Can the Wigner function be determined by properties for translation and   parity transformation on lattice phase space?
  We show that the Fano operator for one dimensional quantum system is uniquely determined by assuming the reasonable behavior under translation and parity transformation on phase space. Contrarily, for the system with lattice phase space the same procedure does not work. 
Minoru Horibe, Takaaki Hashimoto and Akihisa Hayashi