An evaluation formula for Radon-Nikodym derivatives similar to conditional expectations over paths (Q2227946)

A simple formula for conditional Wiener integrals containing time integrals with conditioning functions was introduced by \textit{C. Park} and \textit{D. Skoug} in [Pac. J. Math. 135, No. 2, 381--394 (1988; Zbl 0655.28007)]. And an analogue of Wiener measure space \(C[0,T]\) was introduced by \textit{M. K. Im} and \textit{K. S. Ryu} in [J. Korean Math. Soc. 39, No. 5, 801--819 (2002; Zbl 1017.28007)]. The paper under review is a further work of the author on the Wiener integrals on the Im and Ryu's analogue of Wiener measure space [the author, Trans. Am. Math. Soc. 360, No. 7, 3795--3811 (2008; Zbl 1151.28017); \textit{D. H. Cho}, Filomat 32, No. 18, 6441--6456 (2018; \url{doi:10.2298/FIL1818441C}); J. Korean Math. Soc. 57, No. 2, 451--470 (2020; Zbl 1443.28006)]. In Theorem 4 the author, using the Fourier transform of the process \(W(x,t)=x(t)\) on \(C[0,T]\times [0,T]\), derived a simple evaluation formula for the generalized conditional expectations of functions on \(C[0,T]\). Using Theorem 4, which is one of the main results of this paper, the author evaluated the generalized conditional expectations of various functions in Theorems 5 - 9 which are useful in quantum mechanics and the Feynman integration theory.