Abstract and Applied Analysis

Volume 2012 (2012), Article ID 754217, 16 pages

doi:10.1155/2012/754217

Research Article

Approximation by the -Szász-Mirakjan Operators

N. I. Mahmudov

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, North Cyprus, Mersin 10, Turkey

Received 6 September 2012; Accepted 4 December 2012

Academic Editor: Behnam Hashemi

Copyright © 2012 N. I. Mahmudov. 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.

This paper deals with approximating properties of the q-generalization of the Szász-Mirakjan operators in the case . Quantitative estimates of the convergence in the polynomial-weighted spaces and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by the q-Szász-Mirakjan operators ( ) is of order versus 1/n for the classical Szász-Mirakjan operators.

1. Introduction

The approximation of functions by using linear positive operators introduced via -Calculus is currently under intensive research. The pioneer work has been made by Lupaş [1] and Phillips [2] who proposed generalizations of Bernstein polynomials based on the -integers. The -Bernstein polynomials quickly gained the popularity, see [3–11]. Other important classes of discrete operators have been investigated by using -Calculus in the case , for example, -Meyer-König operators [12–14], -Bleimann, Butzer and Hahn operators [15–17], -Szász-Mirakjan operators [18–21], and -Baskakov operators [22, 23].

In the present paper, we introduce a -generalization of the Szász operators in the case . Notice that different -generalizations of Szász-Mirakjan operators were introduced and studied by Aral and Gupta [18, 19], by Radu [20], and by Mahmudov [21] in the case . Since we define -Szász-Mirakjan operators for , the rate of approximation by the -Szász-Mirakjan operators () is of order , which is essentially better than (rate of approximation for the classical Szász-Mirakjan operators). Thus our -Szász-Mirakjan operators have better approximation properties than the classical Szász-Mirakjan operators and the other -Szász-Mirakjan operators.

The paper is organized as follows. In Section 2, we give standard notations that will be used throughout the paper, introduce -Szász-Mirakjan operators, and evaluate the moments of . In Section 3 we study convergence properties of the -Szász-Mirakjan operators in the polynomial-weighted spaces. In Section 4, we give the quantitative Voronovskaja-type asymptotic formula.

2. Construction of and Estimation of Moments

Throughout the paper we employ the standard notations of -calculus, see [24, 25].

-integer and -factorial are defined by

For integers   -binomial is defined by

The -derivative of a function , denoted by , is defined by

The formula for the -derivative of a product and quotient are

Also, it is known that

If , or and , the -exponential function was defined by Jackson

If , is an entire function and

There is another -exponential function which is entire when and which converges when if . To obtain it we must invert the base in (2.6), that is, :

We immediately obtain from (2.7) that

The -difference equations corresponding to and are

Let be the set of all real valued functions , continuous on , such that is uniformly continuous and bounded on endowed with the norm

Here

The corresponding Lipschitz classes are given for by

Now we introduce the -parametric Szász-Mirakjan operator.

Definition 2.1. Let and . For one defines the Szász-Mirakjan operator based on the -integers

Similarly as a classical Szász-Mirakjan operator , the operator is linear and positive. Furthermore, in the case of we obtain classical Szász-Mirakjan operators.

Moments are of particular importance in the theory of approximation by positive operators. From (2.14) one easily derives the following recurrence formula and explicit formulas for moments ,  .

Lemma 2.2. Let . The following recurrence formula holds

Proof. The recurrence formula (2.15) easily follows from the definition of and as show below:

Lemma 2.3. The following identities hold for all ,  , , and :

where .

Proof. The first identitiy follows from the following simple calculations

The second one follows from the first:

Lemma 2.4. Let . One has

Proof. For a fixed , by the -Taylor theorem [24], we obtain

Choosing and taking into account

we get for that

In other words .

Calculation of , , based on the recurrence formula (2.17) (or (2.15)). We only calculate and :

Lemma 2.5. Assume that . For every there hold

Proof. First of all we give an explicit formula for .

Now we prove explicit formula for the moments , which is a -analogue of a result of Becker, see [26, Lemma 3].

Lemma 2.6. For ,   there holds

where

In particular is a polynomial of degree without a constant term.

Proof. Because of , , the representation (2.29) holds true for with ,  .

Now assume (2.29) to be valued for then by Lemma 2.3 we have

Remark 2.7. Notice that are Stirling numbers of the second kind introduced by Goodman et al. in [8]. For the formulae (2.30) become recurrence formulas satisfied by Stirling numbers of the second type.

3. in Polynomial-Weighted Spaces

Lemma 3.1. Let and be fixed. Then there exists a positive constant such that

Moreover for every one has

Thus is a linear positive operator from into for any .

Proof. The inequality (3.1) is obvious for . Let . Then by (2.29) we have

is a positive constant depending on and . From this follows (3.1). On the other hand

for every . By applying (3.1), we obtain (3.2).

Lemma 3.2. Let and be fixed. Then there exists a positive constant such that

Proof. The formula (2.25) imply (3.5) for . We have

for . If then we get

which by Lemma 2.5 yields (3.5) for .

Let . By applying (2.29), we get

where is a polynomial of degree . Therefore one has

Our first main result in this section is a local approximation property of stated below.

Theorem 3.3. There exists an absolute constant such that

where , and .

Proof. Using the Taylor formula

we obtain that

Now we consider the modified Steklov means

has the following properties:

and therefore

We have the following direct approximation theorem.

Theorem 3.4. For every and ,  , one has

Particularly, if for some , then

Proof. For and

and therefore

Since , we get that

Thus, choosing , the proof is completed.

Corollary 3.5. If ,   and ,  then

This converegnce is uniform on every , .

