A NOTE ON INTEGRAL MODIFICATION OF THE MEYER-KÖNIG AND ZELLER OPERATORS

Guo (1988) introduced the integral modification of 
Meyer-Ko nig and Zeller operators Mˆn and studied the rate of convergence for functions of bounded 
variation. Gupta (1995) gave the sharp estimate for the operators Mˆn. Zeng (1998) gave the exact bound and claimed to improve the results of Guo and Gupta, but there is a major mistake in the paper of Zeng. In the present 
note, we give the correct estimate for the rate of convergence on bounded variation functions.

Theorem 1.1.Let f be a function of bounded variation on [0, 1], x ∈ (0, 1).Then, for all n sufficiently large, where and V b a (g x ) is the total variation of g x on [a, b].
Gupta [3] gave a sharp estimate as in the following theorem.
Theorem 1.2.Let f be a function of bounded variation on [0, 1], x ∈ (0, 1).Then, for all n sufficiently large, Zeng [9] gave the exact bound for Meyer-König and Zeller basis functions and claimed to obtain the sharp estimate over the results of Guo [1] and Gupta [3].Although the bound obtained in [9] is optimum, the main estimate given by Zeng [9] for the operators Mn (f , x) is not correct.Zeng obtained the following theorem.
Theorem 1.3.Let f be a function of bounded variation on [0, 1].Then, for every x ∈ (0, 1) and n sufficiently large, We may remark here that Theorem 1.3 obtained by Zeng [9] has a major mistake because the right-hand side does not converge to zero for sufficiently large n.Also, the remark given before [9,Theorem 4.3] is contradictory to the main theorem [9,Theorem 4.3].This motivated us to give the correct estimate for these operators and in this note we give an improved estimate for the rate of convergence on functions of bounded variation for the operators (1.3).

Auxiliary results.
In this section, we give certain results, which are necessary to prove the main result.Lemma 2.1 [7].Let X 1 ,X 2 ,X 3 ,...,X n be n independent and identically distributed random variables with zero mean and a finite absolute third moment.
Theorem 3.1.Let f be a function of bounded variation on [0, 1].Then, for every x ∈ (0, 1) and n sufficiently large, where V b a (g x ) is the total variation of g x on [a, b]. Proof.First, Now, with the kernel we have Mn sign(t − x), x = (3.4) Using Lemma 2.5, we have Using Lemma 2.4, we get Next, by Lemma 2.3, we get Substituting the value of | Mn (sign(t − x), x)| and proceeding along the lines of [1] for the value of | Mn (g x ,x)|, the theorem follows.This completes the proof of theorem.
Remark 3.2.We remark here that in order to prove the main theorem, the inequality ρ 3 ≤16/(1−x) 3 is used in Theorem 1.1 and the value is used in Theorem 1.2.Zeng [9] used this value and the exact bound and gave the misprinted theorem (Theorem 1.3).Although Zeng obtained the exact bound for Meyer-König and Zeller basis functions, he has not used it correctly to obtain his main result, that is, Theorem 1.3.

4.
Bezier variant of the operators Mn .In this section, we propose the Bezier variant of the integrated Meyer-König and Zeller operators as where It is easily verified that the operators (4.1) are linear positive operators.In particular, for α = 1, operators (4.1) reduce to the operators (1.3).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation