Convergence and norm estimates of Hermite interpolation at zeros of Chevyshev polynomials

In this paper, we investigate the simultaneous approximation of a function f(x) and its derivative \(f^{\prime }(x)\) by Hermite interpolation operator \(H_{2n+1}(f;x)\) based on Chevyshev polynomials. We also establish general theorem on extreme points for Hermite interpolation operator. Some results are considered to be an improvement over those obtained in Al-Khaled and Khalil (Numer Funct Anal Optim 21(5–6): 579–588, 2000), while others agrees with Pottinger’s results (Pottinger in Z Agnew Math Mech 56: T310–T311, 1976).

The main purpose of this paper is to investigate the convergence of Hermite interpolating processes based on the zeros of Chevyshev polynomials. Let \(I=[-1,1]\) and \(C^p(I)\) be the space consisting of the p-times continuously differentiable real-valued function on I. For a given function \(f\in C^1(I)\), define its norm by

where \(\Vert f\Vert _{1}\) denotes the Chevyshev norm on I. For a given nodal matrix \(M=\{x_{0}^{(n)},\ldots ,x_{n}^{(n)}\}\), the Hermite interpolation operator \(H_{2n+1}(f;x)\) is defined to be the unique polynomial of degree at most \(2n+1\) satisfying the conditions: \(H_{2n+1}\left( f;x_{i}^{(n)}\right) =f\left( x_{i}^{n}\right) \), and \(H^{\prime }_{2n+1}\left( f;x_{i}^{(n)}\right) =f^{\prime }\left( x_{i}^{(n)}\right) , \; i=0,1,2,\ldots ,n\). In what follows, all the functions, weight functions, and spaces of functions under consideration are defined in the interval I. Given a positive integer n, and a function f, we denote by \(E_{n}(f)\) the error of best uniform approximation of f by algebraic polynomials of degree at most n. Pottinger (1976) proved that when the nodal matrix M consists of the Chevyshev nodes, the convergence

holds for \(f\in C^2(I)\). There is an extensive literature on Hermite interpolation (see, for example Al-Khaled 2000; Agarwal and Wong 1991; Sababhah et al. 2003, and the references therein) which has been studied from various points of view. Al-Khaled and Khalil (2000) studied the Hermite type interpolation, and gave some estimates for norms of certain interpolation operators on the space of continuously differentiable functions on I. The work in Varma and Prasad (1989) deals with mean convergence of \(H_{2n+1}(f;x)\) in the special case of interpolation based on zeros of the Chevyshev polynomials. It is proved that \(H_{2n+1}(f;x)\) converges in weighted \(L^{p}\) norms to \(f\in C^1\) at the rate of \(E_{2n+1}(f;x)/n\). The Hermite–Fejer interpolating polynomial \(F_{2n+1}(f;x)\) is a polynomial of degree at most \(2n+1\) that agrees with f at the interpolating points, and whose first derivative vanishes there. It is closely related to Hermite interpolating polynomial \(H_{2n+1}(f;x)\). Weighted mean convergence of \(F_{2n+1}(f;x)\) was studied in details in Nevai and Vertesi (1989), and Nevai and Vertesi (1976). In Nevai and Yuan (1994), the authors consider algebraic polynomials of degree at most \(2n-1\) for Hermite interpolation, more precisely, polynomials which interpolate a given function and its derivative at the n zeros of the so-called generalized Chevyshev polynomials. The convergence of the polynomials and their first derivatives is studied for weighted \(L^p\)-norms. Some applications to the Hermite interpolation problem on \([-1,1]\) are illustrated in Berriochoa et al. (2015). In Agarwal (1991), the authors give results for the weighted \(L^p\)-convergence of derivatives of extended Hermite interpolation on the zeros of Chevyshev polynomials plus additional points. Finally, results on convergence and norm estimation of Hermite and other type of interpolation processes in one or several variables are given in Agarwal and Wong (1993), Costabile and DellAccio (2007), and Mastroianni and Milovanovi (2008), and more recently in the framework of scattered data interpolation (Caira et al. 2012; Costabile et al. 2012, 2013; DellAccio and Tommaso 2016). The aim of this paper is to prove results analogous to (2) using different node system and involving endpoints of the interval too. We shall prove that when the nodal matrix M consists of the mixed Chevyshev nodes, the formula (2) also holds for \(f\in C^2[-1,1]\). In the last section, we establish a general theorem on extreme points for Hermite operator that agrees with a known result by Pottinger (1976).

