J. Pseudo-Differ. Oper. Appl. (2013) 4:1–12
DOI 10.1007/s11868-012-0059-4
The conditional Weyl transform and its generalization
Leon Cohen · Patrick Loughlin
Received: 10 August 2012 / Revised: 20 November 2012 / Accepted: 21 November 2012 /
Published online: 21 December 2012
© Springer Basel 2012
Abstract The expectation value of the Weyl transform of a symbol with a state
function equals the phase-space averaging of the symbol with the Wigner distribution.
We define the conditional Weyl transform so that its expectation value equals the
conditional average of the symbol taken with the Wigner distribution. Furthermore,
we generalize to arbitrary operator correspondences and the generalized phase-space
distributions.
Keywords Weyl transforms · Instantaneous quantities · Wigner distribution ·
Generalized phase-space distributions
Mathematics Subject Classification (2000) Primary 47G30; Secondary 81S30
The research was supported by the Office of Naval Research; grant numbers N00014-09-1-0162 (LC) and
N00014-10-1-0053 (PL).
L. Cohen (B )
Department of Physics,
Hunter College of the City University of New York,
695 Park Ave, New York, NY 10021, USA
e-mail: leon.cohen@hunter.cuny.edu
P. Loughlin
Department of Bioengineering,
University of Pittsburgh,
Pittsburgh, PA 15261, USA