## 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

### Keywords:

Fourier-type functionals; integral transform
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