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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 563613, 8 pages
http://dx.doi.org/10.1155/2014/563613
Research Article

Shape-Preserving and Convergence Properties for the -Szász-Mirakjan Operators for Fixed

1School of Mathematical Sciences, BCMIIS, Capital Normal University, Beijing 100048, China
2Jia Huiming Educational Technology Co. (Ltd.), Beijing 100068, China
3School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received 22 February 2014; Accepted 14 April 2014; Published 6 May 2014

Academic Editor: Sofiya Ostrovska

Copyright © 2014 Heping Wang et al. 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.

Abstract

We introduce a -generalization of Szász-Mirakjan operators and discuss their properties for fixed . We show that the -Szász-Mirakjan operators have good shape-preserving properties. For example, are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed , we prove that the sequence converges to uniformly on for each , where is the limit -Bernstein operator. We obtain the estimates for the rate of convergence for by the modulus of continuity of , and the estimates are sharp in the sense of order for Lipschitz continuous functions.

1. Introduction

Let . For each nonnegative integer , the -integer and the -factorial are defined by For integers , the -binomial coefficient is defined by We give the following two -analogues of exponential function : where . Clearly, we have

In [1], Phillips proposed the -Bernstein polynomials: for each positive integer and , the -Bernstein polynomial of is Note that for , is the classical Bernstein polynomial. In [2], II’inskiia and Ostrovska proved that, for each and , the sequence converges to as uniformly on , where The operators are called the limit -Bernstein operators. They also arise as the limit for a sequence of -Meyer-König Zeller operators (see [3]). For results about properties of we refer to [2, 4, 5].

In [6], Aral introduced the following -Szász-Mirakjan operator: for each positive integer and , where , and is a sequence of positive numbers such that . In this paper, we introduce the following -Szász-Mirakjan operator: for each positive integer and , Obviously, the operators are equal to the operators with , . When , the -Szász-Mirakjan operators reduce to the classical Szász-Mirakjan operators.

In recent years, generalizations of linear operators connected with -Calculus have been investigated intensively. The pioneer work has been made by Lupas [7] and Phillips [1] who proposed generalizations of Bernstein polynomials based on the -integers. There are also other important -operators, for example, the two-parametric generalization of -Bernstein polynomials [8], the -Bernstein-Durrmeyer operator [9], -Meyer-König Zeller operators [10], -Bleimann, Butzer and Hahn operators [11], and -Szász-Mirakjan operators [6, 1215]. Among these generalizations, -Bernstein polynomials proposed by Phillips attracted the most attention and were studied widely by a number of authors (see [1, 2, 5, 1624]).

In this paper, we will discuss convergence and shape-preserving properties of the -Szász-Mirakjan operators for fixed . We will show that the operators share good shape-preserving properties such as the variation-diminishing properties, and for each the sequence converges to the function uniformly on , where are the limit -Bernstein operators defined by (6). We also investigate the rate of convergence of the -Szász-Mirakjan operators for fixed . Our results demonstrate that in general convergence properties of the -Szász-Mirakjan operators are essentially different from those for the classical Szász-Mirakjan operators; however, they are very similar to those for the -Bernstein polynomials. Notice that different -generalizations of Szász-Mirakjan operators were introduced and studied by Aral and Gupta [6, 12], by Radu [13], and by Mahmudov [14, 15]. However, our -Szász-Mirakjan operators have better convergence properties than the other -generalizations of Szász-Mirakjan operators for fixed .

The paper is organized as follows. In Section 2, we recall some properties of the -Szász-Mirakjan operators and discuss their shape-preserving properties. In Section 3 we investigate the convergence of for fixed and obtain the rate of convergence of by the modulus of continuity of , and the estimates are sharp in the sense of order for Lipschitz continuous functions.

2. Shape-Preserving Properties of for

In the sequel we always assume that . First we show that the -Szász-Mirakjan operators are the positive linear operators on . Clearly, it suffices to prove that, for , Indeed, for arbitrary , there exist a constant and a such that for all , and for . We choose to be the minimum positive integer greater than . Then, for any , It follows from the Euler identity that This implies that, for , where is a constant independent of and as . This proves (9).

The -Szász-Mirakjan operators possess the end-point interpolation property: They leave invariant linear functions: and are degree-preserving on polynomials; that is, if is a polynomial of degree , then is a polynomial of degree (see [6, Lemma 1] or [25, Theorem 1]).

The following representation of the -Szász-Mirakjan operators , called the -difference form, was obtained in [6, Corollary 4]: where denotes the usual divided difference; that is,

Aral and Gupta discussed the shape-preserving properties of the -Szász-Mirakjan operators in [12, Corollary 3.2]. We say a function on an interval is -convex, , if and all th forward differences are nonnegative. Obviously, a -convex function is nondecreasing and a -convex function is convex. Aral and Gupta obtained that, for an -convex function on , there exists such that is also -convex on for .

