Nonstandard Analysis and Generalized Functions

This application of nonstandard analysis utilizes the notion of the highly-saturated enlargement. These nonstandard methods are applied to the theory of generalized functions (distributions) and demonstrates how such analysis clarifies many aspects of this theory. 1. Additional Modeling Concepts. In what follows, the basic notation and definitions are as they appear in [3]. However, except as mentioned in the appendix of [3], a significant aspect of nonstandard analysis has not been developed fully. The superstructure constructed in [3] is a model for Γ where Γ is the set of all sentences that hold in the structure. Since it is constructed from IR, it is called a model (for real analysis), for apparently every true statement from analysis holds true in the superstructure. The elementary nonstandard structure M = ( H,∈,=) is associated with the standard model M = (H,∈,=) in a slightly special sense. Due to the applications in [3], it was not necessary to discuss M relative to its special properties. This is no longer the case. The structure M is constructed from a (bounded) ultrapower based upon the structure M. [1, p. 15–19], [2], [5, p. 83–88] It is assumed that there are constants that denote every member of H where we do not differentiate between a constant and the object it names. Suppose that J is the index set and that U is an appropriate ultrafilter on J. Let infinite A ∈ H and A contains no individuals in IR. Let f ∈ H and f(j) = A, for each j ∈ J. This is the constant map, constant in two ways both as a mathematical entity and relative to the value being denoted by a constant in our language. Some authors define an injection k, which is denoted by e in [5] and i in [1], such that k(A) = [A], where [A] is the U-equivalence class in the ultrapower that contains the constant map f. To obtain an isomorphic copy of M, we could follow the usual Mostowski collapsing process as described in [5, p. 84–85] that gives the ∗ mapping and let this map be restricted to the domain of all the constant sequence U-equivalence classes in the ultrapower. This leads to an isomorphic copy of M. [5, p 85.] But, care must be taken relative to the interpretation of the * map. It must always be remembered the M is a model of the bounded expressions that hold in M although sometimes the bounding set is not expressly stated in an expression it must be understood that the quantifiers are