Journal of the
Korean Mathematical Society JKMS

In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional $F$ given by \[ F(x)=\hat{\nu}((e_1, x)^{\sim}, \ldots, (e_n, x)^{\sim}), \] where $(e,x)^{\sim}$ denotes the Paley-Wiener-Zygmund stochastic integral with $x$ in a very general function space $C_{a,b}[0,T]$ and $\hat\nu$ is the Fourier transform of complex measure $\nu$ on $\mathcal{B}(\Bbb R^n)$ with finite total variation. We then define two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

Keywords: generalized Brownian motion process, Paley-Wiener-Zygmund stochastic integral, cylinder functional, generalized Fourier-Feynman transform, sequential $\mathcal{P}$-transform, sequential $\mathcal{N}$-transform

MSC numbers: 28C20, 60J65