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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.

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
Full Text: DOI
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