The Study Of Generalizations Of Bernstein Operator

The Bernstein operators are important series of linear operators. Since given by Bernstein in1912, the Bernstein operators have lots of applications in the theory of approximation andcomputational mathematics, because of the refined quality and excellent structure. They are usefultools in researching for different kinds of functions in approximation theory. For about a century’spersistent work by researchers, the Bernstein operators have gotten abundant results in properties,expansions and application. In 1953, Lorentz made comprehensive descriptions and conclusionsabout polynomials in his book [1]. From then on, plenty of tutors did more extensive research aboutthe Bernstein operators. In 1997, the q-Bernstein operators were given by Phillips. Wang Hepingand others developed the theory of q-Bernstein operators. In 2003, Della and others gave Bernsteinoperators with singularities and their linear combinations about point-wise approximation，whichfurther advance the theory of approximation content.The paper extends the Bernstein operators on three parts : the form, the range and the space. Onthe form, the research is about approximation properties when the q-Bernstein operatorsapproximate to the functions. On the range, there are two parts in the research, one hand we do theresearch on weighted approximation of functions on the semi axis, the other hand the research is onweighted approximation of functions with singularities.There are five chapters in this thesis, Chapter1 introduces the development and present situationof Bernstein operator, and the sign of the relevant papers and some common basic theorem, Chapter2 extends the Bernstein operators on the form, the research is about approximation properties whenthe q-Bernstein operators approximate to the functions. And the research gets the convergence rateby two different methods corresponding to the range of q. Chapter 3 extends the Bernstein operatorson the range, the research discuss the weighted approximation to functions with singularities andgets the converse theorem .Chapter 4 gives estimates for weighted approximation of functions onthe semi axis, which extends the result of the paper [33], while Chapter 5 is the conclusion of thetext.