On the Rate of Convergence by Generalized Baskakov Operators

Abstract

We firstly construct generalized Baskakov operators and their truncated sum . Secondly, we study the pointwise convergence and the uniform convergence of the operators , respectively, and estimate that the rate of convergence by the operators is . Finally, we study the convergence by the truncated operators and state that the finite truncated sum can replace the operators in the computational point of view provided that .

1. Introduction

Let , , , and . For a fixed , we introduce the weighted function on by Associated with the above weighted function, we also introduce the polynomial weighted space of all real-valued continuous functions on for which is uniformly continuous and bounded on , and the norm on is defined by the formula Obviously, when , then the above norm is the ordinary norm . Furthermore, for fixed , let be the set of all functions for which () are continuous and bounded on and is uniformly continuous on , where () denote the th order derivative of on .

Let be a function defined on ; Baskakov [1] introduced the sequence of linear positive operators as follows: where is called a Baskakov operator’s kernel, which is defined by

Based on the Baskakov operators, many Baskakov-type operators [2–13] and their multivariate Baskakov operators [11, 14–18] were discussed. Particularly, Gupta and Agarwal studied the Baskakov-Kantorovich operators, Szász-Baskakov operators, and so forth in their recent book [6]. One of the most famous Baskakov-type operators is called generalized Baskakov operators [19–22]. One has where Other modified Baskakov operators are defined as follows [10]: , .

By combining the above operators (5) with (7), we introduce the following class of operators.

Definition 1. For and , other generalized Baskakov-type operators are defined by

The actual construction of Baskakov operator and its various modifications requires estimations of infinite series which in a certain sense restrict their usefulness from the computational point of view. A question naturally arises of whether the Baskakov operators can be replaced by a finite sum. In connection with this question we construct a new family of linear positive operators as follows: where is a sequence of positive numbers such that and denotes the integral part of .

Obviously, when and , the operators (8) are (5), while the operators (9) are degenerated as follows, which are firstly proposed by Walczak [11]: And when , the operators (8) are (7), while the operators (9) can be represented by [12]

For the convenience of discussion in the rest of paper, we use the notation that denotes the remainder term of operators associated with the truncated sum . Consider

This paper focuses on convergence of the operators and their truncated sum . The rest of the paper is organized as follows. In Section 2, we give main lemmas and prove that the remainder term of the operators associated with the truncated sum is convergent to 0 provided that . In Section 3, we state the pointwise convergence and the uniform convergence of the operators on the polynomial weighted space , respectively, which indicate that the rate of convergence by the operators is . Finally, we study the convergence by the truncated operators and state that the finite truncated sum can replace the operators in the computational point of view.

In this paper, for better characterizing the degree of approximation by the generalized Baskakov operators , we introduce the classical modulus of continuity of a function , defined by [23] Here, we give an important property of modulus of continuity, which will be used in the proof of Theorem 6. One has

2. Main Lemmas

In this section, we give some properties of the above operators, which will be used to prove the main theorems.

Lemma 2 (see [22]). If is defined by formula (5) then

From the first equality in Lemma 2, for all , , we have .

Lemma 3 (see [19]). If is defined by formula (5), for fixed , there exist -order algebraic polynomials , , with coefficients depending only on , such that where and denotes the integral part of . Moreover, Here and in the rest of the paper, denotes a positive absolute constant, whose value may change from line to line but is independent of .

For example, when , we have the following 4-order algebraic polynomial: For fixed , obviously, we have

Furthermore, with respect to the above weighted function , the generalized Baskakov operators (5) have the following results, which demonstrate that the weighted function is also important to the generalized Baskakov operators.

Lemma 4 (see [15, 21]). If and weighted function are defined by formula (5) and (1), respectively, for , then there exist positive absolute constants , such that

Now we will give the estimation of .

Lemma 5. For , , is defied by (12), then Furthermore, one has

Proof. By assumption , there is a positive absolute constant , such that , . With the elementary inequality for , , we get So we have Next, we estimate the sum of the last term, since in the last term; for , we remark that Finally, using Hölder inequality with Lemmas 2 and 3, we get the following inequality: Fixing , there exist constants that maybe depend on and constants but are independent of , such that and noticing that , then we can get , .

3. Main Results

In this section, we will study the properties of the operators and give the estimation of degree of approximation by these operators.

Theorem 6. Fix , for every ; then there exists a positive absolute constant , such that where is dependent only on and but is independent of and .

Proof. By assumption, using the modified Taylor formula [10], with Lemma 2 and inequality (14), we get where () denotes the Beta function, . Using the Hölder inequality with Lemmas 2 and 3, we further have Thus, we obtain Because denotes an algebraic polynomial with order at most , there exists a positive absolute constant , such that , while is an at most -order algebraic polynomial with respect to ; that is, there exists a positive absolute constant depending on and , such that .

Remark 7. The result of can be easily obtained by imitating Theorem 6; here we omit it because it will be mentioned in the proof of next theorem.

Theorem 6 is to focus on the pointwise approximation of the operators ; now we will study their uniform approximation.

Theorem 8. Fix ; for every , one has

Proof. From the proof of Theorem 6, for , we can get Using the Hölder inequality with Lemma 3, we obtain For all , we have On the other hand, for , similar to the proof of Theorem 6, we get By Lemma 3, we obtain For all , we further have Combining the above two inequalities (36) and (39), for all and fixed , the desired equality (33) is obtained.

Remark 9. Theorem 8 indicates that the rate of convergence by the operator is .

Corollary 10. Let with some , for all ; then

Finally, we will discuss the convergence of the truncated sum .

Theorem 11. Let with some , for fixed ; then Moreover, the assertion (41) holds uniformly on every rectangle with .

Proof. Notice that Using Corollary 10 and Lemma 5, we easily get the assertion (41).

Remark 12. Theorem 11 demonstrates that the generalized Baskakov operators can be replaced by the truncated operators in a certain sense from the computational point of view.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under Grants nos. 41204041 and 61261043, by the EU FP7 Project EYE2E (269118), LIVCODE (295151), by the Science Research Project of Ningxia Higher Education Institutions of China under Grant no. NGY20140147, and in part by the Science Research Project of the State Ethnic Affairs Commission of China under Grant no. 14BFZ002.