Generalized Analytic Fourier-Feynman Transform of Functionals in a Banach Algebra

We introduce the Fresnel type class . We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra . 1. Introduction Let be a separable Hilbert space and let be the space of all complex-valued Borel measures on . The Fourier transform of in is defined by The set of all functions of the form (1) is denoted by and is called the Fresnel class of . Let be an abstract Wiener space. It is known [1, 2] that each functional of the form (1) can be extended to uniquely by where is a stochastic inner product between and . The Fresnel class of is the space of (equivalence classes of) all functionals of the form (2). There has been a tremendous amount of papers and books in the literature on the Fresnel integral theory and Fresnel classes and on abstract Wiener and Hilbert spaces. For an elementary introduction see [3, Chapter 20]. Furthermore, in [1], Kallianpur and Bromley introduced a larger class than the Fresnel class and showed the existence of the analytic Feynman integral of functionals in for a successful treatment of certain physical problems by means of a Feynman integral. The Fresnel class of is the space of (equivalence classes of) all functionals on of the following form: where and are bounded, nonnegative, and self-adjoint operators on and . In this paper we study the functionals of the form (3) with in a very general function space . The function space , induced by generalized Brownian motion process, was introduced by Yeh [4, 5] and was used extensively in [6–13]. In this paper, we also construct a concrete theory of the generalized analytic Fourier-Feynman transform (GFFT) of functionals in a generalized Fresnel type class defined on . Other work involving GFFT theories on include [6, 7, 9, 12, 13]. The Wiener process used in [1, 2, 14–17] is stationary in time and is free of drift while the stochastic process used in this paper, as well as in [4, 6–13, 18], is nonstationary in time and is subject to a drift . It turns out, as noted in Remark 7 below, that including a drift term makes establishing the existence of the GFFT of functionals on very difficult. However, when and on , the general function space reduces to the Wiener space . 2. Definitions and Preliminaries Let be an absolutely continuous real-valued function on with , , and let be a strictly increasing, continuously differentiable real-valued function with and for each . The generalized Brownian motion process determined by and is a Gaussian process with mean function and covariance function . For more details, see [6, 10,