On the foundations of nonlinear generalized functions I and II

2001 ◽  
Vol 153 (729) ◽  
pp. 0-0 ◽  
Author(s):  
Michael Grosser ◽  
Eva Farkas ◽  
Michael Kunzinger ◽  
Roland Steinbauer
Keyword(s):  
Generalized Functions ◽  
Nonlinear Generalized Functions

GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY

2002 ◽  
Vol 17 (20) ◽  
pp. 2776-2776
Author(s):  
R. STEINBAUER ◽  
M. KUNZINGER
Keyword(s):  
General Relativity ◽  
Riemannian Geometry ◽  
Rapid Development ◽  
Generalized Functions ◽  
Field Equations ◽  
Mathematical Methods ◽  
Colombeau Algebras ◽  
Rigorous Treatment ◽  
General Relativistic ◽  
Nonlinear Generalized Functions

The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed7. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.

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.

scholarly journals Sheaves of nonlinear generalized functions and manifold-valued distributions

2009 ◽  
Vol 361 (10) ◽  
pp. 5177-5177 ◽  
Author(s):  
Michael Kunzinger ◽  
Roland Steinbauer ◽  
James A. Vickers
Keyword(s):  
Generalized Functions ◽  
Nonlinear Generalized Functions

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 A nonlinear theory of distributional geometry

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200642 ◽  
Author(s):  
E. A. Nigsch ◽  
J. A. Vickers
Keyword(s):  
Differential Geometry ◽  
Nonlinear Theory ◽  
Delta Function ◽  
Generalized Functions ◽  
Lie Derivative ◽  
Tensor Fields ◽  
Generalized Metric ◽  
Nonlinear Generalized Functions ◽  
Distributional Geometry ◽  
Conical Metric

This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 ( doi:10.1098/rspa.2020.0640 )) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.

scholarly journals A Convenient Notion of Compact Set for Generalized Functions

2017 ◽  
Vol 61 (1) ◽  
pp. 57-92
Author(s):  
Paolo Giordano ◽  
Michael Kunzinger
Keyword(s):  
Function Spaces ◽  
Generalized Functions ◽  
Distribution Theory ◽  
Compact Set ◽  
Compact Sets ◽  
Smooth Functions ◽  
Test Function ◽  
Compactly Supported ◽  
Nonlinear Generalized Functions

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

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