Poisson Summation Formula for The Space of Functionals

Abstract

In our last work, we formulate a Fourier transformation on the infinite-dimensional space of functionals. Here we first calculate the Fourier transformation of infinite-dimensional Gaussian distribution $\exp(-\pi \xi\int_{-\infty}^{\infty}\alpha ^2(t)dt)$ for $\xi\in{\bf C}$ with Re$(\xi)>0$, $\alpha \in L^2({\bf R})$, using our formulated Feynman path integral. Secondly we develop the Poisson summation formula for the space of functionals, and define a functional $Z_s$, $s\in {\bf C}$, the Feynman path integral of that corresponds to the Riemann zeta function in the case Re$(s)>1$.