Colombeau generalized functions: A theory based on harmonic regularizations

1998 ◽  
Vol 63 (2) ◽  
pp. 275-279 ◽  
Author(s):  
V. M. Shelkovich
Keyword(s):  
Generalized Functions ◽  
Colombeau 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 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.

Algebraic and Geometric Theory of the Topological Ring of Colombeau Generalized Functions

2008 ◽  
Vol 51 (3) ◽  
pp. 545-564 ◽  
Author(s):  
J. Aragona ◽  
S. O. Juriaans ◽  
O. R. B. Oliveira ◽  
D. Scarpalezos
Keyword(s):  
Topological Structure ◽  
Differential Calculus ◽  
Geometric Theory ◽  
Generalized Functions ◽  
Topological Ring ◽  
Main Achievement ◽  
Colombeau Generalized Functions ◽  
Generalized Manifolds ◽  
Algebraic Basis

AbstractWe continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of and introduce several invariants of the ideals of (Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C∞-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

Inversion of Colombeau generalized functions

2013 ◽  
Vol 56 (2) ◽  
pp. 469-500 ◽  
Author(s):  
Evelina Erlacher
Keyword(s):  
Necessary Conditions ◽  
Inverse Function ◽  
Generalized Functions ◽  
Generalized Function ◽  
Colombeau Generalized Functions

AbstractWe introduce different notions of invertibility for generalized functions in the sense of Colombeau. Several necessary conditions for (left, right) invertibility are derived, giving rise to the concepts of compactly asymptotic injectivity and surjectivity. We analyse the extent to which these properties are also sufficient to guarantee the existence of a (left, right) inverse of a generalized function. Finally, we establish several Inverse Function Theorems in this setting and study the relation to their classical counterparts.

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