## LSU Historical Dissertations and Theses

1993

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Hui-Hsiung Kuo

#### Abstract

The mathematical framework of white noise analysis is based on an infinite dimensional analogue of the Schwartz distribution theory. The Lebesgue measure on $\IR\sp{k}$ is replaced with standard Gaussian measures $\mu$ on infinite dimensional spaces. There is an infinite dimensional analogue $({\cal E})\subset L\sp2({\cal E}\sp*,\mu)\subset({\cal E})\sp*$ of a Gel'fand triple ${\cal E}\subset E\subset{\cal E}\sp*$ which is obtained from ${\cal S}(\IR\sp{k})\subset L\sp2(\IR\sp{k})\subset{\cal S}\sp*(\IR\sp{k})$ in a general setup. There are spaces $({\cal E}\sp\beta),({\cal E}\sp\beta)\sp*, \beta\in\lbrack 0,1)$ with $({\cal E}\sp\beta)\subset({\cal E})\subset L\sp2({\cal E}\sp*,\mu)\subset({\cal E})\sp*\subset({\cal E}\sp\beta)\sp*.$. The compositions of Schwartz distributions and Gaussian random variables have been discussed. A new Gel'fand triple ${\cal H}(\IR\sp{k})\subset{\cal H}\sb0(\IR\sp{k})\subset{\cal H}\sp*(\IR\sp{k})$ over $\IR\sp{k}$ plays the key role for characterizing the class of functions F such that the composition $F\ \circ\ (\langle{\cdot},\xi\sb 1\rangle,\...,\langle{\cdot},\xi\sb {k}\rangle),\xi\sb1,\...,\xi\sb{k}\in E$ is in $({\cal E})\sp*$ and the class of functions F such that $F\ \circ\ (\langle{\cdot},\xi\sb 1\rangle,\...,\langle{\cdot},\xi\sb{k}\rangle),\xi\sb 1,\...,\xi\sb{k}\in E$ is in $({\cal E})$. Moreover, the spaces ${\cal H}(\IR\sp{k}), {\cal H}\sp*(\IR\sp{k})$ have been characterized. This work introduces a new Gel'fand triple ${\cal H}\sp\beta(\IR\sp{k})\subset{\cal H}\sbsp{0}{0}(\IR\sp{k})\subset({\cal H}\sp\beta)\sp*(\IR \sp{k})$ to extend these types of results to $({\cal E}\sp\beta)$ and $({\cal E}\sp\beta)\sp*$ with characterizations of the spaces ${\cal H}\sp\beta(\IR\sp{k}), ({\cal H}\sp\beta)\sp*(\IR\sp{k}).$.

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