Approximation by the -Szász-Mirakjan Operators
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.
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  and Phillips  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 , and by Mahmudov  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
-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 , 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.
Remark 2.7. Notice that are Stirling numbers of the second kind introduced by Goodman et al. in . 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
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 .
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.
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