$L_1$ analytic Fourier-Feynman transform on the Fresnel class of abstract Wiener space

Bull. Korean Math. Soc. 1998 Vol. 35, No. 1, 99-117

Jae Moon Ahn Kon-Kuk University

Abstract : Let $(B,H,p_1)$ be an abstract Wiener space and $\Cal F(B)$ the Fresnel class on $(B,H,p_1)$ which consists of functionals $F$ of the form\,: $$ F(x)=\int_H\exp\{i(h,x)^\sim\}\,df(h),\quad x\in B, $$ where $(\cdot,\cdot)^\sim$ is a stochastic inner product between $H$ and $B$, and $f$ is in $\Cal M(H)$, the space of complex Borel measures on $H$. We introduce an $L_1$ analytic Fourier-Feynman transform on $\Cal F(B)$ and verify the existence of the $L_1$ analytic Fourier-Feynman transforms for functionls in $\Cal F(B)$. Furthermore, we introduce a convolution on $\Cal F(B)$, and then verify the existence of the $L_1$ analytic Fourier-Feynman transform for the convolution product of two functionals in $\Cal F(B)$, and we establish the relationships between the $L_1$ analytic Fourier-Feynman transform of the convolution product for two functionals in $\Cal F(B)$ and the $L_1$ analytic Fourier-Feynman transforms for each functional. Finally, we show that most results in [7] follows from our results in Section 3.

Keywords : abstract Wiener space, $L_1$ analytic Fourier-Feynman transform, convolution