Unitarity of Kuo's Fourier-Mehler transform

Un Cig Ji; Nobuaki Obata

doi:10.1142/S0219025704001487

Unitarity of Kuo's Fourier-Mehler transform

Un Cig Ji, Nobuaki Obata

Center for Data-driven Science and Artificial Intelligence (CDS)

Chungbuk National University

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

Kuo's Fourier-Mehler transforms {F_θ} form a one-parameter automorphism group of the space of white noise distributions W*. The adjoint transforms {G_θ = F^θ*} form a one-parameter automorphism group of the space of test white noise functions W but do not admit unitary extensions with respect to the L²-norm induced from the Gaussian measure with variance 1. We prove that {G _θ} admits an extension to a one-parameter group of unitary operators on the L²-space with respect to the Gaussian measure with variance 1/2.

Original language	English
Pages (from-to)	147-154
Number of pages	8
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	7
Issue number	1
DOIs	https://doi.org/10.1142/S0219025704001487
Publication status	Published - 2004 Mar

Keywords

Fourier transform
Fourier-Mehler transform
Gaussian space
Generalized white noise functional
White noise

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abstract = "Kuo's Fourier-Mehler transforms {Fθ} form a one-parameter automorphism group of the space of white noise distributions W*. The adjoint transforms {Gθ = Fθ*} form a one-parameter automorphism group of the space of test white noise functions W but do not admit unitary extensions with respect to the L2-norm induced from the Gaussian measure with variance 1. We prove that {G θ} admits an extension to a one-parameter group of unitary operators on the L2-space with respect to the Gaussian measure with variance 1/2.",

keywords = "Fourier transform, Fourier-Mehler transform, Gaussian space, Generalized white noise functional, White noise",

author = "Ji, {Un Cig} and Nobuaki Obata",

note = "Funding Information: U.C.J. was supported by grant (No. R05-2002-000-00142-0) from the Basic Research Program of the Korea Science & Engineering Foundation. N.O. was supported in part by Strategic Information and Communications R&D Promotion Scheme of the Ministry of Public Management, Home Affairs, Posts and Telecommunications, Japan.",

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N2 - Kuo's Fourier-Mehler transforms {Fθ} form a one-parameter automorphism group of the space of white noise distributions W*. The adjoint transforms {Gθ = Fθ*} form a one-parameter automorphism group of the space of test white noise functions W but do not admit unitary extensions with respect to the L2-norm induced from the Gaussian measure with variance 1. We prove that {G θ} admits an extension to a one-parameter group of unitary operators on the L2-space with respect to the Gaussian measure with variance 1/2.

AB - Kuo's Fourier-Mehler transforms {Fθ} form a one-parameter automorphism group of the space of white noise distributions W*. The adjoint transforms {Gθ = Fθ*} form a one-parameter automorphism group of the space of test white noise functions W but do not admit unitary extensions with respect to the L2-norm induced from the Gaussian measure with variance 1. We prove that {G θ} admits an extension to a one-parameter group of unitary operators on the L2-space with respect to the Gaussian measure with variance 1/2.

KW - Fourier transform

KW - Fourier-Mehler transform

KW - Gaussian space

KW - Generalized white noise functional

KW - White noise

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Unitarity of Kuo's Fourier-Mehler transform

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Keywords

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