General summability methods in the approximation byBernstein-Chlodovsky operators

In this master thesis, the approximation properties of Bernstein-Chlodovsky operators has been investigated by using methods in summability theory, especially regular summability matrices, and rate of convergences in the approximation have been computed. As is known, the Weierstrass Approximation Theorem states that any function that is continuous on a closed interval [a,b] can be approximated uniforomly by polynomials. The first original version of this theorem was introduced by Weierstrass in 1885. Later, Bernstein gave another proof of this theorem in 1912, which was based on the construction with the polynomials. The idea of this approach has been applied by many researchers and this situation has enabled to determine new and effective approximation operators. In 1937, Chlodovsky generalized Bernstein's polynomials to approximate the functions defined in the interval [0;+∞). Later, many studies in this field have been conducted in the literature so far. However, in almost all of these studies, the following limit condition on a given sequence (bn) of positive real numbers lim(bn/n)=0 n→∞ is needed to achieve the approximation. In this thesis, it is aimed to weaken this limit condition with the help of regular summability methods. We should remind that regular summability methods, such as the method of arithmetic mean convergence, preserve the usual convergence as well as able to sum many sequences that are not the classical convergent. Therefore, the approximation theorems obtained with summability methods take the classical results one step further. Although summability methods are frequently used in the approximation by positive linear operators in the literature, no approach has yet been made on the approximation by Bernstein-Chlodovsky operators. In this thesis, it is aimed to fill this gap in the literature. In the thesis, first of all, the approximation properties of the classical Bernstein-Chlodovsky operators will be reminded, then a modification of them will be defined with the help of summability methods, and then more general and strong approximation results for this new operators will be reached. Rate of convergences in the approximation will also be calculated. For this, the concept of modulus of continuity, which is an important tool in approximation theory, will be used. An application that does not satisfy the classical situation but allows an approximation according to this new modification will be given and the results will be graphically observed. Another aim of the thesis will be on extending the obtained results to multivariable functions. After giving a general approximation theorem here, the situation of approximation to functions of two variables will be supported with graphics. Finally, the results obtained in the thesis will be discussed and possible future research on the topic will be evaluated.