A nonstandard definition of finite order ultradistributions (Q1961745)

The Silva axioms for finite order distributions [cf. \textit{J. Sebastião e Silva}, ``Obras de J. S. Silva'', Portugal INIC, Vol. 3, pp. 183-196], seem to have initiated the authors to deal with non-standard ultradistributions of finite order and the Fourier transforms mapping \({\mathcal D}_{\text{fin}}'\) (space of all Schwartz distributions of finite order) into \({\mathcal Z}_{\text{fin}}'\) (space of all finite order ultradistributions), hence they conceived the idea that, from the known non-standard representation of finite order distributions it should be possible to derive a non-standard representation for finite order ultradistributions, and in achieving this the authors have slightly re-phrased the definition of \({^\Xi C_\infty}\) (a non-standard model for the axiomatic definition of finite order distributions on \(\mathbb{R}\), as proposed by Silva; \(C_\infty\) is a quotient set). The notations/definitions, re-phrased are: \({\mathcal D}\) is the Schwartz space of all infinitely differentiable functions of compact support and \({^*{\mathcal D}}\) is the non-standard extension of \({\mathcal D}\); \({^s{\mathcal D}}\) denotes the \({^*C_b}\) submodule of \({^sC^\infty}\) (\({^*C^\infty}(\mathbb{R})\) is the internal set of all infinitely *differentiable functions on \({^*\mathbb{R}}\) and \({^3C^\infty}\) is the external set of all functions \(F\) in \({^3C^\infty}\), which are finite valued; \({^*\mathbb{R}_b}\) is the set of all finite hyperreal numbers) which comprises all infinitely *differentiable functions of hypercompact support, which are finite valued and S-continuous on \({^*\mathbb{R}_b}\). The Silva axioms and related definitions given are exhaustive, which is followed by a theorem to assist further analysis of the problem. The second section deals with the title of the article, beginning with definitions of differential operators of \(\infty\)-order; the inverse Fourier transform in \({^\Xi C_\infty}\), an important lemma and a corollary. The main result is established, which concludes the article as well, in this section in the form of a statement: The inverse Fourier transform of a \(^\Xi\)distribution (i.e. the members of the quotient \({^\Xi C_\infty}\equiv{^\Xi C_\infty}(\mathbb{R})= {^*{\mathbf D}^\infty}\{^sC^\infty\}/ \Xi\)) in \({^\Xi C_\infty}\) is representable as a finite sum of (standard) \(\infty\)-order derivative of internal functions in \({^*{\mathcal Z}}\), whose standard parts are continuous functions of polynomial growth. The references quoted are very useful for furthering the research and it is firmly believed that we may extend such work to Boehmians [cf. \textit{T. K. Boehme}, Trans. Am. Math. Soc. 176, 319-334 (1973; Zbl 0268.44005)] and Boehmian spaces.