Let \(u_{n}(x)=\frac{\sin ((2n+1)/2)\theta }{\sin \theta /2}\), where \(x=\cos \theta \), be the Chevyshev polynomials. We call the zeros of \(w(x)=(1-x^2)u_{n}(x): x_{k}^{(n)}=\cos \theta _{k}^{(n)}, \;\theta _{k}^{(n)}=k\pi /n, \; k=0,1,2,\ldots \), the nodes of the Chevyshev polynomials (see, Rivlin 1974). It is clear that

and

Passing a direct calculations, we obtain that

For the sake of convenience, hereafter we replace \(x_{k}^{(n)}, \theta _{k}^{(n)}\) by \(x_{k}, \theta _{k}\). For \(f(x) \in C^1(I)\), the Hermite interpolation operators based on the Chevyshev nodes are

where

Using (5)–(8) we can get

Now, by (10), (11), we have

In view of \(|\sin n \theta |\le n |\sin \theta |\), it is clear that

Again by a known result: \(\sum\nolimits_{k=1}^{n-1} \ell _{k}^2(x) \le 2\), Rivlin (1974) we get that

On the other hand, from (3), (6) and \( |\sin n \theta |\le n |\sin \theta |\), we have

Before proving our main results, the following elementary inequalities will be useful:

Lemma 1

The following estimates hold for \(a_{k}=\frac{1}{2\left( (n+1)-(n+2)\sin ^2 \theta _{k}/2\right) }=\frac{1}{n+(n+2)\cos \theta _{k}}\)

1. 1.

   \(|a_{k}|\le \frac{1}{n}, \; 1 \le k \le [n/2], or\;\; 0 \le \theta _{k} \le \pi /2\)

2. 2.

   \(|a_{k}| \le \left| \frac{1}{n-(n+2)n/k}\right| ,\;\; [n/2]+1\le k \le n-1, \;or\; \pi /2<\theta _k<\pi .\)

Also we shall require.

Lemma 2

The following inequality hold true for all \(n \ge 1.\)

The first result in this paper is

Theorem 1

For \(f(x) \in C^1[-1,1],\) the estimation \(| H_{2n+1}(f;x)| =\mathcal{O}(n^2)\) holds true as \(n \rightarrow \infty .\)

Proof

By (1), (9), it follows that

To estimate \(J_{1}\), by (10) and (11), it follows that

We shall proceed to estimate each of these terms: For \(J_{1}^{(1)}\), by (14), we have

For \(J_{1}^{(4)}\), we have

For \(J_{1}^{(2)}\), we have

where in the last inequality we used \(|w(x) \le 2\). Now using the first part of Lemma 1, and \(\sin ^2\theta _{k}/2=\sin ^2\theta _{k}/(4\cos ^2 \theta _{k}/2)\), we obtain

Using the Jordan inequality: \(\sin \theta \ge 2\theta /\pi , \; 0 \le \theta \le \pi /2\), we have

For \(J_{1}^{(3)}\), we have

Now by the second part of Lemma 1, and \(\sin ^2\theta _{k}/2=\sin ^2 \theta _{k}/\left( 4 \cos ^2\theta _{k}/2\right) \) we obtain

Since \(\sin \theta _{k}=\sin \theta _{k-n}\), we can apply Jordan inequality again, so that

But,

Therefore,

Combining the estimates of \(J_{1}^{(1)},J_{1}^{(2)},J_{1}^{(3)}\) and \(J_{1}^{(4)}\), there is a constant \(K_{1}\) such that

To estimate \(J_{2}\), by (13), (15) and Lemma 2, we have

Because

and \(\sin \theta _{n-k}=\sin \theta _{k}\), using the Jordan inequality: \(\sin \theta \ge 2\theta /\pi \) for \(0 \le \theta \le \pi /2\), we obtain

