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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 217464, 12 pages
Asymptotic Formulae via a Korovkin-Type Result
1Departamento de Matemáticas, Universidad de Jaén, Campus Las Lagunillas s/n., 23071 Jaén, Spain
2Department of Mathematics, Technical University, Str. C. Daicoviciu 15, 400020 Cluj-Napoca, Romania
Received 31 December 2011; Accepted 15 February 2012
Academic Editor: Gabriel Turinici
Copyright © 2012 Daniel Cárdenas-Morales et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a sort of Korovkin-type result that provides a tool to obtain asymptotic formulae for sequences of linear positive operators.
This paper deals with the approximation of continuous functions by sequences of positive linear operators. In this setting, on studying a sequence of operators, say , one usually intends to prove firstly that the sequence defines an approximation process; that is, for each function of some space, converges to , in a certain sense as tends to infinity; afterwards, one searches for quantitative results that estimate the degree of convergence, and finally one measures the goodness of the estimates, mainly through inverse and saturation results. An outstanding tool to achieve these saturation results is given by the asymptotic formulae that provide information about the so-called optimal degree of convergence. The most representative expression of this type was stated for the classical Bernstein operators by Voronovskaja in  and reads as follows: for and , if exists, then This work dwells upon this type of expressions.
A classical key ingredient to prove an asymptotic formula for a sequence of positive linear operators is Taylor's theorem. From it, the formula appears after some minor work if one is able to find easy-to-use expressions for the first moments of the operators, namely, ( and denote the monomials and ). On the contrary, if the calculation of the moments gets complicated, then a drawback appears.
The main purpose of this paper is to present a tool that helps to overcome these situations. The basic ideas behind the main result (Theorem 2.1 below) lies in [2, Chapter 5], the novelty here being that instead of using the Taylor formula of the function to be approximated, we consider the Taylor formula of for a certain function .
It is the intention of the authors that the paper offers a clear and quick procedure to obtain asymptotic expressions for a wide variety of sequences of linear positive operators.
2. The Main Result
Let be any -times continuously differentiable function on , such that , and for . We denote by the function and denotes the usual th differential operator, though we keep on using the common notation and for the first and second derivatives of a function .
Although we restrict our attention to the space of all continuous functions defined on , the following main result of the paper remains valid for any compact real interval with the obvious modifications.
Theorem 2.1. Let be a sequence of linear positive operators, and let be fixed. Let us assume that there exist a sequence of positive real numbers (as ) and two functions , being strictly positive on , such that for all Then for each , twice differentiable at the point ,
Remark 2.2. Notice that for , identity (2.2) becomes , which is obviously fulfilled if the operators preserve the constants. This property is satisfied by most classical sequences of linear operators (see, e.g., ), among them being the ones that we study in the present paper.
Moreover, for , identity (2.2) becomes, respectively, These are, actually, the identities that we will explicitly use throughout the paper. However, we have decided to write the hypotheses of the theorem as in (2.2) for the sake of brevity and to put across that we can think of the result as if it were of Korovkin type, since we can guarantee the convergence of and obtain its limit for any , whenever we have it for four test functions.
Proof. The classical Taylor theorem, applied to the function , yields for that
where is a continuous function which vanishes at 0. Equivalently we can write for
where . Applying the operator and then evaluating at the fixed point , we obtain the equality
Now we subtract from both sides and multiply by to get
Taking into account the basic identities
and using (2.4), we derive that
The proof will be over once we prove that as .
To this purpose let and let be an open set containing such that for , . Then if we define , we have that for all On the other hand, vanishes on , so there is a constant such that for all ,
Finally, the linearity and positivity of allow us to write, from (2.11) and (2.12), from where taking limits and using again (2.4), This ends the proof, as and are strictly positive and was arbitrary.
The first two applications correspond to sequences of operators recently introduced in [4, 5]. They represent respective modifications of the classical Bernstein operators and the well-known modified Meyer-König and Zeller operators (see ) which, instead of preserving the linear functions, hold fixed and . Thus they are inside this new line of work which originated with the paper  and found further development in a long list of papers (see, e.g., [8–14]).
The aforementioned preserving property usually makes it quite difficult to compute the first moments and consequently to obtain asymptotic formulae. Here our result, applied with and , enters the scene.
Two further applications with different values of are presented afterwards in less detail.
3.1. Modified Bernstein Operators Which Preserve
This section deals with the following sequence of operators presented in  (we use the same notation) as a byproduct of some interesting results, defined for and as As pointed out in , this provides an example of a sequence of positive linear polynomial operators that preserve and and represents an approximation process for functions .
The presence of the square root in the definition makes it difficult to obtain easy-to-handle expressions for the moments , and consequently to obtain an asymptotic formula. We will apply our theorem to get it, though we first prove a quantitative result missed in .
Proposition 3.1. Let , , and let . Then
Proof. From the usual quantitative estimate in terms of stated in , we can write Now, for , , so and . Thus On the other hand, for , , so and using the equalities we get from which the result follows.
Corollary 3.2. For all and , whenever exists,
Proof. We will apply the theorem with , and . As the operators preserve the constants, it suffices to check that (2.4) holds true.
First computations yield Thus for the quantities appear after some calculations using the following identities:
Finally we are in a position to apply Theorem 2.1 and then prove the corollary, since we show below that assumptions in (2.4) are fulfilled:
3.2. The Modified Meyer-König and Zeller Operators
For , , and , we consider the operators defined as for and .
It turns to be another example of a sequence of positive linear operators that preserve and and represents an approximation process for functions .
Here again, the presence of the square root in the definition makes it difficult to obtain easy-to-handle expressions for , , and consequently to obtain by usual means an asymptotic formula. With this aim we shall apply our theorem.
Corollary 3.3. For all and , whenever exists,
Proof. We will apply the theorem with and . As the operators preserve the constants, it suffices to check that (2.4) holds true.
Direct computations with the use of mathematical software (Mathematica) give Now, using again (3.10) we can write and then the following identities which prove the corollary:
3.3. The Modified Bernstein Operators Which Preserve a General Function
Let be any function fulfilling the same properties as the general function considered in the paper. For , , and , we consider the following operators defined from the classical Bernstein operators : This sequence of linear operators was studied by the authors in . Here we show a nice way to obtain its asymptotic formula.
Corollary 3.4. For all and , whenever exists,
Proof. It follows the same pattern as the proof of the previous corollaries. It suffices to make use of the following identities which one can obtain directly from the corresponding ones for the Bernstein operators (i.e., ; see, e.g., ):
3.4. The Modified Bernstein Operators Which Preserve
This section deals with the family of sequences of positive linear polynomial operators , presented in  and defined for and as
The first two elements and are, respectively, the Bernstein operators and those ones studied in Section 3.1. The operator holds fixed the functions and .
The next corollary deals with and shows an application of Theorem 2.1 with .
Corollary 3.5. For all and , whenever exists,
Proof. It follows the same pattern as Corollary 3.2 although some more cumbersome calculations, which we have carried out with the use of Mathematica, are required. We detail below the identities that allow us to end the proof. We make use of the notation For the quantities , required to apply Theorem 2.1, appear after some calculations using the following identities:
Finally, motivated by the well-known Voronovskaja formula for the classical Bernstein operators and by the results in Corollaries 3.2 and 3.5, we close this section and the paper stating the following conjecture.
Conjecture 3.6. For all , and , whenever exists,
The authors thank the referees for their suggestions. The first two authors are partially supported by the Junta de Andalucía (Research Group FQM-0178) and by Universidad de Jaén and Caja Rural de Jaén (Project UJA 2009/12/07).
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