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Published online: 2008-09-30
The local structure of nonstandard representatives of distributionsHans Vernaeve (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 , 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