\(^\S \)Department of Mathematics and Computer Science, University of Oradea, Universitatii str. no. 1, 410087 Oradea, Romania, e-mail :

lcoroianu@uoradea.ro

\(^\ast \)Department of Mathematics and Computer Science, University of Oradea, Universitatii str. no. 1, 410087 Oradea, Romania, e-mail : galso@uoradea.ro

\(^\bullet \)The work of both authors has been supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0861.

1 Introduction

Based on the Open Problem 5.5.4, pp. 324-326 in [ 14 ] , in a series of recent papers we have introduced and studied the so-called max-product operators attached to the Bernstein polynomials and to other linear Bernstein-type operators, like those of Favard-Szász-Mirakjan operators (truncated and nontruncated case), see [ 1 ] , [ 3 ] , Meyer-König and Zeller operators, see [ 4 ] , Baskakov operators, see [ 6 ] , [ 7 ] and Bleimann-Butzer-Hahn operators, see [ 5 ] .

For example, in the recent paper [ 2 ] , starting from the linear Bernstein operators \(B_{n}(f)(x)=\sum _{k=0}^{n}b_{n,k} (x)f(k/n)\), where \(b_{n,k}(x)={\binom {n}{k}}x^{k}(1-x)^{n-k}\), written in the equivalent form

and then replacing the sum operator \(\Sigma \) by the maximum operator \(\bigvee \), one obtains the nonlinear Bernstein operator of max-product kind

where the notation \(\bigvee _{k=0}^{n}b_{n, k}(x)\) means \(\max \{ b_{n, k}(x) ; k\in \{ 0, ..., n\} \} \) and similarly for the numerator.

For this max-product operator, nice approximation and shape preserving properties were found in the class of positive valued functions, in e.g. [ 2 ] , [ 12 ] .

In other two recent papers [ 9 ] and [ 10 ] , this idea is applied to the Lagrange interpolation based on the Chebyshev nodes of second kind plus the endpoints, and to the Hermite-Fejér interpolation based on the Chebyshev nodes of first kind respectively, obtaining max-product interpolation operators which, in general, (for example, in the class of positive Lipschitz functions) approximates essentially better than the corresponding Lagrange and Hermite-Fejér interpolation polynomials.

Let \(I=[a,b]\), \(a{\lt}b\) and \(f:[a,b]\rightarrow \mathbb {R}\). The max-product Lagrange interpolation operator on equidistant knots attached to the function \(f\) is given by (see [ 11 ] )

where \(x_{n,k}=a+(b-a)k/n\) for all \(n\in \mathbb {N}\) and \(k\in \{ 0,1,...,n\} \) and

for all \(x\in I\), \(n\in \mathbb {N}\) and \(k\in \{ 0,1,...,n\} .\) Note that \(L_{n}^{(M)}(f)\) is a well defined function. Indeed, using the fundamental Lagrange polynomials,

we observe that we can rewrite \(l_{n,k}(x)\), \(x\in I\), in the form

where

Then, since for any \(x\in I\) we have \(\sum \nolimits _{i=0}^{n}p_{n,i}(x)=1\) it follows the existence of \(i(x)\in \{ 0,1,...,n\} \) such that \(p_{n,i(x)}(x){\gt}0\) and noting that \(c_{n,i(x)}{\gt}0\) it easily results that \(l_{n,i(x)}(x){\gt}0\) and this implies that \(\bigvee \limits _{k=0}^{n}l_{n,k}(x){\gt}0 \) for all \(x\in I\), which means that indeed \(L_{n}^{(M)}(f)\) is a well defined function on \([a, b]\).

The max-product operator \(L^{(M)}_{n}(f)(x)\) is continuous on \([a, b]\) and has the interpolation properties \(L^{(M)}_{n}(f)(x_{n,j}) = f(x_{n, j})\) for all \(j\in \{ 0, 1, ... , \} \).

Also, according to Corollary 3.2, (i), in [ 11 ] , for positive valued functions, i.e. for \(f:[a, b]\to \mathbb {R}_{+}\), it satisfies the Jackson-type estimate

where \(\omega _{1}\left(f ; \tfrac {b-a}{n}\right)_{[a, b]}\) denotes the modulus of continuity of \(f\) on \([a, b]\). This estimate for the Lagrange max-product operator essentially improves for positive valued functions the order of approximation by the classical Lagrange interpolation polynomials on equidistant nodes, when as it is well-know, we can also have a very pronounced divergence phenomenon in \([a, b]\) (see e.g. Chapter 4 in the book [ 17 ] , see also [ 16 ] , [ 8 ] ).

It is worth noting that saturation and local inverse results for \(L^{(M)}_{n}(f)(x)\) were obtained in [ 13 ] .

