Conditional Fourier-Feynman transforms and conditional convolution products (Q2708261)

An \(L_p\) analytic Fourier-Feynman transform for \(1\leq p\leq 2\) was developed by \textit{M. D. Brue} [``A functional transform for Feynman integrals similar to the Fourier transform'' (Thesis, University of Minnesota), Minneapolis (1972)], \textit{R. H. Cameron} and \textit{D. A. Storvick} [Mich. Math. J. 23, 1-30 (1976; Zbl 0382.42008)], and \textit{G. W. Johnson} and \textit{D. L. Skoug} [Mich. Math. J. 26, 103-127 (1979; Zbl 0409.28007)]. \textit{D. L. Huffman} and the authors defined a convolution product for functionals on a Wiener space and obtained various results involving and relating the Fourier-Feynman transform and the convolution product [Trans. Am. Math. Soc. 347, No.~2, 661-673 (1995; Zbl 0880.28011); Rocky Mt. J. Math. 27, No.~3, 827-841 (1997; Zbl 0901.28010); Mich. Math. J. 43, No.~2, 247-261 (1996; Zbl 0864.28007)]. In [Int. J. Math. Math. Sci. 20, No.~1, 19-32 (1997; Zbl 0982.28011)], they also worked with a generalized Fourier-Feynman transform and a generalized convolution product using ideas and results from \textit{D. M. Chung} and the authors [Mich. Math. J. 40, No.~2, 377-391 (1993; Zbl 0799.60049)]. In this paper, the authors define a generalized conditional Fourier-Feynman transform and a generalized conditional convolution product and obtaine several interesting relationships between them. In particular, they show that the conditional transform of the conditional convolution product is the product of conditional transforms.