Generalized Fourier-Feynman transforms, convolution products, and first variations on function space
type: Item
instance of: scholarly article
title: Generalized Fourier-Feynman transforms, convolution products, and first variations on function space
zbMATH Open document ID: 1202.60133
DOI: 10.1216/RMJ-2010-40-3-761
author: Seung Jun Chang
author: Jae Gil Choi
author: David Skoug
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-07-23T00:00:00Z
review text: Let \(Y\) be a generalized Brownian motion process, i.e.\ a Gaussian process with an absolutely continuous mean function \(a(t)\), \(t\in [0,T]\), with \(a(0)=0\), \(a'\in L^2[0,T]\), and covariance function \(r(s,t)=\min\{b(s),b(t)\}\), \(s,t\in [0,T]\) where \(b\) is a strictly increasing \(C^1\)-function with \(b(0)=0\). The authors are interested in functionals of the form \[ F(x) = f(\langle\alpha_1,x \rangle, \ldots, \langle\alpha_n,x \rangle) \] where \(\langle\alpha,x \rangle = \int_0^T \alpha(t)\,dx(t)\) is the Paley-Wiener-Zygmund stochastic integral for \(x\in C_{a,b}[0,T]\). (Unfortunately, the authors fail to define this function space in the paper). The authors introduce the \textit{generalized analytic Fourier-Feynman transform}, the \textit{generalized convolution product} and the \textit{generalized first variation} for such functionals \(F\). The main part of the paper is devoted to the study of various relationships if any two or any three distinct operations are combined.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 5760093
zbMATH Keywords: Generalized Brownian motion
zbMATH Keywords: generalized analytic Feynman integral
zbMATH Keywords: generalized Fourier-Feynman transform
zbMATH Keywords: convolution product
zbMATH Keywords: first variation
reviewed by: R. L. Schilling
MaRDI profile type: Publication
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full work available at URL: rmj-2010-40-3-761
OpenAlex ID: W2043782615
description: scientific article; zbMATH DE number 5760093