Therefore,

Again, by (13), we have \(|(x-1)\ell _{0}^2(x)|+|(x+1)\ell _{n}^2(x)|\le |x-1|+|x+1|=2, \; (-1<x<1)\). Therefore,

From this and (17), we obtain the desired result. \(\square \)

Theorem 2

For \(f(x) \in C^1[-1,1]\) , the estimation \(|H^{\prime }_{2n+1}(f;x)| =\mathcal{O}(n^2)\) holds true as \(n\rightarrow \infty \).

Proof

By (9) and \(\frac{\hbox {d}}{\hbox {d}x}\sum _{k=0}^n (v_{k}(x) \ell _{k}^2(x))=0\), we have

Since \(|f(x)-f(x_{k})|\le \max _{|x|\le 1}|f^{\prime }(x)||x-x_{k}|\), the above formula implies that

For \(I_{1}\), by (12), we know that

Again by (13), we have

Similarly, we can show that \(I_{1}^{(2)}\le 4n^2+4n+2.\) For \(I_{1}^{(3)}\), with the aid of the first part of Lemma 1, we have

For \(I_{1}^{(4)}\), with the aid of the second part of Lemma 1, we have

Since \(\sin \theta _{k}=\sin \theta _{k-n}\), we can apply Jordan inequality to obtain

Summarizing these estimates for \(I_{1}\), we get \(I_{1}\le \mathcal{O}(n^2), (n \rightarrow \infty )\). For \(I_{2}\), using the fact that \(I_{2} =\sum _{k=1}^{n-1} \ell _{k}^2(x) \le 2\) and (2.12) we get \(I_{2} \le 4\). For \(I_{3}\),

Note that \(|a_{0} w^{\prime }(x)|\le 1\), so that

and

For \(I_{3}^{(3)}\), we have,

Similar to the estimate in \(I_{3}^{(2)}\), we can show that \(I_{3}^{(4)}=|v_{n}(x) \ell _{n}^2(x)|\le 4n^2+4n+3.\) For \(I_{3}^{(5)}\): By using \(|w^{\prime }(x)|\le 2(2n+1)\) and part one of Lemma 1, we obtain

For \(I_{3}^{(6)}\), we have

Since \(\sin {\theta _{k}}=\sin \theta _{k-n}\), we can apply Jordan inequality to obtain

It is easy to show that \(I_{3}^{(7)}\le 2+ \frac{\pi ^2\sqrt{2}}{3}(2n^2+7n+4)\) and \(I_{3}^{(8)}\le 2 +K_{2}(2n^2+7n+4)\) for some constant \(K_{2}\). Combining all the estimates for \(I_{3}^{(1)}\) till \(I_{3}^{(8)}\), we observe that \(\parallel I_{3} \parallel ={{\mathcal {O}}}(n^2)\). For \(I_{4}\):

In view of \(\sum _{k=0}^n \ell _{k}^2(x) \le 4\); and using Lemma 2, we have

Therefore,

Combining all the estimates of \(I_{1}, I_{2}, I_{3}, I_{4}\), we complete the proof of Theorem 2. \(\square \)

The norm estimates obtained in the previous section are used to prove convergence of the interpolation functions in the \(C^2\)-norm, and that error estimates are given in terms of the modulus of continuity of the second derivative. Before we state our theorem, we define \(E_{n}^1(f)=\inf _{p\in H_{n}} \Vert f-p\Vert _{1}\) where \({{\mathcal {P}}}_{n}\) is the space of all polynomials of degree \(\le n\). (this definition is stated in Pottinger 1978, p. 272).

Theorem 3

If \(f(x) \in C^2[-1,1]\) and \(\omega (f^{(2)};\delta )\) is the modulus of continuity of \(f^{(2)}\) , then

Hereafter \({\bar{M}}\) denote different constants which are independent of n.

