A relation of the Wigner quasiprobability to the Bargmann representation of
pure and mixed states by convolution is derived and generalized to the mai
n class of quasiprobabilities and its inversion is given. The derivation us
es a realization of the abstract group SU(1, 1) by second-order differentia
tion and multiplication operators for a pair of complex conjugated variable
s and disentanglement of exponential functions of these operators by group-
theoretical methods. Examples for the calculation of the Wigner quasiprobab
ility via the Bargmann representation of states demonstrate the action of t
his relation. A short collection of different basic representations of the
Wigner quasiprobability is given. An appendix presents results for the dise
ntanglement of SU(1, 1)-group operators by products of special operators in
different ordering.