Approximation Strategy by Positive Linear Operators

Wei Xiao Science School, Nanjing University of Science and Technology, 200 Xiao Ling Wei Street, Nanjing 210094, China Correspondence should be addressed to Wei Xiao; chinaxiao@sina.com Received 31 March 2013; Accepted 28 June 2013 Academic Editor: Satit Saejung Copyright © 2013 Wei Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Two important techniques to achieve the Jackson type estimation by Kantorovich type positive linear operators in L spaces are introduced in the present paper, and three typical applications are given.

When the kernel function K , () of ( 2) is taken as then the operator   (, ) is the well-known Shepard operator one can check [6,7], for example.When the kernel K , () in ( 2) is taken as then we obtain the Kantorovich-Bernstein operator which has been studied most widely among the positive linear operators of the form (2) (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]).Interested readers could also refer to the related papers for the other similar operators.This paper will take the above three typical operators (9), (11), and (13) as examples to illustrate two quantitative methods on   approximation.Over discussion, we find out that the Jackson order in   spaces to approximate () ∈   [0,1] by the operators in (3) or ( 5) is decided completely by the kernels {K , ()}  =1 , or by the kernel function   (, ).Therefore, on applying this idea, we need only to investigate the properties of the kernels to obtain the magnitude of the Jackson order of the corresponding operators, which seems to be a different approach from the past   approximating methods.

Notations and Terminologies
In this section, we give all preliminary notations and terminologies.For () ∈  .
To understand clearly   approximation by the positive linear operators, we need to make analysis of the kernels corresponding to the operators.Hence, some new terminologies on the kernels will be given and explained by some examples.
then {Φ()} ∞ =1 is called a local domination of K  ().Like in Definition 1, a locally dominated geometrical sequence of K  () and a locally dominated -arithmetic sequence can be defined similarly.
The conceptions of Definitions 1 and 2 will be illustrated by the following examples.
Remark 10.We make a brief discussion on the above definitions.

Elementary Approximation Technique (II)
In Section 3, by applying the -functional, we obtain the Jackson type estimation in   spaces for  > 1.However, the Jackson constant in that case must depend upon , and thus we cannot establish corresponding result in  1 space!In this section, we will exhibit another efficient technique in   spaces which will be used to obtain Jackson constant independent of ! Theorem 12. Let () ∈   [0,1] , 1 ≤  < ∞, an  be given with 0 <  < 1 and the positive linear operators   (, ) defined by (3).If the kernels K  () with (1) are dominated globally by {Φ()} ∞ =1 , and for some 0 <  < 1 satisfy the following conditions: holds, where  > 0 is an absolute constant.
To prove Theorem 12, we first give two lemmas.
Proof.This Lemma is proved in [7]; we give a sketch here for self-completeness.Due to the symmetries on  and , as well as on  and , we need only to prove the lemma under  ≥ .  ( where Note that Furthermore, condition (ii) implies that This means However, from Lemma 13, which leads to Combining ( 62) with (43), we get This, with (53), finishes Theorem 12.
For  1 space, we have the following result while conditions of Theorem 12 can be loosed.Proof.The argument of proof is similar, and we can just repeat the corresponding parts of the proof of Theorem 12.
(1) If the kernels possess good properties, the conditions of Theorem 12 can be easily verified on the terminology of domination.For instance, if the kernels have geometric order, then the corresponding conditions of Theorem 12 are obviously satisfied (see the next section).
(2) There exists essential difference between Theorem 11 and Theorem 12. Theorem 11 requires weaker conditions than Theorem 12 does, but the latter obtains stronger result (the Jackson order is complete up to 1/, and the Jackson constant is independent of !); we will make further illustrations in the coming section.

Applications
This section illustrates how to apply Theorems 11 and 12 to estimate   approximation.To check the efficiency of two techniques on   approximation by Kantorovich type positive linear operators, three examples will be exhibited.Those positive linear operators come from three different categories: rational Müntz operators from rational Müntz systems; the Shepard operators from general real rational function systems; and Bernstein polynomials from the polynomial system.Moreover, in our point of view, they represent three different types: positive linear operators with kernels of geometric order, positive linear operators of arithmetical order, and positive linear operators of local geometric order.It is because the kernels have different domination properties or different speeds of {Φ()} that the   approximations by the corresponding positive linear operators possess different Jackson orders.
To show the key role of the global (or local) domination on the kernels, the condition (ii) of Theorem 11 will be further explicated to the following lemma.

Rational Müntz Approximation.
Rational Müntz Approximation has been researched in [5], shows the application of Theorem 12, and simplifies the proof of [5].Write where span{ where   is an absolute constant depending only upon  (independent of !).

Conclusions
On the above discussions, the positive linear operators used in   approximation can be classified according to the properties of their kernels.We have three categories: kernels with geometrical order (such as the rational Müntz operators), kernels with arithmetic order (such as the Shepard operators), and kernels with local arithmetic order (such as the Bernstein operators).In another word, for characterizing the Jackson type estimate in   spaces by the Kantorovich type operators (3) or (5), it always plays an essential role how well the kernels of the operators under study behave.
This section gives one of the approximation techniques in   spaces by the Kantorovich type operators (3).We mainly apply -functional and maximum principle to obtain the Jackson type estimation in   spaces.