Generalized Weyl transformation
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.
scientific article; zbMATH DE number 4132901
scientific article