Communications of the
Korean Mathematical Society CKMS

Huffman, Park and Skoug introduced various results for the $L_p$ analytic Fourier-Feynman transform and the convolution for functionals on classical Wiener space which belong to some Banach algebra ${\mathcal S}$ introduced by Cameron and Storvick. Also Chang, Kim and Yoo extended the above results to an abstract Wiener space for functionals in the Fresnel class ${\mathcal F}(B)$ which corresponds to ${\mathcal S}$. Recently Kim, Song and Yoo investigated more generalized relationships between the Fourier-Feynman transform and the convolution product for functionals in a generalized Fresnel class ${\mathcal F}_{A_1,A_2}$ containing ${\mathcal F}(B)$. In this paper, we establish various interesting relationships and expressions involving the first variation and one or two of the concepts of the Fourier-Feynman transform and the convolution product for functionals in ${\mathcal F}_{A_1,A_2}$.

Keywords: abstract Wiener space, generalized Fresnel class, analytic Feynman integral, Fourier-Feynman transform, convolution, first variation

MSC numbers: 28C20