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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 824058, 5 pages
http://dx.doi.org/10.1155/2013/824058
Research Article

Optimal Simultaneous Approximation via -Summability

1Departamento de Matemáticas, I.E.S. Virgen del Carmen, 23008 Jaén, Spain
2Departamento de Matemáticas, Universidad de Jaén, Campus Las Lagunillas s/n., 23071 Jaén, Spain

Received 22 May 2013; Accepted 15 August 2013

Academic Editor: Zhongxiao Jia

Copyright © 2013 Francisco Aguilera 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.

Abstract

We present optimal convergence results for the mth derivative of a function by sequences of linear operators. The usual convergence is replaced by -summability, with being a sequence of infinite matrices with nonnegative real entries, and the operators are assumed to be m-convex. Saturation results for nonconvergent but almost convergent sequences of operators are stated as corollaries.

1. Introduction

The notion of almost convergence of a sequence introduced by Lorentz [1] in 1948 entered the Korovkin-type approximation theory (see [2]) through the papers of King and Swetits [3] and Mohapatra [4]. A step forward was given by Swetits [5] in 1979 who applied in the theory the more general notion of -summability that Bell [6] had introduced a few years earlier.

After Swetits, within a shape preserving approximation setting and using as well -summability, one finds in the literature two recent papers of the authors, [7, 8], where they studied, on one hand, qualitative and quantitative Korovkin-type results, and on the other, results on asymptotic formulae. In this paper we continue this line of work which naturally takes us to the topic of saturation. Indeed, after having established an asymptotic formula, a natural way to keep on is to study optimal results to control the goodness of the approximation errors. Here saturation enters the picture. Now, before detailing our specific aim, we present the general framework of the paper which includes the definition of -summability.

Let be a sequence of infinite matrices with nonnegative real entries; then a sequence of real numbers is said to be -summable to if (whenever the series below converges for all and ) Notice that -summability extends classical convergence, matrix summability, the Cesaro summability, and almost convergence amongst others.

Now, let , let denote the space of all -times continuously differentiable functions on the real interval , let denote the usual th differential operator, and finally, let be a sequence of linear operators fulfilling the following properties:(P0) for each and , is -summable to , or equivalently converges to as tends to infinity, uniformly in , (P1) each is -convex; that is, it maps -convex functions onto -convex functions; recall that a function is said to be -convex whenever for all , (P2) there exist a sequence of real positive numbers and three strictly positive functions , and defined on with such that for , -times differentiable in some neighborhood of a point , uniformly in .

The asymptotic formula (3) informs us that the order of convergence of towards is not better than if the right-hand side of (3) is different from . Thus, is called the optimal order of convergence, and those functions that possess it form the saturation class. As for our specific aim with this paper, the results of Section 2 give us information about this saturation class, while Section 3 is devoted to state a sort of converse result of asymptotic formulae. We follow the line of two respective papers of two of the authors, namely [9, 10], which at the same time have their foundations on two outstanding papers of Lorentz and Schumaker [11] and Berens [12]. The last section of the paper contains some applications. Now we close this one with some remarks and notation that we will use throughout the paper.

Firstly we point out that if (P1) fulfills and on , then for all , .

Secondly, if we consider a bounded subinterval and fix a point , it is well known that the functions , , and form in an extended complete Tchebychev system (see [13]). Moreover is a fundamental system of solutions of the second-order differential equation in the unknown (see the right-hand side of (3)) that follows: Besides .

In this respect, we refer the reader to [11] to recall the class , , formed by those functions , differentiable on , fulfilling where . Notice that if , then amounts to the fact that belongs to the classical class .

Finally, if is a double sequence of real numbers such that uniformly in and is another sequence of real numbers with , then we use the notation to indicate that

2. Saturation Results

In this section we obtain local saturation results in the approximation process of towards . Firstly we state without proof three lemmas; Lemma 1 coincides with [10, Lemma 1], Lemma 2 follows the same pattern as [10, Lemma 2], and finally Lemma 3 is a very direct consequence of (P1).

Lemma 1. Let be a bounded open subinterval of . Let and such that , and . Then there exist a real number , a solution of the differential equation (4) on , say , and a point such that and, for all , .

