[Paper Review] The space of tempered distributions as a k-space

Kei Harada, Hayato Saigo|arXiv (Cornell University)|Sep 8, 2010

Mathematical Analysis and Transform Methods15 references3 citations

TL;DR

This paper establishes that the space of tempered distributions 𝒮′ equipped with the strong topology is a k-space, enabling a unified topological framework for infinite-dimensional harmonic analysis. By leveraging the k-space structure, the authors prove key results on weak convergence of probability measures via characteristic functions, including a new proof of Donsker's invariance principle for the white noise measure using infinite-dimensional characteristic functions.

ABSTRACT

In this paper, we investigate the roles of compact sets in the space of tempered distributions $\mathscr{S}^{\prime}$. The key notion is "k-spaces", which constitute a fairly general class of topological spaces. In a k-space, the system of compact sets controls continuous functions and Borel measures. Focusing on the k-space structure of $\mathscr{S}^{\prime}$, we prove some theorems which seem to be fundamental for infinite dimensional harmonic analysis from a new and unified view point. For example, the invariance principle of Donsker for the white noise measure is shown in terms of infinite dimansional characteristic functions.

Motivation & Objective

• To investigate the role of compact sets in the space of tempered distributions 𝒮′ using the topological concept of k-spaces.
• To establish that 𝒮′ with the strong topology is a k-space, thereby enabling a unified framework for infinite-dimensional harmonic analysis.
• To apply the k-space structure to prove fundamental results on weak convergence of probability measures on 𝒮′.
• To re-derive Donsker's invariance principle for the white noise measure using infinite-dimensional characteristic functions.

Proposed method

• Utilizes the definition of a k-space: a topological space where a set is closed iff its intersection with every compact set is closed.
• Applies the fact that Hilbert spaces and inductive limits of Hilbert spaces (like 𝒮′) are k-spaces, and uses the strong topology on 𝒮′.
• Employs the compactness of embedding maps between dual Sobolev spaces H_{-n} to H_{-n-2} to construct neighborhoods of zero.
• Uses the inductive limit topology to show that a set F ⊂ 𝒮′ is closed if F ∩ K is closed for every compact K ⊂ 𝒮′.
• Applies Prohorov’s theorem and equicontinuity of characteristic functions to characterize weak convergence of measures on 𝒮′.
• Uses Minlos’ theorem and Fubini’s theorem to relate characteristic functions to measures via positive definite continuous functions.

Experimental results

Research questions

• RQ1 Is the space of tempered distributions 𝒮′ a k-space under the strong topology, and what are the implications for its topological structure?
• RQ2 How does the k-space property of 𝒮′ enable new characterizations of weak convergence of probability measures on 𝒮′?
• RQ3 Can Donsker’s invariance principle for the white noise measure be re-derived using characteristic functions in infinite-dimensional settings?
• RQ4 What is the relationship between equicontinuity of characteristic functions and tightness of measures on 𝒮′?

Key findings

• The space of tempered distributions 𝒮′ equipped with the strong topology is a k-space, as shown via the inductive limit structure of Hilbert spaces H_{-n} and the closedness of images of bounded convex sets under continuous embeddings.
• A subset F ⊂ 𝒮′ is closed if and only if F ∩ K is compact for every compact K ⊂ 𝒮′, confirming the k-space property.
• Weak convergence of a sequence of probability measures {μ_n} on 𝒮′ is equivalent to the pointwise convergence of their characteristic functions {ŵ(φ)} to ŵ(φ) for all φ ∈ 𝒮.
• The sequence of measures P_n induced by the scaled process X^{(n)} converges weakly to the white noise measure, with characteristic functions converging to exp(−½∫φ(t)²dt).
• Equicontinuity of characteristic functions at zero is equivalent to uniform tightness and relative compactness of the sequence {μ_n} on 𝒮′.
• The invariance principle of Donsker for the white noise measure is re-proven using the k-space structure and infinite-dimensional characteristic functions.