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A note on the rate of convergence for Chebyshev-Lobatto and Radau systems

Elías Berriochoa
/ Alicia Cachafeiro
• Corresponding author
• Departamento de Matemática Aplicada I, E. Ingeniería Industrial, Universidad de Vigo, 36310 Vigo, Spain
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Jaime Díaz
/ Eduardo Martínez
Published Online: 2016-03-29 | DOI: https://doi.org/10.1515/math-2016-0015

Abstract

This paper is devoted to Hermite interpolation with Chebyshev-Lobatto and Chebyshev-Radau nodal points. The aim of this piece of work is to establish the rate of convergence for some types of smooth functions. Although the rate of convergence is similar to that of Lagrange interpolation, taking into account the asymptotic constants that we obtain, the use of this method is justified and it is very suitable when we dispose of the appropriate information.

MSC 2010: 41A05; 65D05; 41A25; 65M60

1 Introduction

The nodal systems related to the Jacobi polynomials play an important role in the theory of Hermite interpolation on the bounded interval, (see [13]). The Hungarian interpolatory school, beginning with Fejér, has used these systems for Lagrange and Hermite interpolation. Szegő, who is one of the more important references in these subjects, proves for Hermite interpolation that the generalized step polynomials converge to continuous functions uniformly on [-1+ε,1-ε], for every ε > 0, when the nodes are the zeros of the Jacobi polynomials, with parameters α and β. Moreover, if α ≥ 0 and the function is merely continuous in [-1,1], then the step polynomials are in general divergent at x = 1, and a similar result holds for β ≥ 0 and x = −1, (see [4]). P. Szász improves these results of convergence by adding the endpoints to the nodal system and by using them as Lagrange data points, (see [5, 6]). The most important, among these nodal systems, is that corresponding to the Chebyshev polynomials of the second kind joined with the endpoints ±1. This set of points is usually called Chebyshev-Lobatto nodal system. Other useful systems are the so called Chebyshev-Radau nodal systems, which correspond to the zeros of the Chebyshev polynomials of the third and fourth kind joined with the points −1 and 1, respectively. Since the derivative at the endpoints is not prescribed, this approach improves the results of convergence but it does not solve a proper Hermite interpolation problem. Nevertheless, Hermite and Hermite-Fejér interpolation problems with extended nodal systems are interesting problems that have been subject of study for several researchers, obtaining algorithms for computing the interpolation polynomials and results of convergence. Indeed, barycentric formulas presented in [7] were improved in [8] and barycentric methods for more general Hermite interpolation problems can be seen in [9]. The convergence of the Hermite-Fejér process has been proved for continuous functions using the Chebyshev-Lobatto nodal systems and the rate of convergence was obtained in terms of the modulus of continuity, (see [10, 11]). The study of the convergence of the Hermite interpolants for continuous functions using Chebyshev-Lobatto and Chebyshev-Radau nodal systems can be seen in [8]. The technique used in [8] is based on the idea to pass the problem to the unit circle by the Szegő transformation $x=\frac{z+\overline{z}}{2},$ to apply the convergence result of Hermite-Fejér interpolation for continuous functions on the circle given in [12], and then to recover the convergence results for the interpolants on the interval [-1, 1].

Hermite interpolation problems have also been studied with more general nodal systems such as normal and strongly normal point systems, that were introduced by G. Grünwald. The zeros of certain Jacobi polynomials satisfy this last condition. Other important sets of zeros of orthogonal polynomial that were used as nodal points are those corresponding to Legendre and ultraspherical polynomials. In the case of unbounded intervals, some results about convergence of interpolation polynomials were obtained by using as nodes the zeros of orthogonal polynomials with respect to the weights of Hermite, Laguerre, Sonin-Markov and Freud-type.

