#### Abstract

Starting from the study of the Shepard nonlinear operator of max-prod type by Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324β326, the Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form (with an unexplicit absolute constant ) and the question of improving the order of approximation is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions including, for example, that of concave functions, we find the order of approximation , which for many functions is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.

#### 1. Introduction

Starting from the study of the Shepard nonlinear operator of max-prod-type in [1, 2], by the Open Problem in a recent monograph [3, pages 324β326, 5.5.4], the following nonlinear Bernstein operator of max-prod type is introduced (here means maximum):

where , for which by a very complicated method in [4, Theorem ], an upper estimate of the approximation error of the form (with unexplicit absolute constant) is obtained. Also, by Remark in the same paper [4], the question if this order of approximation could be improved is raised.

The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of and it can easily be extended to other max-prod operators of Bernstein type. Then, one proves by a counterexample that in a sense, for arbitrary this order of approximation with respect to cannot be improved, giving thus a negative answer to a question raised in [4, Remark , ]. However, for subclasses of functions including, for example, that of concave functions, we find the order of approximation , which for many functions, is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are presented.

Section 2 presents some general results on nonlinear operators, in Section 3 we prove several auxiliary lemmas, Section 4 contains the approximation results, while in Section 5 we present some shape-preserving properties. The paper ends with Section 6 containing some conclusions concerning the comparisons between the max-product and the linear Bernstein operators.

#### 2. Preliminaries

For the proof of the main results, we need some general considerations on the so-called nonlinear operators of max-prod kind. Over the set of positive reals, , we consider the operations (maximum) and ββproduct. Then has a semiring structure and we call it as Max-Product algebra.

Let be a bounded or unbounded interval, and

The general form of (called here a discrete max-product-type approximation operator) studied in the paper will be

or

where ,, and , for all . These operators are nonlinear, positive operators and moreover they satisfy a pseudolinearity condition of the form

In this section we present some general results on these kinds of operators which will be useful later in the study of the Bernstein max-product-type operator considered in Section 1.

Lemma 2.1 (see [4]). Let be a bounded or unbounded interval: and let , be a sequence of operators satisfying the following properties: (i)if satisfy then for all ; (ii) for all .
Then for all , and we have

Proof. Since it is very simple, we reproduce here the proof in [4]. Let . We have , which by conditions (i)-(ii) successively implies , that is, .
Writing now and applying the above reasonings, it follows that , which, combined with the above inequality, gives .

Remark 2.2. () It is easy to see that the Bernstein max-product operator satisfies the conditions in Lemma 2.1, (i), (ii). In fact, instead of (i), it satisfies the stronger condition: Indeed, taking in the above equality , , it easily follows that .
() In addition, it is immediate that the Bernstein max-product operator is positive homogenous, that is, for all .

Corollary 2.3 (see [4]). Let , be a sequence of operators satisfying conditions (i)-(ii) in Lemma 2.1 and in addition being positive homogenous. Then for all , and one has where , for all , for all ,, and if is unbounded, then we suppose that there exists , for any , .

Proof. The proof is identical with that for positive linear operators and because of its simplicity, we reproduce it what follows. Indeed, from the identity it follows (by the positive homogeneity and by Lemma 2.1) that Now, since for all we have replacing the above, we immediately obtain the estimate in the statement.

An immediate consequence of Corollary 2.3 is as follows.

Corollary 2.4 (see [4]). Suppose that in addition to the conditions in Corollary 2.3, the sequence satisfies , for all . Then for all , and one has

#### 3. Auxiliary Results

Since it is easy to check that for all , notice that in the notations, proofs and statements of the all approximation results, that is, in Lemmas 3.1β3.3, Theorem 4.1, Lemmas 4.2β4.4, Corollaries 4.6, 4.7, in fact we always may suppose that . For the proofs of the main results, we need some notations and auxiliary results, as follows.

For each and let us denote

It is clear that if then

and if then

Also, for each , and let us denote

and for each , and let us denote

Lemma 3.1. Let : (i)for all , one has (ii)for all , one has

Proof. (i) The inequality is immediate.
On the other hand,
which proves (i).
(ii) The inequality is immediate.
On the other hand,
which proves (ii) and the lemma.

Lemma 3.2. For all and one has

Proof. We have two cases: 1) 2) .Case 1. Since clearly the function is nonincreasing on , it follows that which implies Case 2. We get which immediately implies that Since , the conclusion of the lemma is immediate.

