Finite dimensional Hida distributions
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 the respect of having introduced this triple instead of the Gel'fand triple \({\mathcal S}({\mathbf R}^k)\subset L^2({\mathbf R}^k)\subset{\mathcal S}^*({\mathbf R}^k)\) used before, the authors extend their previous results in \textit{I. Kubo} [Notes Control Inf. Sci. 49, 156-166 (1983; Zbl 0529.60066)], \textit{H.-H. Kuo}, \textit{J. Potthoff} and \textit{L. Streit} [Nagoya Math. J. 121, 185-194 (1991; Zbl 0717.60051)] and \textit{I. Kubo} and \textit{H.-H. Kuo} [`Fourier transform and cylindrical Hida distributions', in ``Kallianpur Festschrift'' (1992; Zbl 0783.60043)]. In fact, it turns out that the finite-dimensional elements in \(({\mathcal E})^*\) (resp. \(({\mathcal E})\)) are just the composition of the elements in \({\mathcal H}^*({\mathbf R}^k)\) (resp. \({\mathcal H}({\mathbf R}^k)\)) and \(\langle\cdot,e_1\rangle,\dots, \langle\cdot, e_k\rangle\), where \(\{e_1,\dots, e_k\}\) are orthonormal in \(E\). Mention is also made of the Fourier transform of the element in \({\mathcal H}^*({\mathbf R}^k)\) together with its relationship with the Fourier transform of the finite-dimensional element in \(({\mathcal E})^*\), and of the generalized Itô formula for \(F(B(t))\) with \(F\in{\mathcal H}^*({\mathbf R}^k)\) and \(B(t)\) a Brownian motion on \({\mathbf R}^k\).
scientific article; zbMATH DE number 763999