BERNSTEIN OPERATOR AND IT’S VARIOUS MODIFICATIONS

Abstract

This thesis focuses on the convergence rate and error behaviour for different versions of the Bernstein operator, a crucial tool in approximation theory. The instrumental application of the first of these was made by Sergei Natanovich Bernstein in 1912 when he was proving Weierstrass’s approximation theorem constructively. The result, which implies that on a closed interval any continous function can be approximated uniformly by polynomials, is one of the most important results in mathematical analysis and provides a basis for much of modern approximation theory. In this regard we start our exploration with some basic findings, lemmas’ etc. elucidating charac teristics and properties of various versions of the Bernstein operator we will discussed below. Then we g0 more deeply into understanding basics. First, we thoroughly introduce what a classical Bernstein operator is all about; its birth, fundamental attributes and specific conditions where it can converge to an intended function as highlighted in the paper. Thus, we will include here the comprehensive examinations into graphic presentations of distinct degrees within Bernstein polynomials themselves. This makes it easier to see how these functions approach other given functions from visualization as their order grows bigger. To this, we examine the behavior of convergence for Bernstein polynomials with different degrees by considering the corresponding errors and how these errors diminish with an increase in polynomial degree. Such preliminary information allows a detailed examination on several adjustments to be made on the Bernstein operator. These changes aim at making the Bernstein operator more useful in practical applications by either reducing approximation errors or increasing convergence rates. We also present graphical representations which illustrate their behavior and properties of error and convergence features of each case. Next are some particular types of Bernstein operators like Bernstein-Kantorovich, α-Bernstein, Bernstein-Chlodovsky, Λ-Bernstein, and Bernstein-Durrmeyer variations and more, that we will look into later. We study individual characteristics of each adjustment highlighting subtle mathematical differences from classical Bernstein operator. Our aim is to determine whether amendments provided higher convergence rates or lesser approximation errors when compared to other operators for a given order polynomial. This comparison analysis must be done in order to choose the best operator for a certain application as well as ensuring its maximum efficiency. This study also investigates the various real-life applications of the Bernstein operator and its vari ations apart from theoretical work. These operators are not limited to abstract mathematical creations but have many practical uses in a variety of areas. For instance, the exactness of these operators’ ap proximation ability can be employed in reconstructing and accurately simulating human facial features during facial surgery. Also, the process helps in improvement of voice recognition systems through speech analysis and modeling. This makes it a more accurate system for recognizing various human voices as well as improving others. These illustrations indicate that use of Bernstein operator or any other form is highly critical since it is required by practicality. The basic idea behind this thesis is to provide detailed knowledge about various forms of Bernstein operator. For example, their conver gence rates could be examined along with an error behavior as well as performance comparisons; thus would help us understand how best to apply them theoretically and practically speaking. It seeks to ad 4 vance approximation theory while enhancing application of these powerful mathematical tools in solving real-world problems.