Lemma 3.3. Let . (i)If is such that , then . (ii)If is such that , then

Proof. (i) We observe that Since the function clearly is nonincreasing, it follows that for all Then, since the condition implies , we obtain
β(ii) We observe that
Since the function is nondecreasing, it follows that for all Then, since the condition implies , we obtain which proves the lemma.

Also, a key result in the proof of the main result is the following.

Lemma 3.4. One has where .

Proof. First we show that for fixed and we have Indeed, the inequality one reduces to after simplifications is equivalent to However, since , the above inequality immediately becomes equivalent to By taking in the inequality just proved above, we get and so on, and so on, From all these inequalities, reasoning by recurrence we easily obtain and so on, finally which proves the lemma.

#### 4. Approximation Results

If represents the nonlinear Bernstein operator of max-product type defined in Section 1, then the first main result of this section is the following.

Theorem 4.1. If is continuous, then one has the estimate where

Proof. It is easy to check that the max-product Bernstein operators fulfill the conditions in Corollary 2.4 and we have where So, it is enough to estimate
Let , where is fixed, arbitrary. By Lemma 3.4 we easily obtain
In all what follows we may suppose that , because for simple calculation shows that in this case, we get , for all . So it remains to obtain an upper estimate for each when is fixed, and . In fact, we will prove that which immediately will implies that and taking in (4.3), we immediately obtain the estimate in the statement.
In order to prove (4.6) we distinguish the following cases: 1) , 2) and, 3) .
Case 1. If then Since , it easily follows that
If then Since by Lemma 3.2 we have , we obtain
If then
Case 2. Subcase a
Suppose first that . We get
Subcase b
Suppose now that . Since the function is nondecreasing on the interval it follows that there exists , of maximum value, such that . Then, for we get and Also, we have Indeed, this is a consequence of the fact that is nondecreasing on the interval and because it is easy to see that By Lemma 3.3, (i) it follows that We thus obtain for any

Therefore, in both subcases, by Lemma 3.1, (i) too, we get .
Case 3. Subcase a
Suppose first that . Then we obtain
Subcase b
Suppose now that . Let be the minimum value such that . Then satisfies and Also, because in this case we have it is immediate that By Lemma 3.3, (ii), it follows that . We obtain for any and

In both subcases, by Lemma 3.1, (ii) too, we get .
In conclusion, collecting all the estimates in the above cases and subcases we easily get the relationship (4.6), which completes the proof.

Remarks. The order of approximation in terms of in Theorem 4.1 cannot be improved, in the sense that the order of is exactly (here is defined in the proof of Theorem 4.1). Indeed, for , let us take , and denote . Then, we can write
Since , we easily get , which implies for all On the other hand,

Because , there exists such that
for all It follows for all Taking into account Lemma 3.1, (i) too, it follows that for all we have , which implies the desired conclusion.
() With respect to the method of the proof in [4], the method in this paper presents, at least, two advantages: it produces the explicit constant in front of and its ideas can be easily used for other max-prod Bernstein operators too, which will be done in several forthcoming papers.

In what follows, we will prove that for large subclasses of functions , the order of approximation in Theorem 4.1 can essentially be improved to .

For this purpose, for any , let us define the functions

Then it is clear that for any and we can write

Also we need the following four auxiliary lemmas.

Lemma 4.2. Let be such that Then where

Proof. We distinguish the two following cases.Caseββ(i).Let be fixed such that . Because by simple calculation we have and , it follows that Caseββ(ii).Let be such that We have two subcases: (a), when evidently and we immediately get (b), when Because it follows , which proves the lemma.

Lemma 4.3. Let be such that Then

Proof. We distinguish the two following cases:Caseββ(i)., when as in Lemma 4.2 we get Caseββ(ii)., when we have two subcases:(a), when as in the case of Lemma 4.2 we obtain (b), when by using the same idea as in the subcase (b) of Lemma 4.2 and taking into account that we obtain which proves the lemma.

Lemma 4.4. Let be such that for all . Then

Proof. Let . If or then and from Lemma 4.2, it follows that If then and from Lemma 4.3, we get which ends the proof.

Lemma 4.5. Let be concave. Then the following two properties hold: (i)the function is nonincreasing; (ii)the function is nondecreasing.

Proof. (i) Let be with . Then which implies that .
(ii) Let be with . Then
which implies .

Corollary 4.6. Let be a concave function. Then

Proof. Let and such that . Let be with . Then From Lemma 4.5, (i), we get that is, . Since , we get It is immediate that for it follows that . Thus we obtain
Now let be with . Then