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# On the foundations of nonlinear generalized functions I and II

### About this Title

**Michael Grosser**, **Eva Farkas**, **Michael Kunzinger** and **Roland Steinbauer**

Publication: Memoirs of the American Mathematical Society

Publication Year:
2001; Volume 153, Number 729

ISBNs: 978-0-8218-2729-1 (print); 978-1-4704-0322-5 (online)

DOI: https://doi.org/10.1090/memo/0729

MathSciNet review: 1848157

MSC: Primary 46F30; Secondary 26E15, 35A08, 35D05, 46G05, 46T30

### Table of Contents

**Chapters**

- Part 1. On the foundations of nonlinear generalized functions I
- 1. Introduction
- 2. Notation and terminology
- 3. Scheme of construction
- 4. Calculus
- 5. C- and J-formalism
- 6. Calculus on $U_\epsilon (\Omega )$
- 7. Construction of a diffeomorphism invariant Colombeau algebra
- 8. Sheaf properties
- 9. Separating the basic definition from testing
- 10. Characterization results
- 11. Differential equations
- Part 2. On the foundations of nonlinear generalized functions II
- 12. Introduction to Part 2
- 13. A simple condition equivalent to negligibility
- 14. Some more calculus
- 15. Non-injectivity of the canonical homomorphism from $\mathcal {G}^d(\Omega )$ into $\mathcal {G}^e(\Omega )$
- 16. Classification of smooth Colombeau algebras between $\mathcal {G}^d(\Omega )$ and $\mathcal {G}^e(\Omega )$
- 17. The algebra $\mathcal {G}^2$; classification results
- 18. Concluding remarks