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Analytic Fourier-Feynman transforms and convolution. (English) Zbl 0880.28011
The analytic Fourier-Feynman transform of a functional $$F$$ on the Wiener space is defined in terms of the limit values as $$\lambda\to-iq$$ of the analytic Wiener integral of the translate of $$F:\int F(\lambda^{-1/2}(x+ y))m(dx)$$, where $$m$$ is the Wiener measure. For special classes of cylindrical functionals, $$A^{(p)}_n$$ and $$A^{(\infty)}_n$$, i.e., $F(x)= f\Biggl(\int^T_0 \alpha_1(t) dx(t),\dots, \int^T_0\alpha_n(t) dx(t)\Biggr)$ with $$f\in L_p(\mathbb{R}^n)$$ and $$C_0(\mathbb{R}^n)$$, respectively, and $$\int^T_0 \alpha_i(t) dx(t)$$ Paley-Wiener-Zygmund stochastic integrals, the Fourier-Feynman transforms are shown to exist and are explicitly calculated as integrals over $$\mathbb{R}^n$$. Properties, such as the fact that the Fourier-Feynman transform of $$A^{(p)}_n$$ is an $$A^{(p')}_n$$, or the usual relation with convolution, are derived.

##### MSC:
 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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