Integration formulas involving Fourier-Feynman transforms via a Fubini theorem. (Q2718441)

The authors established a general Fubini theorem for analytic Feynman integrals in [\textit{T. Huffman, D. Skoug} and \textit{D. Storvick}, J. Korean Math. Soc. 38, 409--420 (2001; Zbl 0984.28008)]. In this paper, they obtained several Feynman integration formulas involving analytic Fourier-Feynman transforms by using the general Fubini theorem. They also establish a general Parseval relation. In a unifying paper [J. Funct. Anal. 47, 153--164 (1982; Zbl 0487.44006)], \textit{Y.-J. Lee} obtained some similar results for several different integral transforms including the Fourier-Feynman transform. However the results in this paper hold for much more general functionals. In fact, Lee required the functional \(F(x+\lambda y)\) to be an entire function of \(\lambda\) ober \(\mathbb C\) for each Wiener path \(x\) and \(y\), whereas \(F\) was not even required to be a continuous function. The classes of functionals studied by \textit{J. Yeh} in [Pac. J. Math. 15, 731--738 (1965; Zbl 0128.33702)] and \textit{Il Yoo} in [Rocky Mt. J. Math. 25, 1577--1587 (1995; Zbl 0855.28006)] for the Fourier-Wiener transform are similar to those used by Lee [loc. cit.].