Varyasyon yarı normu ve operatörlerin varyasyon yarı normunda yakınsaklıkları

This thesis is a survey on the property of variation seminorm and on some approximation properties of operators not only in normed spaces but also in variation seminorm. It is indicated in this thesis that the variation detracting property satisfies the approximation.One of the fundamental problems of analysis is to approximate a given function f in some sense or other by functions having certain properties, and generally, by functions which have better properties than f. This thesis consists of five chapters. In the first chapter, we have an introduction and give some necessary definitions and concepts which is used in this thesis.In the second chapter,$~$ we give the fundamental definitions and theorems providing the base of approximation theory including Bernstein operators and Szasz-Mirakjan operators. We introduce Bernstein operators and Szasz-Mirakjan operators and we give their convergence. We show the rate of convergence of these operators. In the third chapter,$~$ we give the definition of finite variation, some important theorems about it. We introduce the Stieltjes integral and BV[a,b]-space and its norm property and TV[a,b]-space and its seminorm property. In chapter four, we investigate the variation detracting property of Bernstein Polynomials and the convergence of the linear positive operator, Bernstein polynomials, in variation seminorm and the rate of its convergence in variation seminorm. In chapter five, we investigate the variation detracting property of Szasz-Mirakjan operator.We show the convergence of Szasz-Mirakjan operators in variation seminorm and its rate of convergence in variation seminorm.