Representing distributions by nonstandard polynomials

One of the features of Nonstandard Analysis is, that it lowers the rank of mathematical concepts. A striking illustration of this is Robinson’s famous characterization of compactness, with quantifiers running over numbers instead of over coverings. This chapter presents reducing classical Schwartzian distributions to nonstandard functions, a simplification which might be useful in nonlinear operations with distributions. A singular distribution is not regular. Two examples will be considered in the sequel: the delta distribution and the Principal Value Distribution P1x. Robinson proved that in an enlargement every distribution is regular. The chapter shows how to represent a distribution T by a “small” kernel {P_a,P_ß,…} of nonstandard polynomials.