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Generalized Fourier-Feynman transforms, convolution products, and first variations on function space. (English) Zbl 1202.60133
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 generalized analytic Fourier-Feynman transform, the generalized convolution product and the 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.

##### MSC:
 60J65 Brownian motion 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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##### References:
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