Rosinger nowhere-dense generalized function algebra

In the general framework of generalized function algebras developed by E.E. Rosinger [a3], [a4], [a5], [a6], a distinguished role is played by ideals in the sequence algebra , an open subset of , which are defined by vanishing properties. Given a family of subsets of , stable under finite unions, one considers the ideal in determined by those for which there is a such that for all , vanishes near eventually, that is, there are a and a neighbourhood of such that for . The nowhere-dense generalized function algebra is obtained when is the class of nowhere-dense, closed subsets of . The space contains via the constant imbedding. It has two distinguishing features. First, the family forms a flabby sheaf, and in a certain sense the smallest flabby sheaf containing , see [a2]. Secondly, the algebra of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of can be imbedded into .

In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets (no growth restrictions near ) can be interpreted as global solutions in by means of a suitable regularization method. The space of distributions (cf. also Generalized functions, space of) is imbedded in any algebra of the form by a general procedure [a4] using an algebraic basis.

Further generalizations of the ideal to include larger exceptional sets as well as applications to non-smooth differential geometry can be found in [a1]; non-linear Lie group actions on generalized functions using the framework of are studied in [a7].

Replacing the single differential algebra with chains of algebras using the spaces in the place of at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of -functions as well as the derivative of at each fixed level .

See also Generalized function algebras.

References

[a1]	A. Mallios, E.E. Rosinger, "Space-time foam dense singularities and de Rham cohomology" Acta Applic. Math. (to appear) MR1847884 Zbl 1005.46020
[a2]	M. Oberguggenberger, E.E. Rosinger, "Solution of continuous nonlinear PDEs through order completion" , North-Holland (1994) MR1286940 Zbl 0821.35001
[a3]	E.E. Rosinger, "Distributions and nonlinear partial differential equations" , Springer (1978) MR0514014 Zbl 0469.35001
[a4]	E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) MR0590891 Zbl 0447.35001
[a5]	E.E. Rosinger, "Generalized solutions of nonlinear partial differential equations" , North-Holland (1987) MR0918145 Zbl 0635.46033
[a6]	E.E. Rosinger, "Nonlinear partial differential equations, an algebraic view of generalized solutions" , North-Holland (1990) Zbl 0717.35001
[a7]	E.E. Rosinger, "Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert's fifth problem" , Kluwer Acad. Publ. (1998)