Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Abstract

Abstract Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $$ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $$ A1,A2 than the Fresnel class $$ \mathcal{F} $$(B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form $$ F\left( x \right) = G\left( x \right)\psi \left( {\left( {\vec e,x} \right)^ \sim } \right) $$, where G∈$$ \mathcal{F} $$(B)and Ψ = ψ + ϕ with ψ ∈ L 1(ℝn) and ϕ is the Fourier transform of a complex Borel measure of bounded variation on ℝn. We also prove a translation theorem for the analytic Feynman integral of the above functionals.