Infinitesimal Fourier Transformation for The Space of Functionals

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.