Remark 3.6. Theorem 3.4 shows the rate of approximation by the -Szász-Mirakjan operators () is of order versus for the classical Szász-Mirakjan operators.

4. Convergence of -Szász-Mirakjan Operators

An interesting problem is to determine the class of all continuous functions such that converges to uniformly on the whole interval as . This problem was investigated by Totik [27, Theorem 1] and de la Cal and Cárcamo [28, Theorem 1]. The following result is a -analogue of Theorem 1 [28].

Theorem 4.1. Assume that is bounded or uniformly continuous. Let

One has, for all and ,

Therefore, converges to uniformly on as , whenever is uniformly continuous.

Proof. By the definition of we have

Thus we can write

Finally, from the inequality

we obtain

In order to complete the proof we need to show that we have for all and ,

Indeed we obtain from the Cauchy-Schwarz inequality:

showing (4.2), and completing the proof.

Next we prove Voronovskaja type result for -Szász-Mirakjan operators.

Theorem 4.2. Assume that . For any the following equality holds

for every .

Proof. Let be fixed. By the Taylor formula we may write

where is the Peano form of the remainder, , and . Applying to (4.10) we obtain

By the Cauchy-Schwartz inequality, we have

Observe that . Then it follows from Corollary 3.5 that

Now from (4.12), (4.13), and Lemma 2.5 we get immediately

The proof is completed.

References

A. Lupaş, “A q-analogue of the Bernstein operator,” in Seminar on Numerical and Statistical Calculus, vol. 9, pp. 85–92, University of Cluj-Napoca, 1987.

G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol. 4, no. 1–4, pp. 511–518, 1997.

A. II'inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of Approximation Theory, vol. 116, no. 1, pp. 100–112, 2002.

S. Ostrovska, “q-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123, no. 2, pp. 232–255, 2003.

S. Ostrovska, “On the limit q-Bernstein operator,” Mathematica Balkanica. New Series, vol. 18, no. 1-2, pp. 165–172, 2004.

S. Ostrovska, “On the Lupaş q-analogue of the Bernstein operator,” The Rocky Mountain Journal of Mathematics, vol. 36, no. 5, pp. 1615–1629, 2006.

S. Ostrovska, “Positive linear operators generated by analytic functions,” Proceedings of the Indian Academy of Science, vol. 117, no. 4, pp. 485–493, 2007.

T. N. T. Goodman, H. Oruç, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,” Proceedings of the Edinburgh Mathematical Society, vol. 42, no. 1, pp. 179–190, 1999.

T. K. Boehme and R. E. Powell, “Positive linear operators generated by analytic functions,” SIAM Journal on Applied Mathematics, vol. 16, pp. 510–519, 1968.

N. I. Mahmudov, “Korovkin-type theorems and applications,” Central European Journal of Mathematics, vol. 7, no. 2, pp. 348–356, 2009.

V. S. Videnskii, “On some classes of q-parametric positive linear operators,” in Selected Topics in Complex Analysis, vol. 158 of Operator Theory: Advances and Applications, pp. 213–222, Birkhäuser, Basel, Switzerland, 2005.

T. Trif, “Meyer-König and Zeller operators based on the q-integers,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 29, no. 2, pp. 221–229, 2000.

O. Doğru and O. Duman, “Statistical approximation of Meyer-König and Zeller operators based on q-integers,” Publicationes Mathematicae Debrecen, vol. 68, no. 1-2, pp. 199–214, 2006.

W. Heping, “Properties of convergence for the q-Meyer-König and Zeller operators,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1360–1373, 2007.

A. Aral and O. Doğru, “Bleimann, Butzer, and Hahn operators based on the q-integers,” Journal of Inequalities and Applications, vol. 2007, Article ID 79410, 12 pages, 2007.

N. I. Mahmudov and P. Sabancıgil, “q-parametric Bleimann Butzer and Hahn operators,” Journal of Inequalities and Applications, vol. 2008, Article ID 816367, 15 pages, 2008.

N. Mahmudov and P. Sabancigil, “A q-analogue of the Meyer-König and Zeller operators,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 1, pp. 39–51, 2012.

A. Aral and V. Gupta, “The q-derivative and applications to q-Szász Mirakyan operators,” Calcolo, vol. 43, no. 3, pp. 151–170, 2006.

A. Aral, “A generalization of Szász-Mirakyan operators based on q-integers,” Mathematical and Computer Modelling, vol. 47, no. 9-10, pp. 1052–1062, 2008.

C. Radu, “On statistical approximation of a general class of positive linear operators extended in q-calculus,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2317–2325, 2009.

N. I. Mahmudov, “On q-parametric Szász-Mirakjan operators,” Mediterranean Journal of Mathematics, vol. 7, no. 3, pp. 297–311, 2010.

O. Agratini and C. Radu, “On q-Baskakov-Mastroianni operators,” Rocky Mountain Journal of Mathematics, vol. 42, no. 3, pp. 773–790, 2012.

N. I. Mahmudov, “Statistical approximation of Baskakov and Baskakov-Kantorovich operators based on the q-integers,” Central European Journal of Mathematics, vol. 8, no. 4, pp. 816–826, 2010.

V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.

T. Ernst, “The history of q-calculus and a new method,” U.U.D.M. Report 16, Departament of Mathematics, Uppsala University, Uppsala, Sweden, 2000.

M. Becker, “Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces,” Indiana University Mathematics Journal, vol. 27, no. 1, pp. 127–142, 1978.

V. Totik, “Uniform approximation by Szász-Mirakjan type operators,” Acta Mathematica Hungarica, vol. 41, no. 3-4, pp. 291–307, 1983.

J. de la Cal and J. Cárcamo, “On uniform approximation by some classical Bernstein-type operators,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 625–638, 2003.