Paley-Wiener theorem for white noise analysis
type: Item
instance of: scholarly article
title: Paley-Wiener theorem for white noise analysis
zbMATH Open document ID: 0961.46033
published in: Journal of Functional Analysis
publication date: 2001-05-29T00:00:00Z
review text: 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 \(\zeta\),…
Mathematics Subject Classification ID: 46F25
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 1484763
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: distributional Fourier transform
zbMATH Keywords: infinite-dimensional white noise space
zbMATH Keywords: Gaussian measure
zbMATH Keywords: test functions
zbMATH Keywords: generalized functions
zbMATH Keywords: \(S\)-transform
zbMATH Keywords: compact subsets
author: A. I. Stan
reviewed by: Dragisa Mitrović
MaRDI profile type: Publication
full work available at URL: jfan.2000.3571
OpenAlex ID: W2089288875
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DOI: 10.1006/JFAN.2000.3571
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description: scientific article; zbMATH DE number 1484763