The plan of the paper goes as follows. In Section 2 an interesting strong localization result for the Lagrange max-product operator \(L_{n}^{(M)}\) is obtained. At the end of the section and as consequences of this localization result, a local direct result and an interesting local shape preserving property are proved. Section 3 contains comparisons with some linear interpolation operators of rational type.

It is worth noting in Section 2 the strong localization result expressed by Theorem 2.1, that shows that if the continuous strictly positive functions \(f\) and \(g\) coincide on a subinterval \([\alpha , \beta ]\subset [0, 1]\), then for sufficiently large values of \(n\), \(L_{n}^{(M)}(f)\) and \(L_{n}^{(M)}(g)\) coincide on subintervals sufficiently close to \([\alpha , \beta ]\). Clearly, Corollary 2.4 shows that \(L_{n}^{(M)}(f)\) is very suitable to approximate continuous functions which are constant on some subintervals. Namely, if \(f\) is a continuous strictly positive function which is constant on some subintervals \([\alpha _{i}, \beta _{i}]\), \(i=1, ..., p\), of \([a, b]\), then for sufficiently large \(n\), \(L_{n}^{(M)}(f)\) takes the same constant values on subintervals sufficiently close to each \([\alpha _{i}, \beta _{i}]\), \(i=1, ..., p\).

2 Localization Results

Let \(\ l_{n,k}\) denote the fundamental Lagrange polynomials attached to the knots \(x_{n,k}=k/n\), \(k\in \{ 0,1,...,n\} \), \(n\in \mathbb {N}\).

The main result of this section is the following localization result.

Let us choose arbitrary \(x\in \lbrack c,d]\) and for each \(n\in \mathbb {N}\) let \(j_{x}\in \{ 0,1,...,n\} \) ( \(j_{x}\) depends on \(n\) too, but there is no need at all to complicate on the notations) be such that \(x\in \lbrack j_{x}/n,(j_{x}+1)/n].\) Then we know that

where \(J_{n}(x)=\{ k\in \{ 0,1,...,n\} :l_{n,k}(x){\gt}0\} \) and \(l_{n,k}\), \(k\in \{ 0,1,...,n\} \) are the Lagrange fundamental polynomials attached to the knots \(x_{n,k}=k/n\), \(k\in \{ 0,1,...,n\} .\) Since \(x\in \lbrack c,d]\cap \lbrack j_{x}/n,(j_{x}+1)/n]\) and since \(a{\lt}c{\lt}d{\lt}b\) it is immediate that for \(n\geq n_{0}\) where \(n_{0}\) is chosen such that \(1/n_{0}{\lt}\min \{ c-a,d-b\} \), we obtain \(a{\lt}j_{x}/n{\lt}b\) which gives \(na{\lt}j_{x}{\lt}nb\) for all \(n\geq n_{0}\) (indeed, if we would suppose that there exists \(n{\gt}n_{0}\) which does not satisfy the previous double inequalities, then we would easily get a contradiction).

It is important to notice here that \(n_{0}\) does not depend on \(x.\) From the inequalities \(na{\lt}j_{x}{\lt}b\) it follows that if \(n\geq n_{0}\) then for any \(x\in \lbrack c,d]\) there exists \(\alpha _{x}\in \lbrack a,b]\) such that \(j_{x}=n\alpha _{x}\).

In what follows, it will serve to our purpose to use the sequence \((a_{n})_{n\geq 1}\), \(a_{n}=\sqrt{n}\). For this sequence there exists \(n_{1}\in \mathbb {N}\) such that \(na-a_{n}{\gt}0\) for all \(n\geq n_{1}.\)

Our intention is to prove as an intermediate result, that there exists an absolute constant \(N_{0}\in \mathbb {N}\) which does not depend of \(x\in \lbrack c,d]\) such that for any \(n\geq N_{0}\) and \(x\in \lbrack c,d]\) we have \(\bigvee \limits _{k=0}^{n}l_{n,k}(x)f(\tfrac {k}{n})=\bigvee \limits _{k\in I_{n,x}}l_{n,k}(x)f(\tfrac {k}{n})\) where \(I_{n,x}=\{ k\in J_{n}(x):j_{x}-a_{n}\leq k\leq j_{x}+a_{n}\} .\) In order to obtain this conclusion, for \(n\geq \max \{ n_{0},n_{1}\} \) let us choose \(k\in J_{n}(x)\setminus I_{n,x}\). We have two cases: i) \(k+a_{n}{\lt}j_{x}\) and ii) \(j_{x}+a_{n}{\lt}k.\)

Case i) Firstly, note that \(j_{x}\in J_{n}(x)\), because \(sign(l_{n, j}(x))=(-1)^{n-j}\cdot (-1)^{n-j}=1\). Noting that \(k/n{\lt}(j_{x}-a_{n})/n\) and \(nx\geq j_{x},\) we get   

