Banach Journal of Mathematical Analysis

Translation theorems for the Fourier–Feynman transform on the product function space Ca,b2[0,T]

Seung Jun Chang, Jae Gil Choi, and David Skoug

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In this article, we establish the Cameron–Martin translation theorems for the analytic Fourier–Feynman transform of functionals on the product function space Ca,b2[0,T]. The function space Ca,b[0,T] is induced by the generalized Brownian motion process associated with continuous functions a(t) and b(t) on the time interval [0,T]. The process used here is nonstationary in time and is subject to a drift a(t). To study our translation theorem, we introduce a Fresnel-type class FA1,A2a,b of functionals on Ca,b2[0,T], which is a generalization of the Kallianpur and Bromley–Fresnel class FA1,A2. We then proceed to establish the translation theorems for the functionals in FA1,A2a,b.

Article information

Banach J. Math. Anal., Volume 13, Number 1 (2019), 192-216.

Received: 27 March 2018
Accepted: 27 June 2018
First available in Project Euclid: 6 December 2018

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Zentralblatt MATH identifier

Primary: 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60J65: Brownian motion [See also 58J65] 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

generalized Brownian motion process generalized analytic Feynman integral generalized analytic Fourier–Feynman transform generalized Fresnel-type class translation theorem


Chang, Seung Jun; Choi, Jae Gil; Skoug, David. Translation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$. Banach J. Math. Anal. 13 (2019), no. 1, 192--216. doi:10.1215/17358787-2018-0022.

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