THE APPLICATION OF BERNOULLI TRIALS TO THE THEORY OF APPROXIMATION

In recent years, q-Bernstein polynomials and alpha-Bernstein polynomials have emerged as prominent topics in approximation theory. Numerous studies have examined the convergence criteria of these polynomials, highlighting their importance and usefulness. The main idea underlying the article is that the kernels of these polynomials depend on the probabilities of the variable parameter binomial distribution. Taking advantage of this property, we developed a simplified form of these polynomials regarding binomial dependence. This facilitated the calculation of moments for a binomial variable with variable parameters. This method both simplifies the computational processes and allows us to better understand the convergence properties of these polynomials. By examining these reduced forms, important information has been obtained regarding the structure of the underlying distribution. The findings underscore the versatility and power of q-Bernstein and alpha-Bernstein polynomials in approximation theory and provide a deeper understanding of their mathematical foundations and potential applications.