[Paper Review] General characterization theorems and intrinsic topologies in white noise analysis

Nobuhiro Asai, Izumi Kubo+1 more|ArXiv.org|Apr 12, 2001

Advanced Harmonic Analysis Research12 references4 citations

TL;DR

This paper establishes general characterization theorems for generalized functions and test functions in white noise analysis using a function $ u $ satisfying specific growth and convexity conditions. It introduces an intrinsic topology on the space $[\mathcal{E}]_u$ of test functions and proves characterization theorems via $ S $-transforms, while also providing conditions for white noise operator theory and Wick products.

ABSTRACT

Let $u$ be a positive continuous function on $[0, \infty)$ satisfying the conditions: (i) $\lim_{r o\infty} r^{-1/2}\log u(r)=\infty$, (ii) $\inf_{r\geq 0} u(r)=1$, (iii) $\lim_{r o \infty}\break r^{-1}\log u(r)

Motivation & Objective

• To develop a general framework for characterizing generalized functions and test functions in white noise analysis using a positive continuous function $ u $ with specific analytic and convexity properties.
• To construct an intrinsic topology on the space $[\mathcal{E}]_u$ of test functions based on the Legendre transform of $ u $, ensuring nuclearity and completeness.
• To prove characterization theorems for elements in $[\mathcal{E}]_u$ and its dual $[\mathcal{E}]_u^*$ using their $ S $-transforms under the same assumptions on $ u $.
• To identify conditions under which white noise operator theory and Wick products can be rigorously formulated within the constructed framework.
• To relate the proposed method to recent work by Gannoun et al. [10], clarifying connections and distinctions in the functional analytic setup.

Proposed method

• Construct a Gel'fand triple $[\mathcal{E}]_u \subset (L^2) \subset [\mathcal{E}]_u^*$ using the Legendre transform of a function $ u $ satisfying (i)–(iv): growth rate, positivity, logarithmic sublinear growth, and convexity of $ \log u(x^2) $.
• Define the space $[\mathcal{E}]_u$ as the projective limit of Hilbert spaces $[\mathcal{E}_p]_u$ equipped with norms $ \|\varphi\|_{p,u} = \left( \sum_{n=0}^\infty n! \, u(n) \, |f_n|_p^2 \right)^{1/2} $, where $ \varphi $ has chaos expansion $ \sum_n I_n(f_n) $.
• Establish an intrinsic topology on $[\mathcal{E}]_u$ via seminorms derived from the Legendre transform of $ u $, ensuring the space is a nuclear Fréchet space.
• Prove that the $ S $-transform provides a characterization of elements in $[\mathcal{E}]_u^*$ and $[\mathcal{E}]_u $, linking the abstract space to the generating function $ G_u(r) = \sum_{n=0}^\infty \frac{r^n}{n! u(n)} $.
• Use the joint continuity of multiplication in $[\mathcal{E}]_u$ to ensure algebraic structure, proven via estimates on operator norms and decomposition into dyadic levels.
• Derive conditions for the existence of Wick products and white noise operators by analyzing the growth of $ u(n) $, particularly requiring $ \lim_{n\to\infty} \left( \frac{u(n)}{n!} \right)^{1/n} = 0 $.

Experimental results

Research questions

• RQ1 How can a general class of Gel'fand triples in white noise analysis be constructed using a function $ u $ satisfying specific analytic and convexity conditions?
• RQ2 What is the intrinsic topology of the space $[\mathcal{E}]_u$ of test functions, and how does it relate to the Legendre transform of $ u $?
• RQ3 Can the $ S $-transform be used to fully characterize elements in $[\mathcal{E}]_u $ and its dual $[\mathcal{E}]_u^*$ under the same assumptions on $ u $?
• RQ4 Under what conditions on $ u $ can white noise operator theory and Wick products be consistently defined within this framework?
• RQ5 How does this approach compare to or relate to the CKS-space construction and recent work by Gannoun et al. [10]?

Key findings

• The space $[\mathcal{E}]_u $ is a nuclear Fréchet space equipped with an intrinsic topology defined via the Legendre transform of $ u $, ensuring completeness and continuity of operations.
• A characterization theorem for generalized functions in $[\mathcal{E}]_u^* $ is established via their $ S $-transforms, where the $ S $-transform of $ \varphi \in [\mathcal{E}]_u^* $ is given by $ S[\varphi](\xi) = \sum_{n=0}^\infty \frac{1}{n!} \langle f_n, \xi^{\otimes n} \rangle $, with $ f_n \in \mathcal{E}_0^{\widehat{\otimes}n} $.
• The $ S $-transform provides a topological isomorphism between $[\mathcal{E}]_u $ and a space of entire functions on $ \mathcal{E}_c $, with growth controlled by $ G_u(r) $, the exponential generating function of $ 1/u(n) $.
• The product in $[\mathcal{E}]_u $ is jointly continuous, proven by estimating operator norms and decomposing products into dyadic levels with controlled error terms.
• White noise operator theory and Wick products can be carried out if $ \lim_{n\to\infty} \left( \frac{u(n)}{n!} \right)^{1/n} = 0 $, ensuring sufficient decay of the weight sequence.
• The framework generalizes the CKS-space construction and provides a broader class of test function spaces than previously considered, with a unified treatment of $ S $-transform characterizations and intrinsic topologies.