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 $\xi \in \mathbf{C}$ with Re$(\xi )>0$, $\alpha \in L^2(\mathbf{R})$, using our formulated Feynman path integral. Secondly we develop the Poisson summation formula for the space of functionals, and define a functional $Z_s$, $s\in \mathbf{C}$, the Feynman path integral of that corresponds to the Riemann zeta function in the case Re$(s)>1$.

References in corpus (1)

• Infinitesimal Fourier Transformation for The Space of Functionals