The behavior of the first variation under the Fourier-Feynman transform on abstract Wiener spaces

We construct the behavior of the first variation under the Fourier-Feynman transform and we show that the Wiener integral of the first variation can be nicely behaved under the Fourier-Feynman transform for functionals F(x) = integral(H) exp{i (h, x)(similar to)} d mu(h) in the Fresnel class and the Fourier-Feynman transform of the first variation can be perfectly expressed as the limit of Wiener integrals of the first variation on the abstract Wiener space.