- Research Article
- Open Access
Rate of Convergence of a New Type Kantorovich Variant of Bleimann-Butzer-Hahn Operators
© L. Chen and X.-M. Zeng. 2009
- Received: 28 September 2009
- Accepted: 16 November 2009
- Published: 24 November 2009
A new type Kantorovich variant of Bleimann-Butzer-Hahn operator is introduced. Furthermore, the approximation properties of the operators are studied. An estimate on the rate of convergence of the operators for functions of bounded variation is obtained.
- Positive Integer
- Growth Condition
- Total Variation
- Probability Theory
- Limit Theorem
In 1980, Bleimann et al.  introduced a sequence of positive linear Bernstein-type operators (abbreviated in the following by BBH operators) defined on the infinite interval by
where denotes the set of natural numbers.
Bleimann et al.  proved that as for (the space of all bounded continuous functions on ) and give an estimate on the rate of convergence of measured with the second modulus of continuity of .
where , , and is Lebesgue measure.
which was first considered by Abel and Ivan in . The integrand function in the operator (1.3) has been replaced with new integrand function in the operator (1.2). In this paper we will study the approximation properties of for the functions of bounded variation. The rate of convergence for functions of bounded variation was investigated by many authors such as Bojanić and Vuilleumier , Chêng , Guo and Khan , Zeng and Piriou , Gupta et al. , involving several different operators.
Our main result can be stated as follows.
In order to prove Theorem 1.1, we need the following lemmas for preparation. Lemma 2.1 is the well-known Berry-Esséen bound for the classical central limit theorem of probability theory. Its proof and further discussion can be founded in Feller [8, page 515].
where , , .
If is fixed and is sufficiently large, then
Hence , for sufficiently large.
The proof of is similar.
Let , and , Bojanic-Cheng decomposition yields
Assuming that , for some ( ), then we have
Set , then by (2.4) and using Lemma 2.1, we have
Finally, we estimate .
Noticing and for , we have .
Next, let .
Now, the remainder of our work is to estimate .
Theorem 1.1 now follows from (3.3), (3.9), and (3.29).
This work is supported by the National Natural Science Foundation of China and the Fujian Provincial Science Foundation of China. The authors thank the associate editor and the referee(s) for their several important comments and suggestions which improve the quality of the paper.
- Bleimann G, Butzer PL, Hahn L: A Bernstein-type operator approximating continuous functions on the semi-axis. Indagationes Mathematicae 1980,83(3):255–262. 10.1016/1385-7258(80)90027-XMathSciNetView ArticleMATHGoogle Scholar
- Abel U, Ivan M: A Kantorovich variant of the Bleimann, Butzer and Hahn operators. Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento 2002, 68: 205–218.MathSciNetMATHGoogle Scholar
- Bojanić R, Vuilleumier M: On the rate of convergence of Fourier-Legendre series of functions of bounded variation. Journal of Approximation Theory 1981,31(1):67–79. 10.1016/0021-9045(81)90031-9MathSciNetView ArticleMATHGoogle Scholar
- Chêng F: On the rate of convergence of Bernstein polynomials of functions of bounded variation. Journal of Approximation Theory 1983,39(3):259–274. 10.1016/0021-9045(83)90098-9MathSciNetView ArticleMATHGoogle Scholar
- Guo SS, Khan MK: On the rate of convergence of some operators on functions of bounded variation. Journal of Approximation Theory 1989,58(1):90–101. 10.1016/0021-9045(89)90011-7MathSciNetView ArticleMATHGoogle Scholar
- Zeng X-M, Piriou A: On the rate of convergence of two Bernstein-Bézier type operators for bounded variation functions. Journal of Approximation Theory 1998,95(3):369–387. 10.1006/jath.1997.3227MathSciNetView ArticleMATHGoogle Scholar
- Gupta V, Abel U, Ivan M: Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation. International Journal of Mathematics and Mathematical Sciences 2005,2005(23):3827–3833. 10.1155/IJMMS.2005.3827MathSciNetView ArticleMATHGoogle Scholar
- Feller W: An Introduction to Probability Theory and Its Applications. Volume 2. 2nd edition. John Wiley & Sons, New York, NY, USA; 1971:xxiv+669.MATHGoogle Scholar
- Zeng X-M: Bounds for Bernstein basis functions and Meyer-König and Zeller basis functions. Journal of Mathematical Analysis and Applications 1998,219(2):364–376. 10.1006/jmaa.1997.5819MathSciNetView ArticleMATHGoogle Scholar
- Zeng X-M, Zhao J-N: Exact bounds for some basis functions of approximation operators. Journal of Inequalities and Applications 2001,6(5):563–575. 10.1155/S1025583401000340MathSciNetMATHGoogle Scholar
- Srivastava HM, Gupta V: Rate of convergence for the Bézier variant of the Bleimann-Butzer-Hahn operators. Applied Mathematics Letters 2005,18(8):849–857. 10.1016/j.aml.2004.08.014MathSciNetView ArticleMATHGoogle Scholar
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