Generalized Weyl transformation
type: Item
instance of: scholarly article
title: Generalized Weyl transformation
zbMATH Open document ID: 0692.46041
DOI: 10.1007/BF02006225
author: Ping Xu
author: Min Qian
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1988-01-01T00:00:00Z
review text: The authors study some properties of the Weyl transform W(f). If the classical observable function f is given on the phase space \({\mathbb{R}}\times {\mathbb{R}}\) then W(f) is defined by the formula \[ W(f)=1/\sqrt{2\pi}\iint_{{\mathbb{R}}\times {\mathbb{R}}}(Ff)(p',q')e^{iq'p+ip'q} dq' dp', \] where Ff is the Fourier transform of f. Motivated by the distribution theory they introduce the concepts of a test operator and generalized operator. This enables them to extend the definition of Weyl transformation over the space of temperate distributions on \({\mathbb{R}}\times {\mathbb{R}}\) and spaces \(L^ p({\mathbb{R}}\times {\mathbb{R}})\) with \(1\leq p\leq 2\). Finally it is shown that the Weyl transformation is an isomorphism from the space of test functions to the space of test operators, and also an isomorphism from the space of distributions to the space of generalized operators.
reviewed by: Ludvík Janoš
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 4132901
zbMATH Keywords: Weyl transform
zbMATH Keywords: Fourier transform
zbMATH Keywords: test operator
zbMATH Keywords: generalized operator
zbMATH Keywords: Weyl transformation
zbMATH Keywords: space of temperate distributions
MaRDI profile type: Publication
cites work: An algebra of pseudodifferential operators and the asymptotics of quantum mechanics
cites work: Mathematical Aspects of the Weyl Correspondence
cites work: Quantum harmonic analysis on phase space
cites work: Q3873024
cites work: Q4224233
cites work: Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space
cites work: Quantization and the uniqueness of invariant structures
cites work: The bra and ket formalism in extended Hilbert space
full work available at URL: https://doi.org/10.1007/bf02006225
OpenAlex ID: W2077998131
Recommended article: Q3645412
Recommended article: Weyl transforms
Recommended article: Q4020579
Recommended article: Q5293137
Recommended article: The conditional Weyl transform and its generalization
Recommended article: Q3480738
Recommended article: Q4718868
Recommended article: Q3344927
Recommended article: Weyl-Bessel transforms
Recommended article: Q3432572
description: scientific article; zbMATH DE number 4132901