functional Equations ◽  
Generalized Functions ◽  
Ulam Stability ◽  
Schwartz Distributions ◽  
The Stability

We consider the Hyers-Ulam stability for a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand hyperfunctions.

scholarly journals Further Results on Colombeau Product of Distributions

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions

Results on Colombeau product of distributions and are derived. They are obtained in Colombeau differential algebra of generalized functions that contains the space of Schwartz distributions as a subspace and has a notion of “association” that is a faithful generalization of the weak equality in .

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