Then, denoting the infimum and the supremum of \(\ f\) on \([a, b]\) with \(m_{f}\) and \(M_{f}\) respectively (according to the hypotheses these values are strictly positive), we get that

Since \(\lim \limits _{n\rightarrow \infty }\sqrt{n}\cdot \tfrac {m_{f}}{M_{f}}=\infty ,\) it follows that there exists \(n_{2}\in \mathbb {N}\), \(n_{2}\geq \max \{ n_{0},n_{1}\} \) such that \(\tfrac {l_{n,j_{x}}(x)f(\tfrac {j_{x}}{n})}{l_{n,k}(x)f(\tfrac {k}{n})}{\gt}1\) for all \(x\in \lbrack c,d]\), \(n\geq n_{2}\) and \(k\in \{ 0,1,...,n\} \), \(k{\lt}j_{x}-a_{n}\) (as \(k\not\in I_{n,x}\)). In addition, it is important to notice that \(n_{2}\) does not depend on \(x\in \lbrack c,d]\), but of course it depends on \(f.\)

Case ii) The proof is identical with the proof of the above Case i) and therefore we conclude that there exists an absolute constant \(n_{3}\in \mathbb {N}\) which depends only on \(a,b,c,d,f\) such that

for all \(x\in \lbrack c,d]\), \(n\geq n_{3}\) and \(k\in \{ 0,1,...,n\} \), \(k{\gt}j_{x}+a_{n}\).

Analyzing the results obtained in cases i)-ii), it results that for all \(x\in \lbrack c,d]\), \(n\geq N_{0}\),\(\ N_{0}=\max \{ n_{2},n_{3}\} \) and \(k\in \{ 0,1,...,n\} \), with \(k{\lt}j_{x}-a_{n}\) or \(k{\gt}j_{x}+a_{n}\), we have

Since from the Case i) we know that \(j_{x}\in J_{n}(x)\) and since this easily implies that actually \(j_{x}\in I_{n,x}\), we obtain our preliminary result, that is

where \(I_{n,x}=\{ k\in J_{n}(x):j_{x}-a_{n}\leq k\leq j_{x}+a_{n}\} .\)

Next, let us choose arbitrary \(x\in \lbrack c,d]\) and \(n\in \mathbb {N}\) so that \(n\geq N_{0}.\) If there exists \(k\in \) \(I_{n,x}\) such that \(k/n\notin \lbrack c,d]\) then we distinguish two cases. Either \(k/n{\lt}c\) or \(k/n{\gt}d.\) In the first case we observe that

Since \(\lim \limits _{n\rightarrow \infty }\tfrac {a_{n}+1}{n}=0,\) it results that for sufficiently large \(n\) we necessarily have \(\tfrac {a_{n}+1}{n}{\lt}c-a\) which clearly implies that \(k/n\in \lbrack a,c].\) In the same manner, when \(k/n{\gt}d\), for sufficiently large \(n\) we necessarily have \(k/n\in \lbrack d,b]. \)

Summarizing, there exists \(\widetilde{N}_{1}\in \mathbb {N}\) independent of any \(x\in \lbrack c,d]\), such that

and in addition for any \(x\in \lbrack c,d]\), \(n\geq \widetilde{N}_{1}\) and \(k\in I_{n,x}\), we have \(k/n\in \lbrack a,b].\) Also, it is easy to check that \(\widetilde{N}_{1}\) depends only on \(a,b,c,d,f.\) We thus obtain that

and in addition for any \(x\in \lbrack c,d]\), \(n\geq \widetilde{N}_{1}\) and \(k\in I_{n,x}\), we have \(k/n\in \lbrack a,b].\)

Reasoning for the function \(g\) exactly as in the case of the function \(f\), it follows that there exists \(\widetilde{N}_{2}\in \mathbb {N}\) which depends only on \(a,b,c,d,g\) such that

and in addition for any \(x\in \lbrack c,d]\), \(n\geq \widetilde{N}_{2}\) and \(k\in I_{n,x}\), we have \(k/n\in \lbrack a,b].\) Taking \(\widetilde{n}=\max \{ \widetilde{N}_{1},\widetilde{N}_{2}\} \), we easily obtain the desired conclusion.

We can easily extend the above result to arbitrary intervals, as follows.

