Quantitative Estimates for Certain Positive Linear Operators

The convergence behaviour of linear positive operators has been a prominent topic and many scholars have investigated it using a variety of methodologies for exactness. Approx- imation theory is primarily concerned with the investigation of several direct results such as the asymptotic formula, point-wise convergence and error estimates in simultaneous and ordinary approximation. These findings are necessary from a convergence standpoint. Dun- ham Jackson [103] proved algebraic and trigonometric polynomial direct theorems. Profes- sor Edmund Georg Hermann Landau, Jackson's supervisor, was the one who came up with the Big-O and small-o symbols. The asymptotic behaviour of a function is represented by these symbols. Because a function's rate of growth is known as its order, the letter 'O' or 'o' is utilised. As a result, the order of approximation is represented by these symbols. The crucial theorem proved by Weierstrass in 1885 made it possible to study some sorts of complicated functions by a simpler function with a suitable rate of approximation. Thus, linear positive operator approximation theory is based mostly on the "Weierstrass approxima- tion theorem," which states that "every continuous function defined on a closed and bounded interval may be uniformly approximated by a polynomial". Polynomials are among the most basic functions and computers can instantly evaluate polynomials. The Bernstein polyno- mials are used in the most typical proof of this well-known theorem. Additionally, Borislav Dechev Boyanov [33] was one of the pioneers in the field of mathematics to demonstrate the presence and originality of answers to numerous issues pertaining to approximation the- ory. Academician Blagovest Hristov Sendov, who served as his supervisor, visited Moscow State University in the early 1960s as new, interesting fields in approximation theory began to emerge. During this time, Sergei Natanovich Bernstein was still engaged in mathematical studies.