Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

Chang, Seung Jun; Choi, Jae Gil; Skoug, David

doi:10.1090/s0002-9947-03-03256-2

Cited by 47 publications

(40 citation statements)

References 8 publications

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“…Then L 2 a,b [0, T ] is a separable Hilbert space with inner product defined by Remark 2.1. Recall that above, as well as in [12], [14], [15], and [23], we require that a : [0, T ] → R be an absolutely continuous function with a(0) = 0 and with T 0 |a (t)| 2 dt < ∞. Now throughout this paper, we add the requirement that…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…In this section, we will extend the ideas of [20] to obtain expressions of the generalized analytic Feynman integral and the GFFT of functionals of the form (3.2) when A is no longer required to be nonnegative. To do this, we will introduce definitions and notation analogous to those in [15] and [12].…”

Section: The Gfft Of Functionals In a Banach Algebra F Abmentioning

confidence: 99%

“…Next we give the definition of the generalized Fresnel-type class F a,b [12], [15]). Of course, if we choose a(t) ≡ 0, b(t) = t, A 1 = I (identity operator), and A 2 = 0 (zero operator), then the function space C a,b [0, T ] reduces to the classical Wiener space C 0 [0, T ] and the generalized Fresnel-type class F a,b A 1 ,A 2 reduces to the Fresnel class F(C 0 [0, T ]).…”

Section: )mentioning

confidence: 99%

“…where E anfq [F ] means the generalized analytic Feynman integral of F on C a,b [0, T ] (see [12], [15]), and…”

Section: )mentioning

confidence: 99%

“…On the other hand, the translation theorem for the function space integral and the generalized analytic Fourier-Feynman transform (GFFT) have been developed for the functionals on the very general function space C a,b [0, T ] in [9], [10], and [13]. The function space C a,b [0, T ], induced by the generalized Brownian motion process (GBMP), was introduced by Yeh [24], [25] and used extensively in [8], [12]- [16], and [23] (for the precise definition of GBMP, see [24] and [25]).…”

Section: Introductionmentioning

confidence: 99%

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

Chang¹,

Choi²,

Skoug³

2019

Banach J. Math. Anal.

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In this article, we establish the Cameron-Martin translation theorems for the analytic Fourier-Feynman transform of functionals on the productT ] 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 F a,b A1,A2 of functionals on C 2 a,b [0, T ], which is a generalization of the Kallianpur and Bromley-Fresnel class F A1,A2 . We then proceed to establish the translation theorems for the functionals in F a,b A1,A2 .

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Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: The Gfft Of Functionals In a Banach Algebra F Abmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…where E anfq [F ] means the generalized analytic Feynman integral of F on C a,b [0, T ] (see [12], [15]), and…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chang¹,

Choi²,

Skoug³

2019

Banach J. Math. Anal.

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Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Chang¹,

Choi²

2016

Arch. Math.

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In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.Mathematics Subject Classification. 28C20, 60J65, 46G12.1. Introduction. The purpose of this note is to illustrate an effect of drift of the generalized Brownian motion process (GBMP). To do this, we discuss the theory of analytic Feynman integrals. Frankly speaking, in order to emphasize an effect of drift of GBMPs, we present an example of a bounded functional which is not analytic Feynman integrable on the function space C a,b [0, T ]. The function space C a,b [0, T ] is a probability space induced by a GBMP. Let W ≡ C 0 [0, T ] denote one-parameter Wiener space; this is the space of all real-valued continuous functions x on [0, T ] with x(0) = 0. Let M denote the class of all Wiener measurable subsets of C 0 [0, T ] and let m be the Wiener measure. Then, as is well known, (C 0 [0, T ], M, m) is a complete measure space. The coordinate process W on C 0 [0, T ] × [0, T ] defined by (x, t)

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Relationship Between the Analytic Generalized Fourier–Feynman Transform and the Function Space Integral

Choi

2021

Results Math

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Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

Cited by 47 publications

References 8 publications

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

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

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Relationship Between the Analytic Generalized Fourier–Feynman Transform and the Function Space Integral

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