## Banach Journal of Mathematical Analysis

### Translation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$

#### Abstract

In this article, we establish the Cameron–Martin translation theorems for the analytic Fourier–Feynman transform of functionals on the product function space $C_{a,b}^{2}[0,T]$. The function space $C_{a,b}[0,T]$ 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 $\mathcal{F}_{A_{1},A_{2}}^{a,b}$ of functionals on $C_{a,b}^{2}[0,T]$, which is a generalization of the Kallianpur and Bromley–Fresnel class $\mathcal{F}_{A_{1},A_{2}}$. We then proceed to establish the translation theorems for the functionals in $\mathcal{F}_{A_{1},A_{2}}^{a,b}$.

#### Article information

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

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

https://projecteuclid.org/euclid.bjma/1544086815

Digital Object Identifier
doi:10.1215/17358787-2018-0022

Mathematical Reviews number (MathSciNet)
MR3892340

Zentralblatt MATH identifier
07002038

#### Citation

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. https://projecteuclid.org/euclid.bjma/1544086815

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