This note attempts to complete the Hermite interpolation theory with Chebyshev-Lobatto and Chebyshev-Radau systems and in order to extend the results in [8] to another more wide class of functions, we use a new technique in this paper. First we obtain a new representation for the Hermite interpolation polynomials related to the Chebyshev

polynomials of the first kind. As a consequence, we present some results on the rate of convergence for these extended interpolants when applied to some types of smooth functions. Although the rate of convergence is similar to that of Lagrange interpolation, taking into account the asymptotic constants that we obtain; the use of this method is justified when we have more information in the problem to be solved, that is, if we know the values of the derivatives on the nodal points. Really we have proved that with 2ninterpolation conditions the rate of convergence is $\mathcal{O}\left(\frac{1}{{\left(2n\right)}^{s-1}}\right),$ while by using Lagrange interpolation with n interpolation conditions the rate of convergence is $\mathcal{O}\left(\frac{1}{{n}^{s-1}}\right).$ In both cases, s is a parameter related to the smoothness of the coefficients of functions represented by Chebyshev series. Hence, when we dispose of the appropriate information the use of this method is very suitable. For example, in the numerical solution of differential equations, if the values of the solution and its derivative in these nodal points are known, this type of interpolation could be applied to rebuild the solution.

2 Chebyshev-Lobatto Hermite interpolation. Rate of convergence for smooth functions

Let us consider the nodal system ${\left\{{x}_{j}\right\}}_{j=0}^{n}={\left\{\mathrm{cos}\frac{\pi j}{n}\right\}}_{j=0}^{n}$ that is, x0 = 1, x1, …, xn−1 the zeros of the Chebyshev polynomial of the second kind Un-1(x) and xn = −1. This nodal system is named Chebyshev-Lobatto system and the nodal polynomial is $Nn+1(x)=Un−1(x)(1−x2).$(1)

Now our aim is to obtain results on the rate of convergence when we interpolate some types of smooth functions. So if f is a differentiable function defined on [-1,1] we denote by ℒ2n+1(f, x) a polynomial in the space ℙ2n+1 satisfying $ℒ2n+1(f,xj)=f(xj), ℒ′2n+1(f,xj)=f′(xj) for j=0,⋯,n.$(2)

To reach our goal we use some well known results on the Chebyshev polynomials of the first kind {Tn} and the second kind {Un} that can be seen in [13, 14].

First we examine the auxiliary polynomials, closely related with the nodal system, and defined by ${E}_{0}\left(x\right)=\frac{{\left({N}_{n+1}\left(x\right)\right)}^{2}}{4{n}^{2}\left(x-1\right)}$ and ${E}_{n}\text{\hspace{0.17em}}\left(x\right)=\frac{{\left({N}_{n+1}\left(x\right)\right)}^{2}}{4{n}^{2}\left(x+1\right)}$

The polynomials E0, En ϵ ℙ2n+1 satisfy the following properties:

• (i)

E0(xj) = 0 for j = 0; …, n.

• (ii)

${E}_{0}^{\text{'}}\left(1\right)\text{\hspace{0.17em}\hspace{0.17em}}={E}_{0}^{\text{'}}\left({x}_{j}\right)\text{\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}0$ for j = 1; …, n.

• (iii)

${E}_{n}^{\text{'}}\left(-1\right)=\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}{E}_{n}^{\text{'}}\left({x}_{j}\right)\text{\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}0$ for j = 0; …, n-1.

• (iv)

If x ∈ [-1, 1] then |E(x)|, $|{E}_{n}\left(x\right)|\le \frac{1}{2{n}^{2}}.$

(i) - (iii) can be seen in [8].

(iv) If we take into account the definition for E0 we have $E0(x)=(Un−1(x)(1−x2))24n2(x−1)=(sin(n arccos x)sin(arccos x)(1−x2))24n2(x−1)=(x+1)(sin(n arccos x))24n2,$

from which it follows (iv). One can obtain the same bound for En proceeding in a similar way.□

The following result establishes a new representation for the interpolation polynomial corresponding to the Chebyshev polynomial of the first kind Th, when h ≥ 2n.

Let h be a natural number, h = 2n(l+1)+k, with l and k nonnegative integers, n a positive integer and 0≤k≤2n-1.Then $ℒ2n+1(Th,x)=akTk(x)+b2n−kT2n−k(x)+c0E0(x)+dnEn(x),$(3)

where:

• (i)

If k ≠ 0 then ak = 2 + ℓ, b2n-k = -1-ℓ, c0 = 4(2n2 +3n2ℓ+n22) and dn = (-1)k-1 4(2n2+3n2ℓ+n22)

• (ii)

If k = 0 then a0 = -2ℓ-ℓ2, b2n = (ℓ+1)2, c0 = 0 and dn = 0.

