# Portugaliae Mathematica

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**Volume 65, Issue 3, 2008, pp. 321–337**

**DOI: 10.4171/PM/1815**

Published online: 2008-09-30

The local structure of nonstandard representatives of distributions

Hans Vernaeve^{[1]}(1) Universität Innsbruck, Austria

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* [5], 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. Anwendungen*
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 *Portugal. Math.* 48 (1991), 195–216.

*Keywords: *Nonstandard analysis, generalized functions, distributions

Vernaeve Hans: The local structure of nonstandard representatives of distributions. *Port. Math.* 65 (2008), 321-337. doi: 10.4171/PM/1815