The Bernstein-markov Inequalities And Direct Theorems As Well As Inverse Theorems Of Modified Bernstein Operators

The Bernstein operators are series of linear operators, and they have lots of applications in the theory of approximation and computational mathematics, also there are plenty of achievements corresponding to Bernstein operators. They are useful tools in researching for different kinds of functions in approximation theory since they have refined quality and excellent structure. Researchers pay attention to properties, expansions and application in them and bring about abundant results. In 1953, Lorentz made comprehensive descriptions and conclusions about polynomials in his book"Bernstein polynomials", In 2003, there are a few tutors make expansive research to this, and they give Bernstein operators with singularities and their linear combinations about point-wise approximation, further advance the theory of approximation content,we can see the [4] which Wei Bao-rong ,Zhao Yi give.For f has singularities at endpoints or inner points, two modified Bernstein operators and some kind of combinations of Bernstein operators will be given in this paper, and the main results corresponding to the above forms of Bernstein operators are their weighted Bernstein-Markov equalities and the direct and converse theorems. There are five chapters in this thesis, chapter 1 related to relative conceptions, notes and some fundamental theorems.Chapter 2 discuss the style of Bernstein-Markov inualities of B_n^*( f , x)and B_n ( f , x ) as well as the direct and converse estimation of weighted B_n^*( f , x)and Bn ( f , x )polynomials.While chapter 3 discuss the style of Bernstein-Markov inualities of B_n ^* , r( f , x)and Bn , r( f , x )polynomials,and discuss the direct and converse estimation of weighted B_{n , r-1} ^*( f , x)and B_{n , r-1}( f , x)polynomials.Chapter 4 discuss the style of Bernstein-Markov inualities of B_<sup>*n ( f , x)and B_n ^* , r( f , x)as well as B_n ( f , x ) and Bn , r( f , x ), and also discuss the step-weight functionφabout the direct and inverse estimation of B_n ^*( f , x)and B_{n , r-1}^* ( f , x)as well as Bn ( f , x )and B_{n , r-1}( f , x).The fifth chapter is the conclusion of the text.