Proof

Since Hermite interpolation is a projection operator, we have the standard inequality

where the term \(E_{2n+1}(f)=\min \{\Vert f-p \Vert _{1} \; | \; p \in \mathcal{P}_{n+1}(f)\}\) and \(H_{k}\) is the space of all polynomials of degree at most k. Using Theorems 1 and 2, we have

here \(E^1_{n}(f)=\inf _{p\in H_{n}}\parallel f-p\parallel _{1}\). By (1.1) and the formula \(E_{n}^2(f)=E_{n-2}(f^{(2)})\) (see, Pottinger 1978, p. 272),where \(E_{n-2}(f^{(2)})\) is the best approximation to \(f^{(2)}\), we have

In view of \(f\in C^2[-1,1]\), using Jackson’s theorem (Natanson 1965, p. 86), we have

which completes the proof of Theorem 3. \(\square \)

Let f be a continuously differentiable function on \(I=[-1,1]\). Let \(C^p(I)\) be the space consisting of the \(p-\)times continuously differentiable real-valued functions on I. For \(f \in C(I)\), define \(\Vert f\Vert _{\infty }=\sup _{x\in I}|f(x)|.\) Set \(X=C(I) \times C(I)\) with the usual operations

• \((f_{1},g_{1})+(f_{2},g_{2})=(f_{1}+g_{1},f_{2}+g_{2})\), for all \(f_{i}, g_{i}\in C(I), i=1,2.\)

• \(\lambda (f,g)=(\lambda f,\lambda g)\), where \(\lambda \in \) field, and \(f, g\in C(I)\)

Then X is a vector space. For \((f,g) \in X\), one can define different types of norms. We are interested in two norms:

It is known that X is a Banach space under such norms. Let \({{\mathcal {P}}}_{2n+1}\) denote the space of all polynomials of degree less than or equal to \(2n+1\). Now define the operator

where \(x_{i}\) are some nodes on I, and \(v_{i}(x), \ell _{i}(x)\) are functions associated with some weight function w(x) corresponding to the nodes \(x_{i}'s\). Let

and

where, \(w(x)=\prod _{i=0}^n (x_{i}-x_{j}).\) The operator \(H_{n}\) is the same as the Hermite interpolation operator if \(g=f^{\prime }\) in Eq. (21). Al-Khaled and Khalil (2000), also Pottinger (1976), they studied the Hermite type interpolation and gave some estimates for norms of certain interpolation operators on the space of continuously differentiable functions on I. In this section, we study the Hermite interpolation operator given by Eq. (21), and establish general theorem on extreme points of some interpolation operators. The following known result will be needed in this paper.

Lemma 3

Let \(\{x_{1}, x_{2},\ldots ,x_{k}\} \subset I\) be pairwise distinct real numbers and let \(y_{1}, y_{2},\ldots ,y_{k} \subset R\) be arbitrary real numbers. Then there exists \(f\in C(I)\) such that \(f(x_{i})=y_{i}\).

Proof

The points \((x_{i}, y_{i})\) can be connected, in order with their indices, by a polygonal chain, which is the graph of a continuous function provided that \(x_{1}<x_{2}<\cdots <x_{n}.\) \(\square \)

Note that \(H_{n}\) defined as in Eq. (21) is a finite rank operator. Further:

where \({{\mathcal {B}}}(X)\) is the unit ball of X under the norm (19). Now \((f(x_{0}),f(x_{1}),\ldots ,f(x_{n}))\in {\mathbf{R}}^{n+1}\), where \({\mathbf{R}}^{n+1}=\{(\alpha _{0},\alpha _{1},\ldots ,\alpha _{n}): \alpha _{i} \in {\mathbf{R}} \}\), and for \(z\in {\mathbf{R}} ^{n+1}\) we have \(\Vert z\Vert _{\infty }=\sup _{0 \le i \le n}|z_{i}|\). Similarly for \((g(x_{0}),g(x_{1}),\ldots ,g(x_{n}))\). By Lemma 3, we have

Lemma 4

\(\{f(x_{i}): f \in C(I) \}={\mathbf{R}}^{n+1}\) , where equality is isometry equality.

