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

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

1 Introduction

Based on the Open Problem 5.5.4, pp. 324–326 in [ 8 ] , in a series of recent papers [ 1 , 2 , 3 , 4 , 5 ] , 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), Baskakov operators (truncated and nontruncated case) and Bleimann-Butzer-Hahn operators.

This idea applied, for example, to the linear Bernstein operators \(B_{n}(f)(x)=\sum _{k=0}^{n}p_{n, k}(x)f(k/n)\), where \(p_{n,k}(x)={n\choose k}x^{k}(1-x)^{n-k}\), works as follows. Writing in the equivalent form \(B_{n}(f)(x)=\tfrac {\sum _{k=0}^{n}p_{n, k}(x)f(k/n)}{\sum _{k=0}^{n}p_{n, k}(x)}\) and then replacing everywhere the sum operator \(\Sigma \) by the maximum operator \(\bigvee \) (that is \(\sum _{k=0}^{n}p_{n, k}(x)f(k/n)\) is replaced by the maximum \(\max _{\{ k=0, .., n\} }\{ p_{n, k}(x)f(k/n)\} \) and \(\sum _{k=0}^{n}p_{n, k}(x)\) by \(\max _{\{ k=0, ..., n\} }\{ p_{n, k}(x)\} \)), one obtains the nonlinear Bernstein operator of max-product kind

for which, surprisingly nice approximation and shape preserving properties were found.

For example, it is proved that for some classes of functions (like those of monotonous concave functions), the order of approximation given by the max-product Bernstein operators, are essentially better than the approximation order of their linear counterparts.

The aim of the present paper is to use the same idea to interpolation polynomials. In the case of the Hermite-Fejér kind polynomials based on the Chebyshev nodes of first kind, for example, we will obtain that in the class of Lipschitz functions with positive values, the new obtained interpolation operator has essentially better approximation property than the Hermite-Fejér polynomials.

Thus, let \(f:[-1, 1]\to \mathbb {R}\) and \(x_{n, k}=\cos \left(\tfrac {2(n-k)+1}{2(n+1)}\pi \right)\in (-1, 1)\), \(k\in \{ 0, ..., n\} \), \(-1 {\lt} x_{n, 0} {\lt} x_{n,1} {\lt}...{\lt} x_{n, n}{\lt}1\), be the roots of the first kind Chebyshev polynomial \(T_{n+1}(x)=\cos [(n+1){\rm arccos} (x)]\). Consider the Hermite-Fejér interpolation polynomial of degree \(\le 2n+1\) attached to \(f\) and to the nodes \((x_{n, k})_{k}\),

with

It is well known that \(\sum _{k=0}^{n}h_{n, k}(x)=1\) for all \(x\in \mathbb {R}\) (and that \(h_{n,k}(x)\ge 0\) for all \(x\in [-1, 1]\) and \(k=0, ... ,n\)), which allows us to write

Therefore, applying the max-product method as in the above case of Bernstein polynomials, the corresponding max-product Hermite-Fejér interpolation operator will be given by

The plan of the paper goes as follows: in Section 2 we present some auxiliary results, in Section 3 we prove the main approximation result while in Section 4 we compare the approximation result in Section 3 with those for the linear Hermite-Fejér interpolation polynomials based on the Chebyshev knots of first kind.

2 Auxiliary Results

In all what follows, \(f\) will be considered continuous and with positive values, that is

Firstly, we present a general type approximation result, which in fact is valid for all the max-product type operators (including those of Bernstein type proved in [ 1 ] ).

First it is easy to check that as a consequence of the properties of the operator \(\bigvee \), \(f\le g\) implies \(H_{2n+1}^{(M)}(f)\le H_{2n+1}^{(M)}(g)\) and also we have \(H_{2n+1}^{(M)}(f+g)\le H_{2n+1}^{(M)}(f)+H_{2n+1}(g)\), for all \(f,g\in C_{+}[-1, 1]\).

Further, we have \(f=f-g+g\leq |f-g|+g\), which by the above two properties successively implies \(H_{2n+1}^{(M)}(f)(x)\leq H_{2n+1}^{(M)}(|f-g|)(x)+H_{2n+1}^{(M)}(g)(x)\), that is \(H_{2n+1}^{(M)}(f)(x)-H_{2n+1}^{(M)}(g)(x)\leq H_{2n+1}^{(M)}(|f-g|)(x)\).

