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Fourier-type functionals on Wiener space. (English) Zbl 1252.42010
Let \(S(\mathbb{R}^n)\) be the Schwartz space of \(C^\infty(\mathbb{R}^n)\) functions \(f(u)\), \(u\in\mathbb{R}^n\), decaying at infinity together with all its derivatives faster than any polynomial of \(|u|^{-1}\). All the Laplacians \(\Delta^k f\), \(k= 1,2,\dots\), are elements of \(S(\mathbb{R}^n)\) and so are \(\widehat{\Delta^kf}\). Let \(\alpha_1,\alpha_2,\dots\) be any complete orthonormal set of functions in \(L^2 [0,T]\), let \(C_0[0,T]\) be the Wiener space of continuous real-valued functions \(x\) on \([0,T]\) with \(x(0)= 0\), and let \(\langle\alpha,x\rangle\) be the PWZ stochastic integral. For \(f\in S(\mathbb{R}^n)\) and \(k= 0,1,\dots\) let us consider the Fourier-type functionals \(\Delta^k F(x)= (\Delta^k f)(\langle\alpha, x\rangle)\) and \(\widehat{\Delta^k F}(x)= \widehat{\Delta^kf}(\langle\alpha, x\rangle)\), \(k= 0,1,\dots\) . The authors investigate various properties of these functionals and of their integral transforms.

MSC:
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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