Parseval Theorem and Plancherel Theorem on the Product Abstract Wiener Space
Young Sik Kim · Numerical Functional Analysis and Optimization · 2024
We prove the Parseval theorem and the Plancherel Theorem among the analytic Feynman integral and the Fourier Feynman transform and the convolution for the function F:Bν→C in the Class F(Bν): F(x→)=∫H exp {i∑j=1ν(h,xj)∼}dμ(h),μ∈M(H) on the Product Abstract Wiener Space (H,Bν,mν), where Bν=B×B×⋯×B (ν times) and M(H) is the space of a complex valued countably additive measure μ defined on B(H), the Borel class of a real separable infinite dimensional Hilbert space H.