We obtain the desired conclusion as a direct consequence of the previous theorem. Indeed, firstly to make a distinction, we denote with \(\overline{L}_{n}^{(M)}\) the Lagrange max-product operator attached to functions defined on the interval \([0,1].\) In addition, in what follows, for all all \(n\in \mathbb {N}\) and \(k\in \{ 0,1,...,n\} \) we denote with \(l_{n,k}^{1}\) the fundamental Lagrange polynomials defined on the interval \([0,1].\)

Suppose now that for the two functions \(f, g\in C([a,b])\) we have \(f(x)=g(x)\), for all \(x\in [a^{\prime }, b^{\prime }]\). Let us define the function \(h:[0,1]\rightarrow \mathbb {[}a,b]\), \(h(y)=a+(b-a)y.\) It is immediate that for any \(x\in \lbrack a,b]\) there exists an unique \(y(x)=h^{-1}(x)\in \lbrack 0,1]\) such that \(f(x)=(f\circ h)(y(x))\) and \(g(x)=(g\circ h)(y(x))\).

Then we observe that for any \(x\in \lbrack a,b]\) we have

The above equalities imply

and analogously \(L_{n}^{(M)}(g)(x)=\overline{L}_{n}^{(M)}(g\circ h)(y(x))\), for all \(x\in \lbrack a,b].\)

Then, our result is immediate by applying Theorem 2.1 to \(\overline{L}_{n}^{(M)}(g\circ h)(y(x))\) and \(\overline{L}_{n}^{(M)}(g\circ h)(y(x))\), where recall that \(f\circ h, g\circ h :[0, 1]\to [0, +\infty )\), \(y(x)=h^{-1}(x)\) and \(h:[0, 1]\to [a, b]\), \(h(x)=a+(b-a)x\).

The next direct approximation result is now an immediate consequence of the localization result in Theorem 2.2, as follows.

Let us define the function \(F:[a,b]\rightarrow \mathbb {R}\),

The hypothesis imply that \(F\) is continuous and strictly positive on \([a, b]\) and according to Corollary 3.2 in [ 11 ] it results that

Since by the definition of \(F\) we have \(\omega _{1}(F,\tfrac {b-a}{n})_{[a,b]}\leq \omega _{1}(f,\tfrac {b-a}{n})_{[a,b]}\) and since by the relation (2.1) it easily follows \(\omega _{1}(f,\tfrac {b-a}{n})_{[a,b]}\leq C_{0}(b-a)/n\), we get

Now, let us choose arbitrary \(c,d\in \lbrack a^{\prime },b^{\prime }]\) such that \(a^{\prime }{\lt}c{\lt}d{\lt}b^{\prime }.\) Then, by Theorem 2.2 (applicable to \(f\) and \(F\)) it results the existence of \(\widetilde{n}\in \mathbb {N}\) which depends only on \(a,b,a^{\prime },b^{\prime }c,d,f,F\) such that \(L_{n}^{(M)}(F)(x)=L_{n}^{(M)}(f)(x)\) for all \(x\in \lbrack c,d].\) But since actually the function \(F\) depends on the function \(f\), it is clear that in fact \(\widetilde{n}\) depends only on \(a,b,a^{\prime },b^{\prime }c,d\) and \(f\).

Therefore, for arbitrary \(x\in \lbrack c,d]\) and \(n\in \mathbb {N}\) with \(n\geq \widetilde{n}\) we obtain

where \(C_{0}\) and \(\widetilde{n}\) depend only on \(a,b,a^{\prime },b^{\prime }c,d\) and \(f\).

Now, denoting

we finally obtain

with \(C=\max \{ 2C_{0}(b-a),C_{1}\} \) depending only on \(a,b,c,d\) and \(f\). This proves the corollary. At the end of this section, as a consequence of the localization result in Theorem 2.2 we present a locally constant preserving property.

Let \(g:[a, b]\to \mathbb {R}_{+}\) be given by \(g(x)=\alpha {\gt}0\) for all \(x\in [a, b]\). Since \(f(x)=g(x)\) for all \(x\in [a^{\prime }, b^{\prime }]\) and since obviously \(L_{n}^{(M)}(g)(x)=\alpha \) for all \(x\in [a, b]\), by Theorem 2.2 we easily obtain the desired conclusion.

3 Final Remarks

Let us note that in Hermann-Vértesi [ 15 ] , starting from a Lagrange interpolatory process (convergent or not)

with

new linear interpolatory (rational) operators of the form

are constructed, for which in the case when \(r{\gt}2\) and the knots \(x_{n, k}\) satisfy some special requirements (e.g. some Jacobi knots), the Jackson-type order of approximation

is obtained (see e.g. Theorem 3.2 in Hermann-Vértesi [ 15 ] ).

In other words, for the linear rational construction \(R_{n}(f)(x)\), we get the same order of approximation as for the interpolatory max-product operator (which is piecewise rational)

But clearly that with respect to \(R_{n}(f)(x)\), the max-product rational operator \(L_{n}^{(M)}(f)(x)\) presents several advantages, pointed out by the next remarks.