Approximation By Bernstein-Stancu Operator And Its Generalizations

Since its simplicity in construction, and its good properties to keep the raono-tonicity and convexity of the objective functions, Bernstein operator has always been a most important part of theory of approximation by operators. Bernstein operator has been applied widely in Functional Analysis, Computational Mathe-matics and Learning Theory. The main purpose of the present thesis is to consid-er some approximation properties of Bernstein operator and one of its important generalizations-Kantorovich type Bernstein-Stancu operator. The main results can be read as follows:Chapter Ⅰ. We give a brief survey on the research of Bernstein operator and its generalizations.Chapter Ⅱ. We establish both direct and converse theorems for approximation by Sn,α，β* (f x) on the mobile interval An, which contains both global and ponitwise results. We also generalize a result of Gurhan Icoz ([33]) on approximation by the operator Sn,α，β*(f,x) for functions in C(An).Chapter Ⅲ. Both Gurhan Icoz ([33]) and the second chapter showed that Sn,α，β*(f, x) can be applied to approximate the continuous functions defined on An (a subset of [0,1]). In this chapter, we discover that Sn,α，β*(f.x) actually can also approximate the functions in C [0,1] well and generalize the result of Gurhan Icoz ([33]) from C (An) to C [0,1]. Furthermore, we establish both direct and converse theorems for approximation by Sn,α，β*(f,x), which contains both global and ponitwise results. We also generalize a result of Gurhan Igoz ([33]) on approximation by the operator Sn,α，β*(f, x) for functions in Cr [0,1].Chapter IV. We introduce a Durrmeyer variant of a new type of Bernstein-Stancu operators, and establish both direct and converse theorems for approxima-tion by the new operators.