Writing now \(g=g-f+f\leq |f-g|+f\) and applying the above reasonings, it follows \(H_{2n+1}(g)^{(M)}(x)-H_{2n+1}^{(M)}(f)(x)\leq H_{2n+1}^{(M)}(|f-g|)(x)\), which combined with the above inequality gives

Also, it is immediate that \(H_{2n+1}^{(M)}(f)\) is positive homogenous, that is

Now, since it is clear that \(H_{2n+1}^{(M)}(e_{0})=e_{0}\), where \(e_{0}(x)=1\) for all \(x\), from the identity (for a fixed \(x\in [-1,1]\))

and from the above proved properties of \(H_{2n+1}^{(M)}(f)\), it easily follows

Since for all \(t,x\in [-1,1]\) we have

replacing above we immediately obtain the estimate in the statement.

As in case of the Bernstein type max-product operators, first it will be useful to exactly calculate \(\bigvee _{k=0}^{n}h_{n,k}(x)\) for \(x\in [-1, 1]\). In this sense we have the following result.

First we show that for fixed \(n\in \mathbb {N}\) and \(0\le k{\lt}k+1\le n\), there exists a unique point \(y_{n, k}\in (x_{n, k}, x_{n, k+1})\) such that we have

Indeed, the inequality \(h_{n, k+1}(x)\le h_{n, k}(x), x\in [-1, 1]\) is equivalent to

with \(P_{n, k}(x)=x^{3}-x[2+x_{n, k}\cdot x_{n, k+1}]+(x_{n, k}+x_{n, k+1})\). Therefore, the inequality \(h_{n, k+1}(x)\le h_{n, k}(x), x\in [-1, 1]\) is equivalent to the condition that \(x\in \{ x\in [-1, 1] ; P_{n, k}(x)\ge 0 \} \bigcup \{ x_{n, j} ; j\in \{ 0, 1, ..., n\} , j\not=k+1 \} \).

But, since \(P_{n, k}(-1)=(1+x_{n, k})(1+x_{n, k+1}){\gt}0\), \(P_{n, k}(1)=(x_{n, k}-1)(1-x_{n, k+1}){\lt}0\) and \(P_{n, k}^{\prime }(x)=0\) has the two solutions

it easily follows that \(P_{n, k}(x)\) has at \(z_{1}\) a maximum point, at \(z_{2}\) a minimum point, the equation \(P_{n, k}(x)=0\) has a unique solution \(y_{n, k}\in (z_{1}, z_{2})\) and that \(P_{n, k}(x)\ge 0\) on \([-1, 1]\) if and only if \(x\in [0, y_{n, k}]\).

Now we will prove that in fact \(y_{n, k}\in (x_{n, k}, x_{n, k+1})\). Indeed, this is immediate from the following simple calculation

Therefore, as a first conclusion it follows \((2.1)\).

By taking \(k=0, 1, ..,n-1\) in the inequality (2.1), we get

so on,

so on,

From all these inequalities, reasoning by recurrence we easily obtain:

and so on finally

which proves the lemma.

For the proof of the main results we need some notations and auxiliary results, as follows.

Let us denote \(y_{n,-1}=-1\) and \(y_{n,n}=1.\) Then, for all \(k,j\in \{ 0,1,...,n\} \), and for each \(x\in \lbrack y_{n,j-1},y_{n,j}]\), we denote

We observe that for \(k\geq j+1\) we have \(x_{n,k}-x\geq x_{n,j+1}-y_{n,j}\geq 0\) and it follows that \(M_{k,n,j}(x)=m_{k,n,j}(x)(x_{n,k}-x).\) Also for \(j\geq 1\) and \(k\leq j-1\) we have \(x-x_{n,k}\geq y_{n,j-1}-x_{n,j-1}\geq 0\) and it follows that \(M_{k,n,j}(x)=m_{k,n,j}(x)(x-x_{n,k}).\)

By Lemma 2.2 it immediately follows that

for all \(x\in \lbrack y_{n,j-1},y_{n,j}].\) Multiplying the above inequalities with \(1/h_{n,j}(x)\) we get

Since \(m_{j,n,j}(x)=1\) we immediately obtain the desired conclusion.

