Poisson Summation Formula for the Space of Functionals

In our last work, we formulate a Fourier transformation on the infinitedimensional space of functionals. Here we first calculate the Fourier transformation of infinite-dimensional Gaussian distribution exp ( −πξ ∞ −∞ α 2(t)dt ) for ξ ∈ C with Re(ξ) > 0, α ∈ L2(R), using our formulated Feynman path integral. Secondly we develop the Poisson summation formula for the space of functionals, and define a functional Zs, s ∈ C, the Feynman path integral of that corresponds to the Riemann zeta function in the case Re(s) > 1. 0. Introduction Feynman([F-H]) used the concept of his path integral for physical quantizations. The word ′′physical quantizations′′ has two meanings : one is for quantum mechanics and the other is for quantum field theory. We usually use the same word ′′Feynman path integral′′. However the meanings included in ′′Feynman path integral′′ are two sides, according to the above. One is of quantum mechanics and the other is of quantum field theory. The first Feynman path integral corresponds to a study of functional analysis on the space of functions. For functional analysis, there exist many works from standard analysis and nonstandard analysis. However an approach has been hard from standard analysis or nonstandard analysis to study the space of ′′functionals′′ associating with the second Feynman path integral. In our last paper([N-O2]), we defined a delta functional δ and an infinitesimal Fourier transformation F in the space of functionals as one of generalizations for Kinoshita’s infinitesimal Fourier transformation in the space of functions. Historically, in 1962, Gaishi Takeuchi([T]) introduced an infinitesimal δfunction for the space of functions under nonstandard analysis. In 1988, 1990, Kinoshita([K1],[K2]) defined an infinitesimal Fourier transformation for the space of functions. Nitta and Okada([N-O1],[N-O2]) defined, for funtionals, an infinitesimal Fourier transformation, using a concept of double infinitesimal, and calculated the infinitesimal Fourier transformation for two typical examples. The main idea is to use the concept of double infinitesimals and putting standard parts twice st(st( . )). In our theory, the infinitesimal Fourier transformation of δ, δ, ... , and √ δ, ... can be calculated as constant functionals, 1, infinite, ... , and infinitesimal, ... . Now let H be an even infinite number in ∗R, and L be an infinitesimal lattice