<p>An optimization problem has been formulated and solved to study the deviation of Lipschitz-class functions from the superposition of orthogonal polynomials. The problem is based on a positive operator as a generalization of the Abel–Poisson integral and the biharmonic Poisson integral. The main result of the study is presented as an asymptotic equality for the deviation of Lipschitz functions from the generalized Poisson-type operator constructed. The applicability of the considered operator in optimal decision theory is determined by its membership in the class of positive operators as optimal solutions to boundary-value problems. Moreover, the optimality of the generalized Poisson-type operator is significantly enhanced by the presence of approximation properties for Lipschitz-class functions, which can be used as mathematical models in optimal design problems.</p>