Approximation Of The Bernstein-Kantorovich Operator

The approximation operator for every kind of the objective function is different in Approximation Theory. Kantorovich operator is the generalization of the Bernstein operator. In this paper, the generalization and approximation of the Kantorovich operator and approximation for derivatives of the Bernstein-Kantorovich operator are discussed basing on the approximative properties of the Bernstein operator and its generalization. This dissertation will deal with the approximation problem by a kind of generalized Kantorovich operator in Ba space, the direct theorem and the equivalent theorem of the approximation by the new type of Kantorovich operator are studied. At the same time, we use the classic modulus of continuityω( f ,t), use the relation between the derivatives of Bernstein-Kantorovich and the smoothness of the function it approximates, the approximation degree of derivatives of Bernstein-Kantorovich operator for the derivable functions is estimated, and the direct and converse theorem of approximation are established.Jia-Ding Cao constructs and discusses a generalized Bernstein polynomials C n ( f , S n; x ),the conclusion of necessary and sufficient condition of C n ( f , S n; x )uniform convergent to f∈C[0,1]is lni→m∞Snn= 0has been obtained. Zhang Ting and Xue Yin-chuan made use of this generalized Bernstein polynomials to construct a corresponding Kantorovich operator, and discuss the convergence of this operator in L pspace ,the estimation of the degree of approximation with f∈L p( p> 1) was also obtained.Feng Guo discuss direct theorem and the equivalent theorem of the Kantorovich operator in Ba space. At this dissertation, we use the generalized Kantorovich operator of Zhang Ting and Xue Yin-chuan, to discuss the approximation problem by this generalized Kantorovich operator in Ba space. We advance two theorems of in chapter 2:Theorem2.1(direct theorem): SupposeBa = LP ,L P , L , LPm,L is a sequence of Lebesgue space, pm > 1, m = 1,2,L , is non-negative real number line, if whereis suitably chosen positive number only depending onTheorem2.2 (equivalent theorem): With the condition of Theorem2.1 (λ= 1), if 0 <α< 1the following two conclusions are equivalently: In addition, we use the relation between the derivatives of Bernstein-Kantorovich and the smoothness of the function it approximates, the approximation degree of derivatives of Bernstein-Kantorovich operator for the derivable functions is estimated, and the direct and converse theorem of approximation are established.Theorem 3.1 For f∈Cr[ 0,1],r∈N and r < n, we have whereis suitably chosen positive number only depending on r .Theorem 3.2 For f∈Cr[ 0,1],r∈N, and r < n, r <α< r+ 1,we have...