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.
Monographs on the nonlinear theory of
generalized functions.