Lemma 2. Let and let . Assume that there exists a neighborhood of where . Then

Lemma 3. is a solution of the differential equation (4) in some neighborhood of if and only if

The following two propositions, of interest by themselves, prepare the way to prove the announced results. An important role is played by the notion of convexity with respect to the extended complete Tchebychev system that here we relate to the monotonic convergence of the process and allows us to compare the degree of approximation for two different functions.

Proposition 4. Let ; then (a) is convex with respect to on if and only if for each (b)if for all , then is convex with respect to on .

Proof. (a) Let . Assume that is convex with respect to on and let such that (here denotes the right first derivative operator). Then, from [11, Lemma 2.2], we have that for all , and directly from Lemma 2, if we take such that for all , we derive that or equivalently Finally we apply Lemma 3 to the fuction and obtain the required inequality as follows:
To prove the converse we assume the contrary; that is, that is not convex with respect to on ; then there exist three points , and such that where is the unique function of the space which interpolates at and .
Now we apply Lemma 1 with and and derive the existence of , a solution of and satisfying Let us take such that , and on and apply then Lemma 2 taking into account (15). This yields that After introducing equality (16) we get Finally, multiplying by and applying (P2) we obtain the following inequality which contradicts our assumption: (b) If for , then directly from (P0) we have that and it suffices to use (a) to complete the proof.

Proposition 5. Let and let . Then the following items are equivalent (i) are convex with respect to on ,(ii)for each

Proof. It suffices to apply Proposition 4 replacing by .

With appropriate choices of the function and applying the results of [11], we give two saturation results; the first one is stated in terms of classic Lipschitz spaces, while the second one puts across the relationship with the asymptotic formula.

Theorem 6. Let . Then if only if, on ,

Proof. Take such that and then apply Proposition 5. Thus the result follows directly after using (P2) and [11, Theorem 3.2] taking into account that .

Theorem 7. Let . Then if and only if, almost everywhere on ,

Proof. Take such that and then apply Proposition 5. Thus the result follows directly after using (P2) and [11, Theorem 3.2] taking into account that .

3. Converse Result of the Asymptotic Formula

This section is devoted to give a converse result of the asymptotic formula stated in (3). It turns to be an extension of the results of [12]. A rough statement of the problem would read as follows: under the general framework of the paper, assume the existence of a function such that for , Is -times differentiable at ? is it true that ?

The answer, affirmative in certain sense, represents the content of this section. We will make use of two lemmas. We state them without proof as they resemble closely [10, Lemmas 3, 4].

Lemma 8. Let . If then is convex with respect to on .

Lemma 9. Let and let such that for all ; then

Theorem 10. Let and let a finitely valued function in such that for then almost everywhere on , .

Proof. It follows the same pattern as [10, Theorem 1]. We detail it however for the sake of completeness. Let , and let such that for all For , let and be, respectively, the minor and major functions of with respect to such that whose existence is guaranteed from the theory of Lebesgue integration (see e.g., [14]). In particular it follows that From the assumptions and Lemma 9, if we consider such that for all , we have that hence Now Lemma 8 yields that for each , is convex with respect to on . Letting tend to infinity we derive that is convex respect to on . If we proceed this way with , we conclude that is convex respect to on as well. Hence, in this interval and consequently, almost everywhere on from where the proof follows recalling the definition of at the top of the proof, the one of in (4), and finally using (P2).

4. Applications

In this section we illustrate the use of some of the results of the paper. We will make use of the asymptotic formulae obtained in [8, Section 3] to state some saturation results for the classical Bernstein operators and for a modification of them. Here we consider almost convergence, as a particular case of -summability. We refer the reader to [8, Subsections 3.1, 3.2] for further details.

4.1. Saturation of Bernstein Operators and Almost Convexity

Let .

Corollary 11. Let and ; then if and only if

Corollary 12. Let and ; then if and only if

4.2. Saturation of Modified Bernstein Operators and Almost Convexity

Here we consider the sequence of linear operators given in [8, Subsection 3.2].

Corollary 13. Let and ; then if and only if

Corollary 14. Let and ; then if and only if

Acknowledgment

This work is partially supported by Junta de Andalucía (Research Group FQM-0178).

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