On Approximation by Two Kinds Modi fied Durrmeyer Rational Interpolation Operato

(Department of Mathematics,Inner Mongolia Normal University,Huhhote 010022,China)

§1. Introduction and main results

Letbe the weighted Orlicz spaces,the weight function isForits Orlicz norm is de fined as[1]

or

The Luxemburg norm

is equivalent to the Orlicz norm asholds true.

denotes the modulus of continuity of f in Orlicz spaces,that is

Sun Zhiling[2]discussed the approximation of Nevai-Kantorovich rational interpolation operator in weighted Orlicz spaces.In 2009,Gu Chunhe[3]studied the approximation theorem of the first kind of modi fied Vertesi-Kantorovich operator in Orlicz spaces which is formed by the Lagrange interpolation basis function of the first kind of Chebyshev polynomial.In 2010,Feng Yue[4]studied the approximation theorem of shepard-Kantorovich operator in Orlicz space.In the second chapter of[5],not only the de finition of Shepard-Durrmeyer,Vertesi-Durrmeyer and Nevai-Durrmeyer rational interpolation operators are given by Cheng Wentao,but also the approximation theorems of three kinds of operators in Lp(1≤p<∞)spaces were proved in 2012.Therefore,on the basis of the literature[5],we study the approximation problems of two kinds of Durrmeyer rational interpolation operators inspaces.

Let η(x)be a generalized smooth Jacobi weight function de fined by

where −1 −1,k=1,2,···,q and 0< ψ ∈ LipMλ.Denote bythe corresponding system of orthogonal polynomials associated with the weight function η(x),andthe zero of Pn(x).

For anythe modi fied Nevai-Durrmeyer operator is de fined as

For anyde fine the modi fied Shepard-Durrmeyer operators as

Theorem 1.1Let,then

Write

Here and throughout the whole paper,we use Csto indicate a positive constant depending only on s,Cλ,to indicate a positive constant depending only on λ,and C is an absolute positive constant,their values may vary in de ff erent occurences even in the same line.

§2. Preliminaries

In order to proof theorem,we need the following lemmas.

Let x0=1,xn+1= −1,also xk,k=1,2,···,n,be the zeros of pn(u).Set xk=cosθk,then θk∈ [0,π],k=0,1,···,n+1 and

Lemma 2.1[5,6]For any x∈[xj+1,xj],0≤j≤n,0≤k≤n,we have

Lemma 2.2[5,6]Let x=cosθ,θ∈ [0,π],we have

where

Lemma 2.3For s,λ >1,Dn(f,x),Sn(f,x)are positive bounded linear operators in,that is,for anywe have

Let g(θ)=f(cosθ),by using Jackson operators,we can easily obtain the following lemma.

Lemma 2.4[3]Letbe an even function,then there exists an even trigonometric polynomial TN(θ)with degree ≤ N,such that

Lemma 2.5Let PN(x)be a polynomial whose order is less than or equal to N,and TN(θ)=PN(cosθ),then

ProofLet x=cosθ,t=cosυ,θ,υ ∈ [0,π],because of Dn(f,x)=1,we have

whereare the Hardy-Littlewood extremal maximum function of

By lemma 2.3,we obtain

There is another constant C,satis fiesby lemma 2.3,we obtain

Similarly available

§3.Proof the Theorem

Similarly available

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[2]SUN Zhi-ling.Approximation theorems of modi fied Nevai operators in Orlicz spaces[J].Journal of Inner Mongolia University for the Nationalities(NATURAL SCIENCE EDITION),2007,22(2):121-124.

[3]GU Chun-he.Study on some approximation problems in Orlicz spaces[D].Hohhot:School of Mathematical Sciences,Inner Mongolia Normal University,2009.

[4]FENG Yue.Approximation equivalence theorem of Shepard operator in Orlicz spaces[J].journal of inner mongolia normal university2010,39(6):565-568.

[5]CHENG Wen-tao.Approximation theorems of modi fied rational operators[D].Zhejiang:School of mathematics,Zhejiang Sci-Tech University,2012.

[6]ZHOU Guan-zhen.Approximation of a kind of Nevai-Durrmeyer operators in spaces[J].Analysis in Theory and Applications2005,21(3):294-300.

[7]A.-R.K.RAMAZANOV.On approximation by polynomials and rational functions in Orlicz spaces[J].Analysis Mathematica,1984,10:117-132.