  We investigate discretizations of the integrable discrete nonlinear
Schr\"odinger dynamical system and related symplectic structures. We develop an
effective scheme of invariant reducing the corresponding infinite system of
ordinary differential equations to an equivalent finite system of ordinary
differential equations with respect to the evolution parameter. We construct a
finite set of recurrent algebraic regular relations allowing to generate
solutions of the discrete nonlinear Schr\"odinger dynamical system and we
discuss the related functional spaces of solutions. Finally, we discuss the
Fourier transform approach to studying the solution set of the discrete
nonlinear Schr\"odinger dynamical system and its functional-analytical aspects.
