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Gel’fand triples associated with finite dimensional Gaussian measure. (English) Zbl 0858.46034

Summary: Recently, Kubo and Kuo have characterized the finite-dimensional Hida distributions in \(({\mathcal S})^*\). In the characterization, they have introduced the finite-dimensional Gel’fand triple \({\mathcal H}(\mathbb{R}^k)\subset {\mathcal H}_0(\mathbb{R}^k)\subset {\mathcal H}^*(\mathbb{R}^k)\). We construct a new Gel’fand triple \({\mathcal H}^\beta(\mathbb{R}^k)\subset {\mathcal H}^0_0(\mathbb{R}^k)\subset ({\mathcal H}^\beta)^*(\mathbb{R}^k)\) for \(0\leq \beta<1\) and generalize some of their results. In particular, we will generalize the characterization theorems for the spaces \({\mathcal H}^*(\mathbb{R}^k)\) and \({\mathcal H}(\mathbb{R}^k)\) to the spaces \(({\mathcal H}^\beta)^*(\mathbb{R}^k)\) and \({\mathcal H}^\beta(\mathbb{R}^k)\). The Fourier transform is also extended to the space \(({\mathcal H}^\beta)^*(\mathbb{R}^k)\) for further study of the finite-dimensional distribution theory.

MSC:

46F25 Distributions on infinite-dimensional spaces
46F10 Operations with distributions and generalized functions
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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