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The theory of algebras of generalized functions has been initiated by J.F.Colombeau and others about 20 years ago. Since then it has taken up rapid development and has found numerous applications in different fields of mathematics and mathematical physics. These include nonlinear PDE, distribution theory, nonsmooth differential geometry, nonlinear theory of stochastic processes, symmetries of differential equations, nonstandard analysis, quantum field theory, theory of relativity,...

One of the starting points for the development of algebras of generalized functions has been the problem of multiplication of distributions. By a famous result of L.Schwartz, it is impossible to define an intrinsic multiplication on the space of distributions compatible with pointwise multiplication of continuous functions. However, one can construct differential algebras containing spaces of distributions as linear subspaces and the space of smooth functions as a faithful subalgebra while at the same time possessing optimal permanence properties concerning differentiation. Multiplication of distributions can be handled in this framework.

Today, however, the theory of algebras of generalized functions reaches far beyond this original task. One reason for it's wide applicability lies in the direct and natural approach it takes towards introducing nonlinear concepts into distribution theory: Key notions of this approach are regularization processes for gaining smooth representatives of singular objects and factorization of differential algebras in order to obtain satisfying structures. Both concepts are basic constructions of many branches of modern mathematics.

• H.A.BIAGIONI: A Nonlinear Theory of Generalized Functions.
  Lecture Notes Math. Vol. 1421, Springer-Verlag, Berlin 1990.
• J.F.COLOMBEAU: New Generalized Functions and Multiplication of Distributions.
  North Holland Math. Studies Vol.84, North-Holland, Amsterdam 1984.
• J.F.COLOMBEAU: Elementary Introduction to New Generalized Functions.
  North Holland Math. Studies Vol.113, North-Holland, Amsterdam 1984.
• J.F.COLOMBEAU: Multiplication of Distributions. A tool in mathematics, numerical engineering and theoretical physics.
  Lecture Notes Math. Vol. 1532, Springer-Verlag, Berlin 1992.
• M.GROSSER, M.KUNZINGER, M.OBERGUGGENBERGER, R.STEINBAUER: Geoemtric Theory of Generalized Functions (with Applications to General Relativity).
  Mathematics and its Applications Vol. 537, Kluwer, Dordrecht, 2001.
• R.HERMANN: C-O-R Generalized Functions, Current Algebras and Control.
  Interdisciplinary Mathematics, Volume 30, Math Sci Press 1994
• M.NEDELJKOV, S.PILIPOVIC, D.SCARPALEZOS, The Linear Theory of Colombeau Generalized Functions, Pitman Research Notes in Mathematics 385, Longman 1998.
• M.OBERGUGGENBERGER:Multiplication of Distributions and Applications to Partial Differential Equations.
  Pitman Research Notes in Mathematics Vol. 259, Longman, Harlow, 1992.
• M.OBERGUGGENBERGER, E.E.ROSINGER: Solution of Continuous Nonlinear PDEs through Order Completion.
  North Holland Math. Studies Vol.181, North-Holland, Amsterdam 1994.
• S.PILIPOVIC: Colombeau's Generalized Functions and Pseudo-differential Operators.
  Lectures Math. Sciences Vol. 4, Univ. Tokyo 1994.
• E.E.ROSINGER: Distributions and Nonlinear Partial Differential Equations.
  Lecture Notes Math. Vol. 684, Springer-Verlag, Berlin 1978.
• E.E.ROSINGER: Nonlinear Partial Differential Equations. Sequential and Weak Solutions.
  North Holland Math. Studies Vol.44, North-Holland, Amsterdam 1980.
• E.E.ROSINGER: Generalized Solutions of Nonlinear Partial Differential Equations.
  North Holland Math. Studies Vol. 146, North-Holland, Amsterdam 1987.
• E.E.ROSINGER: Nonlinear Partial Differential Equations. An Algebraic View of Generalized Solutions.
  North Holland Math. Studies Vol.164, North-Holland, Amsterdam 1990.

• A.B.ANTONEVICH, Ya.V. RADYNO, On a general method for constructing algebras of generalized functions.
  Soviet Math. Dokl. 43 (1991), 680 - 684.
• V.V. CHISTYAKOV, The Colombeau generalized nonlinear analysis and the Schwartz linear distribution theory. J. Math. Sci. (New York) 93 (1999), no. 1, 42--133.
• J.F.COLOMBEAU, Multiplication of distributions. Bull. Am. Math. Soc. 23 (1990), 251 - 268.
• Yu.V.EGOROV, A contribution to the theory of generalized functions. Russian Math. Surveys 45:5 (1990), 1 - 49.
• M.GROSSER, M.KUNZINGER, G.HÖRMANN, M.OBERGUGGENBERGER (Eds.), Nonlinear Theory of Generalized Functions, Proceedings of a Workshop held at the Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Oct. - Dec. 1997, CRC Research Notes in Mathematics Vol. 401 (1999).
• M. OBERGUGGENBERGER, Nonlinear theories of generalized functions. In: S. Albeverio, W. A. J. Luxemburg, M. P. H. Wolff (Eds.), Advances in Analysis, Probability, and Mathematical Physics - Contributions from Nonstandard Analysis. Kluwer, Dordrecht 1994, 56 - 74.
• M. OBERGUGGENBERGER, Contributions of Nonstandard Analysis to partial differential equations. In: N.J. Cutland, V. Neves, F. Oliveira, J. Sousa Pinto (Eds.), Developments in Nonstandard Mathematics. Pitman Research Notes Math. Vol. 336, Longman, Harlow 1995, 130-150.
• J.SCHMEELK, A guided tour of new tempered distributions. Foundations of Physics Letters 3 (1990), 403 - 423.