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

Discrete approximations on functional classes for the integrable  nonlinear Schr\"odinger dynamical system: a symplectic finite-dimensional  reduction approach