Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Byoung Kim; Il Yoo; Dong Cho

Open Mathematics (2010)

  • Volume: 8, Issue: 3, page 616-632
  • ISSN: 2391-5455

Abstract

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Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class ℱ 𝒜 1 , 𝒜 2 A1,A2 than the Fresnel class ℱ (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form F x = G x ψ e → , x ∼ , where G∈ ℱ (B)and Ψ = ψ + ϕ with ψ ∈ L 1(ℝn) and ϕ is the Fourier transform of a complex Borel measure of bounded variation on ℝn. We also prove a translation theorem for the analytic Feynman integral of the above functionals.

How to cite

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Byoung Kim, Il Yoo, and Dong Cho. "Fourier-Feynman transforms of unbounded functionals on abstract Wiener space." Open Mathematics 8.3 (2010): 616-632. <http://eudml.org/doc/269381>.

@article{ByoungKim2010,
abstract = {Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class \[ \mathcal \{F\}\_\{\mathcal \{A\}\_1 ,\mathcal \{A\}\_2 \} \] A1,A2 than the Fresnel class \[ \mathcal \{F\} \] (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form \[ F\left( x \right) = G\left( x \right)\psi \left( \{\left( \{\vec\{e\},x\} \right)^ \sim \} \right) \] , where G∈\[ \mathcal \{F\} \] (B)and Ψ = ψ + ϕ with ψ ∈ L 1(ℝn) and ϕ is the Fourier transform of a complex Borel measure of bounded variation on ℝn. We also prove a translation theorem for the analytic Feynman integral of the above functionals.},
author = {Byoung Kim, Il Yoo, Dong Cho},
journal = {Open Mathematics},
keywords = {Abstract Wiener space; Fresnel class; Analytic Feynman integral; Fourier-Feynman transform; Convolution; First variation; Translation theorem; abstract Wiener space; analytic Feynman integral; convolution; first variation; translation theorem},
language = {eng},
number = {3},
pages = {616-632},
title = {Fourier-Feynman transforms of unbounded functionals on abstract Wiener space},
url = {http://eudml.org/doc/269381},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Byoung Kim
AU - Il Yoo
AU - Dong Cho
TI - Fourier-Feynman transforms of unbounded functionals on abstract Wiener space
JO - Open Mathematics
PY - 2010
VL - 8
IS - 3
SP - 616
EP - 632
AB - Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class \[ \mathcal {F}_{\mathcal {A}_1 ,\mathcal {A}_2 } \] A1,A2 than the Fresnel class \[ \mathcal {F} \] (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form \[ F\left( x \right) = G\left( x \right)\psi \left( {\left( {\vec{e},x} \right)^ \sim } \right) \] , where G∈\[ \mathcal {F} \] (B)and Ψ = ψ + ϕ with ψ ∈ L 1(ℝn) and ϕ is the Fourier transform of a complex Borel measure of bounded variation on ℝn. We also prove a translation theorem for the analytic Feynman integral of the above functionals.
LA - eng
KW - Abstract Wiener space; Fresnel class; Analytic Feynman integral; Fourier-Feynman transform; Convolution; First variation; Translation theorem; abstract Wiener space; analytic Feynman integral; convolution; first variation; translation theorem
UR - http://eudml.org/doc/269381
ER -

References

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