View source for Rosinger nowhere-dense generalized function algebra

In the general framework of [[Generalized function algebras|generalized function algebras]] developed by E.E. Rosinger [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], a distinguished role is played by ideals in the sequence algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301602.png" /> an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301603.png" />, which are defined by vanishing properties. Given a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301604.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301605.png" />, stable under finite unions, one considers the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301606.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301607.png" /> determined by those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301608.png" /> for which there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301609.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016011.png" /> vanishes near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016012.png" /> eventually, that is, there are a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016013.png" /> and a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016017.png" />. The nowhere-dense generalized function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016018.png" /> is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016019.png" /> is the class of nowhere-dense, closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016020.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016021.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016022.png" /> via the constant imbedding. It has two distinguishing features. First, the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016023.png" /> forms a [[Flabby sheaf|flabby sheaf]], and in a certain sense the smallest flabby sheaf containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016024.png" />, see [[#References|[a2]]]. Secondly, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016025.png" /> of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016026.png" /> can be imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016027.png" />.

In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016028.png" /> (no growth restrictions near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016029.png" />) can be interpreted as global solutions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016030.png" /> by means of a suitable regularization method. The space of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016031.png" /> (cf. also [[Generalized functions, space of|Generalized functions, space of]]) is imbedded in any algebra of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016032.png" /> by a general procedure [[#References|[a4]]] using an algebraic basis.

Further generalizations of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016033.png" /> to include larger exceptional sets as well as applications to non-smooth differential geometry can be found in [[#References|[a1]]]; non-linear Lie group actions on generalized functions using the framework of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016034.png" /> are studied in [[#References|[a7]]].

Replacing the single differential algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016035.png" /> with chains of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016036.png" /> using the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016037.png" /> in the place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016038.png" /> at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016039.png" />-functions as well as the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016040.png" /> at each fixed level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016041.png" />.

See also [[Generalized function algebras|Generalized function algebras]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Mallios, E.E. Rosinger, "Space-time foam dense singularities and de Rham cohomology" ''Acta Applic. Math.'' (to appear) {{MR|1847884}} {{ZBL|1005.46020}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Oberguggenberger, E.E. Rosinger, "Solution of continuous nonlinear PDEs through order completion" , North-Holland (1994) {{MR|1286940}} {{ZBL|0821.35001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.E. Rosinger, "Distributions and nonlinear partial differential equations" , Springer (1978) {{MR|0514014}} {{ZBL|0469.35001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) {{MR|0590891}} {{ZBL|0447.35001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.E. Rosinger, "Generalized solutions of nonlinear partial differential equations" , North-Holland (1987) {{MR|0918145}} {{ZBL|0635.46033}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.E. Rosinger, "Nonlinear partial differential equations, an algebraic view of generalized solutions" , North-Holland (1990) {{MR|}} {{ZBL|0717.35001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> 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)</TD></TR></table>