The Local Structure of Nonstandard Representatives of Distributions

Abstract

It is shown that the nonstandard representatives of Schwartz-distributions, as introduced by K.D. Stroyan and W.A.J. Luxemburg in their book 'Introduction to the theory of infinitesimals', are locally equal to a finite-order derivative of a finite-valued and S-continuous function. By 'equality', we mean a pointwise equality, not an equality in a distributional sense. This proves a conjecture by M. Oberguggenberger in [Z. Anal. Anwend. 10 (1991), 263-264]. Moreover, the representatives of the zero-distribution are locally equal to a finite-order derivative of a function assuming only infinitesimal values. These results also unify the nonstandard theory of distributions by K.D. Stroyan and W.A.J. Luxemburg with the theory by R.F. Hoskins and J. Sousa Pinto in [Portugaliae Mathematica 48(2), 195-216].