Journal of the
Korean Mathematical Society JKMS

In this paper, we analyze the necessary and sufficient condition introduced in \cite{SJHSF}: that a functional $F$ in $L^2(C_{a,b}[0,T])$ has an integral transform $\mathcal{F}_{\gamma,\beta}F $, also belonging to $L^2(C_{a,b}[0,T])$. We then establish the inverse integral transforms of the functionals in $L^2(C_{a,b}[0,T])$ and then examine various properties with respect to the inverse integral transforms via the translation theorem. Several possible outcomes are presented as remarks. Our approach is a new method to solve some difficulties with respect to the inverse integral transform.

Keywords: generalized Brownian motion process, generalized integral transform, dense subset, inverse integral transform, translation theorem

MSC numbers: Primary 60J65, 28C20, 46E20, 44B20