Infinitesimal Fourier Transformation for The Space of Functionals

TL;DR

This paper develops an infinitesimal Fourier transformation for the space of functionals using nonstandard analysis, extending real numbers and defining functionals on double-meaning lattices, with example calculations.

Contribution

It introduces a novel Fourier transformation framework for functionals within nonstandard analysis, handling double-meaning lattices and their domains.

Findings

Formulation of an infinitesimal Fourier transform for functionals.

Extension of real numbers to nonstandard double-meaning structures.

Calculation of typical examples demonstrating the theory.

Abstract

The purpose is to formulate a Fourier transformation for the space of functionals, as an infinitesimal meaning. We extend ${\bf R}$ to $ ^{\star}(^{\ast}{\bf R})$ under the base of nonstandard methods for the construction. The domain of a functional is the set of all internal functions from a $ ^{\ast}$-finite lattice to a $ ^{\ast}$-finite lattice with a double meaning. Considering a $ ^{\ast}$-finite lattice with a double meaning, we find how to treat the domain for a functional in our theory of Fourier transformation, and calculate two typical examples.