Convolution and Fourier-Feynman Transforms

Huffman, Timothy; Park, Chull; Skoug, David

doi:10.1216/rmjm/1181071896

Cited by 44 publications

(35 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also study an associativity result of the generalized convolution product in Theorem 3.9. Although the associativity of the convolution product for the Fourier-Feynman transform was studied in [4,9], the associativity for the convolution product has not yet been established. In Section 4, we establish various relationships between the generalized convolution product and the first variation, while in Section 5, we obtain relationships involving the integral transform, the generalized convolution product, and the first variation where each concept is used exactly once.…”

Section: θ(T X(t)) Dtmentioning

confidence: 99%

See 1 more Smart Citation

Generalized convolution product for an integral transform on a Wiener space

Kim¹,

Yoo²

2017

Turk J Math

View full text Add to dashboard Cite

Section: θ(T X(t)) Dtmentioning

confidence: 99%

“…Huffman et al [9] and Chang et al [4] studied associativity of the convolution product for the FourierFeynman transform. However, Example 3.8 below shows that the generalized convolution product is not associative, that is, it is not always true that (( …”

Section: Corollary 37mentioning

confidence: 99%

Generalized convolution product for an integral transform on a Wiener space

Kim¹,

Yoo²

2017

Turk J Math

View full text Add to dashboard Cite

“…We then complete this function space to obtain the measure space [1][2][3]19,[22][23][24][27][28][29] if and only if a(t) ≡ 0…”

Section: The Function Space C Ab [0 T ]mentioning

confidence: 99%

“…The functionals in S are defined as a stochastic Fourier transform of complex measures on L 2 [0, T ], and are bounded on C 0 [0, T ]. Other classes of the analytic Feynman integrable functionals on C 0 [0, T ] can be found in [2,19,[22][23][24][27][28][29]. But the 'analytic Feynman integral' cannot be interpreted as the integration in standard measure theory.…”

mentioning

confidence: 99%

“…As mentioned above, the SBMP used in [1][2][3]19,[22][23][24][27][28][29] is stationary in time and is free of drift. However, the stochastic process used in this paper, as well as in [4][5][6][7][8][9][10][11][12][13][14][15][16]18,20,21,25,30], is non-stationary in time and is subject to a drift because In this note, we present an example of a bounded functional which is not generalized analytic Feynman integrable on the function space C a,b [0, T ].…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Chang¹,

Choi²

2016

Arch. Math.

View full text Add to dashboard Cite

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)

show abstract

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

Chang

Choi

Chung

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.

show abstract

Convolution and Fourier-Feynman Transforms

Cited by 44 publications

References 9 publications

Generalized convolution product for an integral transform on a Wiener space

Generalized convolution product for an integral transform on a Wiener space

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

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

Contact Info

Product

Resources

About