Paley-Wiener theorem for white noise analysis
The author generalizes the Paley-Wiener theorem (known from the distributional Fourier transform theory) to an infinite-dimensional white noise space \(({\mathcal E}',\mu)\), where \({\mathcal E}'\) is the dual of nuclear space \({\mathcal E}\) and \(\mu\) is the Gaussian measure on \({\mathcal E}'\). Let \(E\) be a real separated Hilbert space with norm \(\|\cdot\|_0\) and complexification \(E_c\). For each \(p\geq 0\) the space \({\mathcal E}_p= \{f\in E:\|A^pf\|_0< \infty\}\) is a Hilbert space, \(A\) being an operator on \(E\). Then \[ {\mathcal E}= \bigcap_{p\geq 0}{\mathcal E}_p\quad\text{and}\quad {\mathcal E}'= \bigcup_{p\geq 0}{\mathcal E}_{-p}. \] The elements in \(({\mathcal E})\) are called the test functions on \({\mathcal E}'\), the elements in \(({\mathcal E})^*\) are called generalized functions on \({\mathcal E}'\). The bilinear pairing between \(({\mathcal E})^*\) and \(({\mathcal E})\) is denoted by \(\langle\langle\cdot,\cdot\rangle\rangle\). By \((\mathbb{L}^2)\) is denoted the Lebesgue space of functions on \({\mathcal E}'\). Moreover, \[ {\mathcal E}\subset E\subset{\mathcal E}'\quad\text{and} \quad ({\mathcal E})\subset (\mathbb{L}^2)\subset({\mathcal E})^*. \] The infinite-dimensional analogue of the Fourier transform called \(S\)-transform on \(\Phi\) is defined as follows: For any \(\Phi\in({\mathcal E})^*\) it is a function on \({\mathcal E}_i\) given by \[ S\Phi(\zeta)= \langle\langle\Phi: \exp\langle\cdot,\zeta\rangle:\rangle\rangle,\quad \zeta\in{\mathcal E}_c. \] The main results are contained in the four theorems of the same type (only one of them is proved completely). For example, let \(V^n_1= \mathbb{R} e_1+ \mathbb{R} e_2+\cdots+ \mathbb{R} e_{n-1}\). A typical result says that a function \(F:{\mathcal E}_c\to \mathbb{C}\) is the \(S\)-transform of a function \(\varphi\in (\mathbb{L}^2)\) with a compact support (depending upon \(R\)) if and only if the following three conditions hold: 1) \(F\) is continuous; 2) For all \(\zeta\), \(\eta\), \({\mathcal E}_c\), the function \(z\mapsto F(z\zeta+ \eta)\) is analytic; 3) There exists a constant \(C>0\) such that for all \(n\in\mathbb{N}\) and for all \(\zeta_1\in{\mathcal E}_c\), \(\zeta_1\perp V^n_{1,c}\) we have \[ 1/\pi^{n- 1}\int|F(\zeta+ \zeta_1)|^2 e^{-|\zeta|^2_0} d\zeta= C e^{2R|\text{Re }\zeta_1|_p- \text{Re}\langle\zeta_1, \zeta_1\rangle} \] over \(V^n_{1,c}\). A part of the present paper is devoted to a description of the compact subsets of \({\mathcal E}'\) and to some points of the probability. The paper is based on \textit{H.-H. Kuo's} book ``White noise distribution theory'', Boca Raton/FL. (1996; Zbl 0853.60001). The author beliefs that this paper ``opens the gate for further research and a deeper understanding of the infinite-dimensional space in the world of white noise analysis''.
scientific article; zbMATH DE number 1484763
scientific article