  A study on a method for the establishment of a phase space representation of
quantum theory is presented. The approach utilizes the properties of Gaussian
distribution, the properties of Hermite polynomials, Fourier analysis and the
current formulation of quantum mechanics which is based on the use of Hilbert
space and linear operators theory. Phase space representation of quantum states
and wave functions in phase space are introduced using properties of a set of
functions called harmonic Gaussian functions. Then, new operators called
dispersion operators are defined and identified as the operators which admit as
eigenstates the basis states of the phase space representation. Generalization
of the approach for multidimensional cases is shown. Examples of applications
are given.