Actually the coefficients depend on h but we omit it in the notation for the sake of simplicity.

Taking into account that both expressions in the representation (3) belong to ℙ2n+1, we only have to prove that the expression for ℒ2n+1 (Th, x) given in (3) fulfills the corresponding interpolation conditions.

• (i)

Let k ≠ 0. Now the interpolation conditions for ℒ2n+1 (Th, x) are: $Th(xj)=cos⁡(harccosxj)=cosπjkn,for0≤j≤n,Th′(xj)=hsin⁡(harccosxj)sinarccosxj=hsinπjkn1−xj2,for1≤j≤n−1,Th′(1)=h2andTh′(1)=(−1)h−1h2=(−1)k−1h2.$

On the other hand the next relations hold.

For j ∈ {0,…,n} $akTk(xj)+b2n−kT2n−k(xj)+c0E0(xj)+dnEn(xj)=akcos(kπjn)+b2n−kcos((2n−k)πjn)=(ak+b2n−k)cos(πjkn)=cos(πjkn).$

For j ∈ {1,…, n-1} $akT′k(xj)+b2n−kT′2n−k(xj)+c0E′0(xj)+dnE′n(xj)=akksin (kπjn)1−xj2+b2n−k(2n−k)sin((2n−k)πjn)1−xj2=sin (πjkn)1−xj2(kak−b2n−k(2n−k))=hsin(πjkn)1−xj2.$

For j = 0 and j = n it holds $akT′k(1)+b2n−kT′2n−k(1)+c0E′0(1)+dnE′n(1)=akk2+b2n−k(2n−k)2+c0=h2,akT′k(−1)+b2n−kT′2n−k(−1)+c0E′0(−1)+dnE′n(−1)=akk2+b2n−k(2n−k)2+dn=(−1)k−1h2.$

Thus for k ≠ 0 equality (3) is proved.

• (ii)

When k = 0 we have a similar situation. On the one hand we have the interpolation conditions: $Th(xj)=cos⁡(harccosxj)=cos⁡((2+2ℓ)πj)=1,for0≤j≤n,T′h(xj)=hsin⁡(harccosxj)sin(arccosxj)=hsin(2n+2ℓn)πjnsin⁡(arccosxj)=0,for1≤j≤n−1,Th′(1)=h2andTh′(−1)h−1h2=−h2.$

On the other hand the following relations hold.

For j ∈ {0,…,n} $a0T0(xj)+b2nT2n(xj)=a0+b2ncos(2nπjn)=a0+b2n=1.$

For j ∈ {1,…,n-1} $a0T′0(xj)+b2nT′2n(xj)=b2n2nsin(2n arccos xj)sin arccos xj=0.$

For j = 0and for j = nit holds $a0T′0(1)+b2n−kT′2n−k(1)=b2n(2n)2=h2,a0T′0(−1)+b2nT′2n(−1)=−b2n(2n)2=−h2.$

So for k = 0 the statement is also proved. □

Notice that the preceding representation is valid for h ≥ 2n and for h = 2n+1 it gives an alternative representation of the polynomial T2n+1(x)

Let h be a natural number, h = 2n(𝓵 + 1) + k, with 𝓵 and k nonnegative integers, n a positive integer and 0 ≤ k ≤ 2n − 1. Then

• (i)

If k ≠ 0 it holds that$|{\mathcal{L}}_{2n+1}\left({T}_{h},x\right)|\le 4{\ell }^{2}+14\ell +11,\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall }x\in \left[-1,\phantom{\rule{thinmathspace}{0ex}}1\right].$

• (ii)

If k =0 it holds that$|{\mathcal{L}}_{2n+1}\left({T}_{h},x\right)|\le 2{\ell }^{2}+4\ell +1,\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall }x\in \left[-1,\phantom{\rule{thinmathspace}{0ex}}1\right].$

(i) and (ii) are straightforward consequences of the previous representations. □

Now we are in a position to study the rate of convergence of the interpolation polynomials for some kind of smooth functions, (see [15]).

