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On the convergence of a kind of q-gamma operators
Journal of Inequalities and Applications volume 2013, Article number: 105 (2013)
In this paper, we introduce a kind of q-gamma operators based on the concept of a q-integral. We estimate moments of these operators and establish direct and local approximation theorems of the operators. The estimates on the rate of convergence and weighted approximation of the operators are obtained, a Voronovskaya asymptotic formula is also presented.
MSC:41A10, 41A25, 41A36.
In recent years, the applications of q-calculus in the approximation theory is one of the main areas of research. After q-Bernstein polynomials were introduced by Phillips  in 1997, many researchers have performed studies in this field; we mention some of them [1–4].
In 2007, Karsli  introduced and estimated the rate of convergence for functions with derivatives of bounded variation on of new gamma type operators as follows:
In 2009, Karsli, Gupta and Izgi  gave an estimate of the rate of pointwise convergence of these operators (1) on a Lebesgue point of bounded variation function f defined on the interval . In 2010, Karsli and Özarslan  obtained some direct local and global approximation results and gave a Voronoskaya-type theorem for the operators (1). As the application of q-calculus in approximation theory is an active field, it seems there are no papers mentioning the q analogue of these operators defined in (1). Inspired by Aral and Gupta , they defined a generalization of q-Baskakov type operators using q-Beta integral and obtained some important approximation properties, which motivates us to introduce the q analogue of this kind of gamma operators.
Before introducing the operators, we mention certain definitions based on q-integers; details can be found in [8, 9]. For any fixed real number and each nonnegative integer k, we denote q-integers by , where
Also, q-factorial and q-binomial coefficients are defined as follows:
The q-improper integrals are defined as (see )
provided the sums converge absolutely.
The q-beta integral is defined by
where and , .
In particular for any positive integer m, n,
where is the q-gamma function satisfying the following functional equations:
For , and , we introduce a kind of q-gamma operators as follows:
Note that for , become the gamma operators defined in (1).
2 Some preliminary results
In order to obtain the approximation properties of the operators , we need the following lemmas.
Lemma 1 For any , and , we have
Proof Using the properties of q-beta integral, we have
Lemma 1 is proved. □
Lemma 2 The following equalities hold:
Proof From Lemma 1, taking , we get (6) and (7). Since and , using (6), (7), we obtain (8) and (9) easily. □
Remark 1 Note that for , from Lemma 2, we have
which is the moments and central moments of the operators defined in (1).
3 Local approximation
In this section we establish direct and local approximation theorems in connection with the operators .
We denote the space of all real-valued continuous bounded functions f defined on the interval by . The norm on the space is given by .
Further, let us consider Peetre’s K-functional
where and .
For , the modulus of continuity of second order is defined by
By [, p.177] there exists an absolute constant such that
Our first result is a direct local approximation theorem for the operators .
Theorem 1 For , and , we have
where C is a positive constant,
Let us define the auxiliary operators
. The operators are linear and preserve the linear functions
Let . By Taylor’s expansion
and (14), we get
Hence, by (13) and (8), we have
On the other hand, by (13), (4) and (6), we have
Now (13) and (15) imply
Hence taking infimum on the right-hand side over all , we get
By (10), for every , we have
where and are defined in (12). This completes the proof of Theorem 1. □
Remark 2 Let be a sequence satisfying and , we have and , these give us the pointwise rate of convergence of the operators to .
4 Rate of convergence
Let be the set of all functions f defined on satisfying the condition , where is a constant depending only on f. We denote the subspace of all continuous functions belonging to by . Also, let be the subspace of all functions , for which is finite. The norm on is . We denote the usual modulus of continuity of f on the closed interval () by
Obviously, for function , the modulus of continuity tends to zero.
Theorem 2 Let , and be the modulus of continuity on the finite interval , where . Then we have
Proof For and , we have
hence we obtain
For and , we have
From (17) and (18), we get
For and , by Schwarz’s inequality and Lemma 2, we have
By taking , we get the assertion of Theorem 2. □
5 Weighted approximation and Voronovskaya-type asymptotic formula
Now we will discuss the weighted approximation theorem.
Theorem 3 Let the sequence satisfy and as , for , we have
Proof By using the Korovkin theorem in , we see that it is sufficient to verify the following three conditions.
Since and , (20) holds true for and .
Finally, for , we have
since , we get , so the condition of (21) holds for as , then the proof of Theorem 3 is completed. □
Finally, we give a Voronovskaya-type asymptotic formula for by means of the second and fourth central moments.
Theorem 4 Let be a sequence satisfying , and . For , ( is a twice differentiable function in ), 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 using L’Hopital’s rule, we have
Applying to (23), we obtain
By the Cauchy-Schwarz inequality, we have
Since , then it follows from Theorem 3 that
Now, from (24), (25) and Lemma 2, we get immediately
and since , we have
Theorem 4 is proved. □
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This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012) and the Project of the Educational Office of Fujian Province of China (Grant No. JK2011041).
The authors declare that they have no competing interests.
QBC and XMZ carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. QBC and XMZ carried out the immunoassays. QBC and XMZ participated in the sequence alignment. QBC and XMZ participated in the design of the study and performed the statistical analysis. QBC and XMZ conceived of the study, and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
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Cai, Q., Zeng, X. On the convergence of a kind of q-gamma operators. J Inequal Appl 2013, 105 (2013). https://doi.org/10.1186/1029-242X-2013-105
- q-gamma operators
- weighted approximation
- rate of convergence