(i) We observe that for all \(k\ge j+1\) we get

which proves (i).

(ii) For all \(k\le j-1\) we get

which proves (ii).

3 Approximation Results

The main result is the following Jackson-type estimate.

By Theorem 2.1 we have

where \(\varphi _{x}(t)=|t-x|.\) So, it is enough to estimate

Let \(x\in \lbrack y_{n,j-1},y_{n,j}]\), where \(j\in \{ 0,1,...,n\} \) is fixed arbitrary. By Lemma 2.2 we easily obtain

It remains to obtain an upper estimate for each \(M_{k,n,j}(x)\) when \(j\in \{ 0,1,...,n\} \) is fixed, \(x\in \lbrack y_{n,j-1},y_{n,j}]\) and \(k\in \{ 0,1,...,n\} .\) In fact we will prove that

which immediately will imply that

and taking \(\delta _{n}=\tfrac {2\pi }{n+1}\) in \((3.1),\) since \([2\pi ]=6\), from the property \(\omega _{1}(f;\lambda \delta )\leq ([\lambda ]+1)\omega _{1}(f;\delta )\) we immediately obtain the estimate in the statement.

In order to prove \((3.2)\), we distinguish the following cases:

1) \(j=0\) ; 2) \(j=n\) and 3) \(j\in \{ 1,2,...,n-1\} \).

Case 1) By Lemma 2.4, (i), it follows that \(E_{n}(x)=\max \limits _{k=0,1}\{ M_{k,n,0}(x)\} \) for all \(x\in \lbrack -1,y_{n,0}].\)

If \(k=0\) then \(M_{0,n,0}(x)=\left| x_{n,0}-x\right| .\) Since \(x\in \lbrack -1,y_{n,0}]\subseteq \lbrack -1,x_{n,1}]\), we obtain

If \(k=1\) then \(M_{1,n,0}(x)=m_{1,n,0}(x)\left| x_{n,1}-x\right| .\) By Lemma 2.3, it follows that \(m_{1,n,0}\leq 1\) and we obtain

In conclusion we obtain \(E_{n}(x)\leq \tfrac {9\pi ^{2} }{8(n+1)^{2}}\) for all \(x\in \lbrack -1,y_{n,0}].\)

Case 2) By Lemma 2.4, (ii), it follows that \(E_{n}(x)=\max \limits _{k=n-1,n}\{ M_{k,n,n}(x)\} \) for all \(x\in \lbrack y_{n,n-1},1].\)

If \(k=n\) then \(M_{n,n,n}(x)=\left| x_{n,n}-x\right| .\) Since \(x\in \lbrack y_{n,n-1},1]\subseteq \lbrack x_{n,n-1},1]\), we obtain

where we used the obvious equality \(1-x_{n,n-1}=x_{n,1}+1.\)

If \(k=n-1\) then \(M_{n-1,n,n}(x)=m_{n-1,n,n}(x)\left| x_{n,n-1}-x\right| \leq \left| x_{n,n-1}-x\right| =x-x_{n,n-1}\leq 1-x_{n,n-1}\leq \tfrac {9\pi ^{2} }{8(n+1)^{2}}.\)

In conclusion, we obtain \(E_{n}(x)\leq \tfrac {9\pi ^{2} }{8(n+1)^{2}}\) for all \(x\in \lbrack y_{n,n-1},1].\)

Case 3) By Lemma 2.4 it follows that \(E_{n}(x)=\max \limits _{k=j-1,j,j+1}\{ M_{k,n,j}(x)\} \) for all \(x\in \lbrack y_{n,j-1},y_{n,j}].\)

If \(k=j\) then \(M_{j,n,j}(x)=\left| x_{n,j}-x\right| .\) For \(x\in \lbrack y_{n,j-1},y_{n,j}]\subseteq \lbrack x_{n,j-1},x_{n,j+1}]\), we obtain

If \(k=j+1\) then \(M_{j+1,n,j}(x)=m_{j+1,n,j}(x)\left| x_{n,j+1}-x\right| \leq x_{n,j+1}-x.\) Since \(x\in \lbrack y_{n,j-1},y_{n,j}]\subseteq \lbrack x_{n,j-1},x_{n,j+1}]\) it follows that

If \(k=j-1\) then \(M_{j-1,n,j}(x)=m_{j-1,n,j}(x)\left| x_{n,j-1}-x\right| \leq x-x_{n,j-1}\leq x_{n,j+1}-x_{n,j-1}\leq \tfrac {2\pi }{n+1}.\)

Collecting all the estimates obtained above and taking into account that \(9\pi ^{2}/[8(n+1)^{2}]\le 2\pi /(n+1)\) for all \(n\in \mathbb {N}\), we easily get \((3.2)\), which completes the proof.

