scholarly journals Change of Scale Formulas for Wiener Integrals Related to Fourier-Feynman Transform and Convolution

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Bong Jin Kim ◽  
Byoung Soo Kim ◽  
Il Yoo
Keyword(s):  
Banach Algebra ◽  
Wiener Space ◽  
Convolution Product ◽  
Abstract Wiener Space ◽  
Change Of Scale ◽  
Generalized Fresnel Class

Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in Banach algebraSon classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel classF(B)and in a generalized Fresnel classFA1,A2on abstract Wiener space. We express Fourier-Feynman transform and convolution product of functionals inSas limits of Wiener integrals. Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals.

scholarly journals Conditional Fourier-Feynman Transforms with Drift on a Function Space

2019 ◽  
Vol 2019 ◽  
pp. 1-16
Author(s):  
Dong Hyun Cho ◽  
Suk Bong Park
Keyword(s):  
Banach Algebra ◽  
Function Space ◽  
Fourier Transforms ◽  
Convolution Product ◽  
Borel Measures ◽  
Conditional Expectations ◽  
Change Of Scale ◽  
Convolution Products ◽  
Complex Borel Measures ◽  
Generalized Cylinder

In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures on L2[0,T] using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish various changes of scale formulas for the conditional transforms and the conditional convolution products.

scholarly journals A change of scale formula for Wiener integrals of cylinder functions on abstract Wiener space

1998 ◽  
Vol 21 (1) ◽  
pp. 73-78 ◽  
Author(s):  
Young Sik Kim
Keyword(s):  
Feynman Integral ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Wiener Integral ◽  
Feynman Integrals ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
The Relationship

The purpose of this paper is to establish the existence of analytic Wiener and Feynman integrals for a class of certain cylinder functions which is of the form:F(x)=f((h1,x)∼,⋯,(hn,x)∼),    x∈B,on the abstract Wiener space, and to establish the relationship between the Wiener integral and the analytic Feynman integral for such cylinder functions on the abstract Wiener space. We then establish a change of scale formula for Wiener integrals of such cylinder functions on the abstract Wiener space.

scholarly journals A change of scale formula for Wiener integrals of cylinder functions on the abstract Wiener space II

2001 ◽  
Vol 25 (4) ◽  
pp. 231-237 ◽  
Author(s):  
Young Sik Kim
Keyword(s):  
Fourier Transform ◽  
Borel Measure ◽  
Feynman Integral ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
The Fourier Transform ◽  
Complex Valued

We show that for certain bounded cylinder functions of the formF(x)=μˆ((h1,x)∼,...,(hn,x)∼),x∈Bwhereμˆ:ℝn→ℂis the Fourier-transform of the complex-valued Borel measureμonℬ(ℝn), the Borelσ-algebra ofℝnwith‖μ‖<∞, the analytic Feynman integral ofFexists, although the analytic Feynman integral,limz→−iqIaw(F;z)=limz→−iq(z/2π)n/2∫ℝnf(u→)exp{−(z/2)|u→|2}du→, do not always exist for bounded cylinder functionsF(x)=f((h1,x)∼,...,(hn,x)∼),x∈B. We prove a change of scale formula for Wiener integrals ofFon the abstract Wiener space.

Conditional Fourier-Feynman Transform and Convolution Product Over Wiener Paths In Abstract Wiener Space

2003 ◽  
Vol 14 (3) ◽  
pp. 217-235 ◽  
Author(s):  
K. S. Chang ◽  
D. H. Cho ◽  
B. S. Kim ◽  
T. S. Song ◽  
I. Yoo
Keyword(s):  
Wiener Space ◽  
Convolution Product ◽  
Abstract Wiener Space

scholarly journals A Rotation of Admixable Operators on Abstract Wiener Space with Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jae Gil Choi ◽  
Seung Jun Chang
Keyword(s):  
Algebraic Structure ◽  
Wiener Measure ◽  
Wiener Space ◽  
Convolution Product ◽  
Abstract Wiener Space

We investigate certain rotation properties of the abstract Wiener measure. To determine our rotation property for the Wiener measure, we introduce the concept of an admixable operator via an algebraic structure on abstract Wiener space. As for applications, we define the analytic Fourier-Feynman transform and the convolution product associated with the admixable operators and proceed to establish the relationships between this transform and the corresponding convolution product.

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