Multiple Binomial-Type Operators and Their Approximation Properties

In this paper, we introduce multiple binomial-type operators (multiple Bernstein–Sheffer operators) and investigate their approximation properties. These properties are analyzed by means of a Korovkin-type theorem. Furthermore, by using the first and second modulus of continuity together with Peetre’s κ-functional, we establish the rate of convergence for continuous functions and study the approximation behavior on Lipschitz classes. Moreover, we show that, depending on the choice of the generating function H, the operator Bη,ηH yields different special forms. In particular, when H is chosen to be linear, the operator reduces to a multiple Bernstein form, which can be regarded as a generalization of the classical Bernstein operator of degree 2η. In this case, the operators achieve an improved rate of convergence, with an improvement of order 1/2 compared to the classical Bernstein operators. Then, for the chosen H, the Korovkin conditions cannot be satisfied, and hence, the operator does not converge. Finally, with an appropriate choice of H, the operator can be represented as a Bell-type operator.