The goal  of this study is to discretize (approximate) classical solutions of Klein–Gordon equation, represented as absolutely convergent multiple functional series, and to estimate the discretization error.  The research methodology is based on numerous similar studies from the theory of approximations. In this study, the following results were obtained: first, exact order of the smallest discretization error is explicitly written out; Secondly, it is proved that the computing unit, which is the sum of trigonometric polynomials defined on hyperbolic crosses, implements the exact order; thirdly, it is proved that any computing unit constructed according to the N trigonometric Fourier coefficients of  initial conditions does not improve the established exact order of the smallest discretization  error. The significance of this study lies in the fact that the theorem formulated according to the results obtained is new in the problems of discretization of classical solutions of partial differential equations.