On the Convergence Analysis of Bernstein Operators

In recent years, many studies have been conducted to emphasize the convergence criteria of the q-Bernstein polynomial and their importance. A point often overlooked in the literature is that the kernels of these polynomials depend on the probabilities of the variable parameter binomial distribution. Taking advantage of this property, we develop a simplified form of this polynomial for the binomial variable with variable parameters. This approach not only simplifies the computational processes but also sheds light on the convergence properties of this polynomial. By examining this reduced form, an upper bound for convergence is determined. The findings highlight the versatility and power of the q-Bernstein polynomial in approximation theory and provide a deeper understanding of the mathematical foundations and potential applications of the polynomial.