Let $mathcal{F}(B)$ be the Fresnel class on an abstract Wiener space (B, H, ${omega}$) which consists of functionals F of the form : $$F(x)={int}_H;{exp}{i(h,x)^{sim}}df(h),;x{in}B$$ where $({cdot}{cdot})^{sim}$ is a stochastic inner product between H and B, and $f$ is in $mathcal{M}(H)$, the space of all complex-valued countably additive Borel measures on H. We introduce the concepts of an $L_p$ analytic Fourier-Feynman transform ($1{leq}p{leq}2$) and a convolution product on $mathcal{F}(B)$ and verify the existence of the $L_p$ analytic Fourier-Feynman transforms for functionls in $mathcal{F}(B)$. Moreover, we verify that the Fresnel class $mathcal{F}(B)$ is closed under the $L_p$ analytic Fourier-Feynman transform and the convolution product, respectively. And we investigate some interesting properties for the $n$-repeated $L_p$ analytic Fourier-Feynman transform on $mathcal{F}(B)$. Finally, we show that several results in [9] come from our results in Section 3.