By Lemma 3, we obtain

where the supremum is taken over all \((a_{i}), (b_{i})\) in the unit ball of \({\mathbf{R}}^{n+1}\). Hence we can look at \(H_{n}\) as an operator

Let \(Y={\mathbf{R}}^{n+1}\times {\mathbf{R}}^{n+1}\), then Y is a finite dimensional normed space. Hence \(H_{n}\) attains its norm at some point in \({{\mathcal {B}}}(Y)\). Again \({{\mathcal {P}}}_{2n+1}\) is finite dimensional, thus \({{\mathcal {P}}}_{2n+1}\) is reflexive, so \(\mathcal{P}_{2n+1}=\left[ ({{\mathcal {P}}}_{2n+1})^{*}\right] ^{*}\). Let \(G=\mathcal{P}_{2n+1}^{*}\). From the theory of tensor products of Banach spaces (see, Khalil 1990), we have

Taking \(E=Y, F={{\mathcal {P}}}_{2n+1}^{*}\), we get

Thus, we can look to \(H_{n}\) as a linear functional in \((Y{\hat{\otimes }} G)\). But a bounded linear functional on a Banach space, if it attains its norm, it is attains at some extreme point (see, for example Khalil 1990). Consequently:

Lemma 5

\(H_{n}\) attains its norm at some extreme point of \({\mathbf{R}}^{n+1}\times {\mathbf{R}}^{n+1}.\)

Proof

Follows from the fact that \(Ext(E{\hat{\otimes }} F)=Ext(E){\otimes }Ext(F).\) Now, if E and F are Banach spaces, and \(W=E \times F\), with the \(\infty -\)norm. Then \((x,y) \in W\) is extreme if and only if \(\Vert x \Vert =\Vert y\Vert =1\), and \(x \in Ext(E), y\in Ext(F)\). But extreme points of \({\mathbf{R}}^{n+1}\) are those of the form \((x_{i})=\mp 1\) for all i. Hence \(\square \)

Lemma 6

\(\Vert H_{n}\Vert =\sup \Vert \sum _{i=0}^n v_{i}(x)\ell _{i}^2(x) a_{i}+\sum _{i=0}^n (x-x_{i})\ell _{i}^2(x)b_{i}\Vert \) where the supremum is taken over all \(a_{i}=\mp 1\) and \(b_{i}=\mp 1\) , for all i.

Also, since \(v_{i}(x) \ge 0\) and \(\ell _{i}^2(x)>0\), we get

Theorem 4

Thus, to estimate (or calculate) \(\parallel H_{n}\parallel \) we have to work with (22). A simple consequence of equation (22) is:

Theorem 5

\(\Vert H_{n}\Vert \le 5.\)

Proof

Since \(\sum _{i=0}^n v_{i}(x) \ell _{i}^2(x)=1\) and \(|x-x_{i}|\ell _{i}^2(x) \le 2 \ell _{i}^2(x)\), using the fact that \(\sum _{i=0}^n \ell _{i}^2(x) \le 2\), the result follows, which agrees with Pottinger’s estimate (Pottinger 1976). \(\square \)

The fundamental goal of this paper is to prove convergence of the interpolation functions in the \(C^2\)-norm, where the error estimates are given in terms of the modulus of continuity of the second derivative. Some results agrees with Pottinger’s estimate (Pottinger 1976), and some are considered to be an improvement over those obtained in Al-Khaled and Khalil (2000). We also establish a general theorem on extreme points for Hermite operator.

KA carried out the calculations in the manuscript, and planned, guiding project development and manuscript revision. MA carried out the retrospective review, manuscript design, and also completed the final draft of the manuscript. Both authors gave their final consent of the published version and agreed to be accountable for all aspects of the work. Both authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the reviewers for their careful and meticulous reading of the paper. The authors also are gratefully acknowledges the valuable comments and suggestions on the last section made by Professor Roshdi Khalil.

Competing interests

The authors declare that they have no competing interests.

Affiliations

1. Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan

   • Kamel Al-Khaled
   •  & Marwan Alquran

Corresponding author

Correspondence to Kamel Al-Khaled.

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