In this section we also study the shape-preserving properties of the operators . We use a completely different method from the one in [12], and our results hold for all . In order to state the results, we introduce some notations.

For any real sequence , finite or infinite, we denote by the number of strict sign changes in . For , where is an interval, we define to be the number of sign changes of ; that is, where the supremum is taken over all increasing sequences and for all positive integers .

Let be a positive linear operator on . We say that is variation-diminishing if, for all functions , we have

A function on , is called a modulus of continuity if is continuous, nondecreasing, and semiadditive and . We denote by the class of continuous functions on satisfying the inequality , where is the modulus of continuity of . Note that if is a concave modulus of continuity, then is nonincreasing on . Also, if is a nondecreasing function such that and is nonincreasing on , then is a modulus of continuity.

Our main results of this section can be formulated as follows.

Theorem 1. (i) The operators are variation-diminishing on .
(ii) If a function is -convex on , then the functions are also -convex on . Specially, if a function is nondecreasing (nonincreasing) on , then are also nondecreasing (nonincreasing) on and if is convex (concave) on , then so are .
(iii) If a function is convex on , then , .
(iv) If is a modulus of continuity, then implies that, for each , ; if is concave, then, for each , .
(v) If is a concave modulus of continuity, then, for each , is also a concave modulus of continuity and .
(vi) If is a nonnegative function such that is nonincreasing on , then, for each , is nonincreasing also.

Proof. (i) Let be an interval, . We assume that, for a real sequence , the power series converges to the function on . By means of the well-known Descartes’ rule of sign it is easy to prove that Obviously, if for any and for , then It follows that which implies that are variation-diminishing.
(ii) The operators possess the end-point interpolation property and are degree-preserving on polynomials and variation-diminishing. Then, (ii) follows from [26, Lemma 15].
(iii) It follows from [27, p. 281] that if a positive operator on reproduces linear functions, then for any convex function and for any . Since are the positive linear operators and reproduce linear functions, we obtain (iii).
(iv) From [26, Corollary 8], we know that if a positive linear operator on    is variation-diminishing and reproduces linear functions, then, for all and , Thus, if , then where and denote the least concave majorant of and , respectively. It is well known that for each modulus of continuity there exists a concave modulus of continuity such that for . Thence, and furthermore if is concave, which means (iv) holds.
(v) From (i) we know that, for a concave modulus of continuity and each , the function is nondecreasing and concave on , where . We also have . This means that is a concave modulus of continuity. The inequality follows directly from (iii).
(vi) Since, for any constant , we get that is nondecreasing or nonincreasing on , where . For any , , we have , and thus . Hence, which implies that is nonincreasing on .
Theorem 1 is proved.

3. The Rate of Convergence for the -Szász-Mirakjan Operators for Fixed

The approximation properties of the sequence in weighted spaces as were investigated in [6, Theorem 2] and [25, Theorem 6]. The obtained results are similar to the ones of the classical Szász-Mirakjan operators. However, there are few results about convergence properties of for fixed . This section is devoted to discussing the convergence properties of the -Szász-Mirakjan operators for fixed .

We set Formerly, for and each , converges to , converges to , and as . Indeed, the above conclusion holds. We have the following stronger results.

Theorem 2. Let . Then, we have where . This estimate is sharp in the following sense of order: for each , , there exists a function which belongs to the Lipschitz class such that where is a positive constant independent of .

Remark 3. It follows from (30) that, for , uniformly on as . Since , if and only if is linear on (see [2, Theorem 6]), we get that the sequence converges to uniformly on if and only if is linear on .

Remark 4. It should be emphasized that the proof of Theorem 2 requires estimation techniques involving the infinite product. Also, it is a little more difficult than the one used for -Bernstein polynomials (see [23]), since .

Proof. Since the operators and reproduce linear functions, we get that, for , where and are defined by (27) and (28), respectively. By means of (32) and (33), direct calculations give that For , we have For , it follows that where in the first equality we used (32); in the last inequality we used the inequality for any ; in the last equality we used (32) and (34).
Now for , by (32) and (33), we have Since we get by (32) In order to estimate , we need to estimate . We have
We note that It follows that where in the fourth inequality we used (32) and (33); in the last inequality we used (34) and (33). We estimate . We have where . Using the inequality ,, we get that It follows that This deduces that, for , and thence We conclude from (39) and (47) that, for , Hence, (30) follows from (35), (36), and (48).
At last we show that the estimate (30) is sharp. For each , suppose that is a continuous function, which is equal to zero in and , equal to in , and linear in the rest of . It is easy to see that . We set . Then, , and for sufficiently large , we have The proof of Theorem 2 is complete.

Conflict of Interests

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

Acknowledgments

The authors were supported by the National Natural Science Foundation of China (Project no. 11271263), the Beijing Natural Science Foundation (1132001), and BCMIIS.

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