4 Comparison with the Hermite-Fejér Polynomials

Firstly we present a brief history on the order in approximation by the Hermite-Fejér polynomials, \(H_{2n+1}(f)(x)\). Denoting \(A_{n+1}(f)=\| H_{2n+1}-f\| \), where \(\| \cdot \| \) is the uniform norm on \(C[-1,1]\), a famous result of Fejér [ 7 ] states that \(lim_{n\to \infty }A_{n+1}(f)=0\), for all \(f\in C[-1, 1]\). The first estimate of the rate of convergence, \(A_{n+1}(f)=O\left(\omega _{1}\left(f ; \tfrac {1}{\sqrt{n+1}}\right)\right)\), obtained by T. Popoviciu [ 12 ] , was improved by E. Moldovan to \(A_{n+1}(f)=O\left(\omega _{1}\left(f ; \tfrac {ln (n+1)}{n+1}\right)\right)\) in [ 11 ] , where \(ln(n)\) denotes the logarithm of \(n\). In Xie Hua Sun and Dechang Jiang [ 17 ] , it was proved that above, \(\omega _{1}\) can be replaced by the Diztzian-Totik modulus \(\omega _{1}^{\varphi }\).

In a sense, the two previous results are the best possible, because for \(g(x)=|x|\) we have \(|H_{2n+1}(g)(0)-g(0)|\ge c_{1}\tfrac {ln(n+1)}{n+1}, n\in \mathbb {N}\), with \(c_{1}{\gt}0\) independent of \(n\). In fact, by R. Bojanic [ 6 ] , if \(f\in Lip_{1}\) then the order \(O(ln(n+1)/(n+1))\) cannot be improved.

On the other hand, as it was remarked in the book of J. Szabados and P. Vértesi [ 14 ] , p. 168, Theorem 5.1, the above order is not the best possible for \(g(x)=|x|^{\delta }\), \(0{\lt}\delta {\lt}1\), the correct estimate being of order \(1/n^{\delta }\). This remark also follows from the equivalence proved by Theorem 2.3 in [ 17 ] ,

Other good estimates were obtained, for example, in R. Bojanic [ 6 ] for the uniform approximation, and in J. Prasad [ 13 ] , which generalizes the estimate of P. Vértesi in [ 16 ] for the pointwise approximation. Also, the saturation order \(\tfrac {1}{n}\), was proved by J. Szabados in [ 15 ] .

Now, from Theorem 3.1, we easily get that the order of approximation obtained by the max-product interpolation operator \(H_{2n+1}^{(M)}(f)(x)\) for the positive function \(f\in Lip_{1}[-1, 1]\), is essentially better than that given by the Hermite-Fejér interpolation polynomials, \(H_{2n+1}(f)(x)\). Indeed, in this case by Theorem 3.1 we get that \(\| H_{2n+1}^{(M)}(f)-f\| \le \tfrac {c}{n+1}\), while by [ 6 ] we have \(\| H_{2n+1}(f)-f\| \sim \tfrac {ln(n+1)}{n+1}\). Here \(a_{n}\sim b_{n}\) means that there exists \(c_{1}, c_{2} {\gt}0 \) independent of \(n\), such that \(c_{1}b_{n}\le a_{n}\le c_{2}b_{n}\) for all \(n\in \mathbb {N}\).

Finally, let us mention that in Hermann-Vértesi [ 9 ] , some linear interpolatory rational operators are constructed, for which a Jackson-type order of approximation is obtained and, in addition, a saturation result is obtained. It remains an open question to prove a saturation result for the nonlinear max-product Hermite-Fejér operator in the present paper, possibly by using some ideas in [ 9 ] .