Finite dimensional Hida distributions
type: Item
instance of: scholarly article
title: Finite dimensional Hida distributions
zbMATH Open document ID: 0892.46045
author: Izumi Kubo
author: Hui-Hsiung Kuo
published in: Journal of Functional Analysis
publication date: 1998-08-03T00:00:00Z
review text: The title of the paper speaks of the composition of the Schwartz distributions and Gaussian random variables in white noise analysis. Let \(E\) be a real separable Hilbert space and \(A\) a positive selfadjoint operator in \(E\) such that \(A^{-\alpha}\), for some \(\alpha>0\), is a trace class operator. Let \({\mathcal E}\subset E\subset{\mathcal E}^*\) be the Gel'fand triple arising from \(E\) and \(A\). Let \(\mu\) be the standard Gaussian measure on \({\mathcal E}^*\) and let \((L^2)= L^2(\mu)\). By the Wiener-Itô decomposition theorem with the second quantization operator \(\Gamma(A)\) one can construct a Gel'fand triple \(({\mathcal E})\subset (L^2)\subset({\mathcal E})^*\). An element \(\varphi\) in the spaces \(({\mathcal E})^*\) and \(({\mathcal E})\) is called a Hida distribution and testing functional, respectively. It is said to be finite-dimensional if there exists a finite-dimensional subspace \(V\) (say, spanned by \(e_1,\dots, e_k\)) of \(E\) such that it belongs to the \(({\mathcal E})^*\)-closure of polynomials of \(\langle\cdot,e_1\rangle,\dots, \langle\cdot, e_k\rangle\). Theorems are given of characterization of the finite-dimensional element in \(({\mathcal E})^*\) and \(({\mathcal E})\), and of approximation of the element in \(({\mathcal E})^*\) and \(({\mathcal E})\) by finite-dimensional ones. A crucial ingredient is the Gel'fand triple \({\mathcal H}({\mathbf R}^k)\subset {\mathcal H}_0({\mathbf R}^k)\subset {\mathcal H}^*({\mathbf R}^k)\) arising from the standard Gaussian measure on \({\mathbf R}^k\) and the operator \(e^{-tL}\) with the Ornstein-Uhlenbeck operator \(L= \Delta- \sum^c_{j= 1} u_j\partial/\partial u_j\) on \({\mathbf R}^k\). \({\mathcal H}^*({\mathbf R}^k)\) properly includes the space \({\mathcal S}^*({\mathbf R}^k)\) of temperated distributions on \({\mathbf R}^k\), but there is no inclusion relation between \({\mathcal H}({\mathbf R}^k)\) and the space \({\mathcal S}({\mathbf R}^k)\) of rapidly decreasing functions. In …
Mathematics Subject Classification ID: 46F25
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 60G20
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 763999
zbMATH Keywords: Schwartz distributions
zbMATH Keywords: Gaussian random variables
zbMATH Keywords: white noise analysis
zbMATH Keywords: Gel'fand triple
zbMATH Keywords: standard Gaussian measure
zbMATH Keywords: Wiener-Itô decomposition theorem
zbMATH Keywords: second quantization operator
zbMATH Keywords: Hida distribution
zbMATH Keywords: Ornstein-Uhlenbeck operator
zbMATH Keywords: Fourier transform
zbMATH Keywords: generalized Itô formula
reviewed by: Takashi Ichinose
MaRDI profile type: Publication
full work available at URL: jfan.1995.1022
OpenAlex ID: W1967285843
DOI: 10.1006/JFAN.1995.1022
description: scientific article; zbMATH DE number 763999