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 $\mathbf{R}$ to ${}^{\star }{(}^{\ast }\mathbf{R})$ 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.