A nonstandard definition of finite order ultradistributions
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A nonstandard definition of finite order ultradistributions J SOUSA PINTO and R F HOSKINS* Departamento de Matem~itica, Universidade de Aveiro, Aveiro, Portugal * Mathematics Department, De Montfort University, Leicester, UK MS received 22 December 1998; revised l0 March 1999
Abstract. Taking into account that finite-order ultradistributions are inverse Fourier transforms of finite-order distributions a nonstandard representation is obtained for one-dimensional finite-order ultradistributions.
Keywords.
Distributions; ultradistributions; nonstandard analysis.
1. Introduction 1.1 The Silva axioms forfinite-order distributions Taking the notions of continuous function and derivative as primitive, the axiomatic definition of (finite-order) distributions on the real line, due to Silva [5], may be given as follows: DEFINITION 1.1 Distributions on R may be characterized as the elements of a linear space E for which two linear maps are defined,
t:C(R)--*E
and
D:E-*E
such that
Axiom 1. t is the injective inclusion: that is, every function in C = C(R) is a distribution on R.
Axiom 2. To each distribution v on R there corresponds a distribution Dr, called the derivative of v, such that if v = t ( f ) , w i t h f E C 1 = CI(R), then Dv = t ( f ) . Axiom 3. If v is a distribution on R then there exists a continuous function f E C and a natural number r E N0 = {0, 1 , 2 , . . . } such that we have v = Drt(f). Axiom 4. Given anyf, g E C and a natural number r E N0, the equality Drt(f) = D't(g) holds if and only i f f - g is a polynomial of degree < r. Silva has given an abstract model for these axioms as follows: on the Cartesian product N0 x C define an equivalence relation, say [], by
(r,f) [] (s, g) ~=~3",eNo[m >_ r, s A ( f f ~ - y - ff~-Sg) E II",], where 1-I., denotes the set of all complex-valued polynomials of degree less than m. and ffk is the kth iterated indefinite integral operator with origin at a E R. If we denote the 389
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quotient set by Coo - c o ( R ) = No x c / D ,
then * Coo is a model for the Silva axioms 1-4, and 9 every model for the Silva axioms is isomorphic to Co. In particular 7:r -- 7~r ), the space of all Schwartz distributions of finite order is isomorphic to C~. For a detailed account of the theory of Schwartz distributions from this point of view the recent text by Campos Ferreira [1] is recommended. 1.2 A nonstandard model for the Silva axioms In an earlier paper [2] a nonstandard model for the Silva axioms for distributions on the line has been given, using the simple ultrapower model *R = R ~ / , ~ for the hyperreals (where t~ = {1, 2 , . . . } ) . For a fuller account, the original paper may be consulted; a general introduction to the concepts and notation peculiar to nonstandard analysis is to be found in, for example, [4]. *C~176 is the internal set of all infinitely *differentiable functions on *~. Thus *C~176 - *C~162 = {F -- [(f~)n~] :f~ E C~ for nearly all n E t~}. We denote by sc~ ==_s c ~ the (external) set of all functions F in *C~176 wh
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