Fourier-Feynman transforms, convolutions and first variations on the space of abstract Wiener space valued continuous functions
The article is devoted to properties of the Fourier-Feynman transform on a Banach algebra associated with Wiener's space. An abstract Wiener space \((H,B,\nu )\) was taken and the Banach space \(C_0(B)\) of all continuous bounded functions \(x: [0,T]\to B\) with \(x(0)=0\) was considered. A Borel measure \(m_B\) on \(C_0(B)\) was induced by a bounded cylinder distribution associated with the measure \(\nu \). The analytic Wiener and Feynman integrals on \(C_0(B)\) relative to \(m_B\) were defined as usual. Then the \(L_p\) analytic Fourier-Feynman transforms \(T^{(p)}_q(F)\) of functionals \(F\) on \(C_0(B)\) were defined for \(1\leq p<\infty \). Further, the classes \(S''_{n,B}\) and \(S''_B\) of analytic Feynman integrable functionals on \(C_0(B)\) were introduced in the paper. They correspond to Cameron-Storvick's classes \(S''_n\) and \(S''\). The class \(S''_B\) was equipped with a Banach algebra structure. The \(L_p\) analytic Fourier-Feynman transform was studied on \(S''_{n,B}\) and \(S''_B\). Its properties concerning convolutions and first variations of functionals were studied. The authors have established various relationships among Fourier-Feynman transform, convolution and first variation of functionals in the Banach algebra \(S''_B\), which were demonstrated in several theorems.
scientific article; zbMATH DE number 5760094
scientific article