THE O-TRANSFORMATION OF CERTAIN POSITIVE LINEAR OPERATORS ALEANDRU LUPA

The intention of this paper is to describe a const,'uction method for a new sequence of linear positive operators, which enables us to get a pointwise order of approximation regarding the polynomial sun'tmator operators which have "best" properties of approximation.

1.The aim of this paper can be described in the following way" Starting wih a sequence A (A,) of approximation operators, we construct by means of the so called 0 transformation a new sequence of operators B (B=) O(A).With the known properties of A we get the corresponding properties of the sequence B O(A).We also prove, that if A is the sequence of (C, 1) means of Chebyshev series, the polynomials (Bf), f C(I), furnish a pointwise order of approximation similar to the best order of approxi- mation.Let H,, n 0, be the linear space of all algebraic polynomials with real coefficients of degree 5 n and T,(t) cos(, arccos t) the n th Chebyshev polynomial, n 0.
For h II2 it is B,h B,,h and so we obtain the lower bound in (2.8).
Other upper bounds for 6, were obtained by J.D.Cao and H.H.Gonska [5].
The upper estimate from theorem 2.2 enables us to write where L, is one of the operators Bn or/.

xE I,
Remark: One knows that, for (b,,) E T '+ the Fejr inequality [6] holds /3.< cos nEIN n+2' __1 for an arbitrary n a similar extremal In the case of Jacobi polynomials R(.''), a, > /3 _> , problem is solved in [8].For an even n the problem is considered in ([1], p.68).However, for all linear positive operators B (B.) generated by polynomial sequences b (b.) P+ one has r,(B) >_ 2sin r (2.15) Let us present a short proof of Fejr's inequality (2.15).If h E II.+x, then it is easy to observe that <l,h) --c,h(xa,,,), s= [] +1, x, r(n + 2) sin n+2 3. A polynomial sequence a (a.) belongs to the class Pa if and only if i) a E P+ and ii) for each n E ]N there exists at least a root zo(n) of a,, in I.
We denote Zo zo(n + 1) and remind that a(x,t) (r .

.,(t)dt
It is clear that the positivity of the tran.lationoperator certifies the fact that b (b,,) E g '+.If l" T ' T '+ is the mapping (an) (b,,).b,, I)eing as in (3.1), we write b l(a).

Journal of Applied Mathematics and Decision Sciences
Special Issue on Intelligent Computational Methods for Financial Engineering

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 If we use the convolution product L(I) x L(I) (I) -1 then our aim is to construct some al)l)roximation operators A,, .H,,, n E IN, such that lira II.fA,fllx O, .fX.

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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