白噪声分析中的一般特征定理和内禀拓扑
General characterization theorems and intrinsic topologies in white noise analysis
摘要：
令$u$是$[0, \infty)$上的正连续函数，满足以下条件：（i）$\lim_{r\to\infty} r^{-1/2}\log u(r)=\infty$，（ii）$\inf_{r\geq 0} u(r)=1$，（iii）$\lim_{r\to \infty}\break r^{-1}\log u(r)<\infty$，（iv）函数$\log u(x^{2}), x\geq 0$是凸的。利用\cite {akk 3}中讨论的$u$的Legendre变换，构造了一个Gel'fand三元组$[\ce]_{u} \subset (L^{2}) \subset [\ce]_{u}^{*}$.在相同的假设下，证明了$[\ce]_{u}^{*}$中广义函数和$[\ce]_{u}$中测试函数的$S$ -变换的特征定理.此外，我们给出了测试函数空间$[\ce]_{u}$的一个内在拓扑，并证明了测度的一个特征定理。我们简要地提到我们的方法和最近的工作由Gannoun等人。\cite{ghor}之间的关系。最后给出了白噪声算子理论和Wick乘积的实现条件。
英文摘要：
Let $u$ be a positive continuous function on $[0, \infty)$ satisfying the conditions: (i) $\lim_{r\to\infty} r^{-1/2}\log u(r)=\infty$, (ii) $\inf_{r\geq 0} u(r)=1$, (iii) $\lim_{r\to \infty}\break r^{-1}\log u(r)<\infty$, (iv) the function $\log u(x^{2}), x\geq 0$, is convex. A Gel'fand triple $[\ce]_{u} \subset (L^{2}) \subset [\ce]_{u}^{*}$ is constructed by making use of the Legendre transform of $u$ discussed in \cite {akk3}. We prove a characterization theorem for generalized functions in $[\ce]_{u}^{*}$ and also for test functions in $[\ce]_{u}$ in terms of their $S$-transforms under the same assumptions on $u$. Moreover, we give an intrinsic topology for the space$[\ce]_{u}$ of test functions and prove a characterization theorem for measures. We briefly mention the relationship between our method and a recent work by Gannoun et al.\cite{ghor}. Finally, conditions for carrying out white noise operator theory and Wick products are given.
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