Fourier-Feynman transforms, convolutions and first variations on the space of abstract Wiener space valued continuous functions
type: Item
instance of: scholarly article
title: Fourier-Feynman transforms, convolutions and first variations on the space of abstract Wiener space valued continuous functions
zbMATH Open document ID: 1204.28021
DOI: 10.1216/RMJ-2010-40-3-789
author: Teuk Seob Song
author: Kun Soo Chang
author: Byoung Soo Kim
author: Il Yoo
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-07-23T00:00:00Z
review text: The article is devoted to properties of the Fourier-Feynman transform on a Banach algebra associated with Wiener's space. An abstract Wiener space \((H,B,\nu )\) was taken and the Banach space \(C_0(B)\) of all continuous bounded functions \(x: [0,T]\to B\) with \(x(0)=0\) was considered. A Borel measure \(m_B\) on \(C_0(B)\) was induced by a bounded cylinder distribution associated with the measure \(\nu \). The analytic Wiener and Feynman integrals on \(C_0(B)\) relative to \(m_B\) were defined as usual. Then the \(L_p\) analytic Fourier-Feynman transforms \(T^{(p)}_q(F)\) of functionals \(F\) on \(C_0(B)\) were defined for \(1\leq p<\infty \). Further, the classes \(S''_{n,B}\) and \(S''_B\) of analytic Feynman integrable functionals on \(C_0(B)\) were introduced in the paper. They correspond to Cameron-Storvick's classes \(S''_n\) and \(S''\). The class \(S''_B\) was equipped with a Banach algebra structure. The \(L_p\) analytic Fourier-Feynman transform was studied on \(S''_{n,B}\) and \(S''_B\). Its properties concerning convolutions and first variations of functionals were studied. The authors have established various relationships among Fourier-Feynman transform, convolution and first variation of functionals in the Banach algebra \(S''_B\), which were demonstrated in several theorems.
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 5760094
zbMATH Keywords: Wiener space
zbMATH Keywords: Feynman integral
zbMATH Keywords: Fourier-Feynman transform
zbMATH Keywords: first variation
zbMATH Keywords: Banach algebra \(S''_B\)
zbMATH Keywords: convolution
reviewed by: Sergey V. Ludkovsky
MaRDI profile type: Publication
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full work available at URL: rmj-2010-40-3-789
OpenAlex ID: W1978117730
description: scientific article; zbMATH DE number 5760094