Let f be a function defined on.[-1; 1] by $f\left(x\right)=\sum _{k=0}^{\infty }{a}_{k}{T}_{k}\left(x\right)\text{\hspace{0.17em}}with\text{\hspace{0.17em}}|{a}_{k}|\le K\frac{1}{{k}^{s}}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\ne 0$, s ≥ 4 and K a positive constant. Then ℒ 𝓛2n+1(f,.) uniformly converges to f on [-1; 1] with rate of convergence $\mathcal{O}\left(\frac{1}{{n}^{s-1}}\right)$.

We decompose f as follows f = f1, 2n+1 + f2, 2n+1 with ${f}_{1,2n+1}=\sum _{k=0}^{2n+1}{a}_{k}{T}_{k}$ and ${f}_{2,2n+1}=\sum _{k=2n+2}^{\infty }{a}_{k}{T}_{k}$. Since ℒ2n + 1(f1, 2n + 1,..) = f1, 2n+1, we study the behavior of ℒ2n + 1(f2, 2n+1,..) − f2, 2n+1.

If we denote by Hn, s the generalized harmonic number defined by ${H}_{n,\text{\hspace{0.17em}}s}=\sum _{k=1}^{n}\frac{1}{{k}^{s}}$ and by $H\left(s\right)=\sum _{k=1}^{\infty }\frac{1}{{k}^{s}}$ then $|f2, 2n+1(x)|≤∑k=2n+2∞|ak|≤∑k=2n+2∞k1ks=k(H(s)−H2n+1,s)≤k(s−1)(2n+1)s−1,$(4)

where the last inequality comes from the classical method of the integral for approximating the sum of the series.

Proceeding in a similar way we obtain $|ℒ2n+1(f2,2n+1,..)|≤∑k=2n+2∞|ah||ℒ2n+1(Th,.)|≤∑ℓ=0∞∑k=02n−1|a2n(ℓ+1)+k||ℒ2n+1(T2n(ℓ+1)+k,..)|≤∑ℓ=0∞∑k=12n−1|a2n(ℓ+1)+k||ℒ2n+1(T2n(ℓ+1)+k,..)|︸∗+∑ℓ=0∞|a2n(ℓ+1)||ℒ2n+1(T2n(ℓ+1),.)|︸∗∗$(5)

and taking into account the previous corollary we get $∗≤∑ℓ=0∞∑k=12n−1k(2n(ℓ+1)+k)s(4ℓ2+14ℓ+11)≤∑ℓ=0∞2nk(2n(ℓ+1))s(4ℓ2+14ℓ+11)≤ k(2n)s−1∑ℓ=0∞4ℓ2+14ℓ+11(ℓ+1)s≤11k(2n)s−1∑ℓ=0∞1(ℓ+1)s−2$(6)

and $∗∗≤ ∑ℓ=0∞k(2n(ℓ+1))s(2ℓ2+4ℓ+1)≤ k(2n)s∑ℓ=0∞2ℓ2+4ℓ+1(ℓ+1)s≤2k(2n)s∑ℓ=0∞1(ℓ+1)s−2.$(7)

Hence the interpolation error can be bounded as follows: $|f(x)−ℒ2n+1(f,x)| = |f2,2n+1(x)−ℒ2n+1(f2,2n+1,x)|≤|f2,2n+1(x)|+|ℒ2n+1(f2,2n+1,x)|$

and using (4), (5), (6) and (7), the result is obtained. □

It is well known that the Chebyshev-Fourier coefficients of functions in L2 converge to zero, and for smooth functions they behave like in Proposition 2.5. Indeed some kind of smooth functions satisfy the preceding requirements. For example, it is easy to conclude that a function with the third derivative of bounded variation on. [-1, 1] fulfills the hypothesis of Proposition 2.5. It can also be proved that functions s times continuously differentiable on. [-1, 1], with s ≥ 4, fulfill the hypothesis of the preceding Proposition. Another interesting question is that we can weaken the hypothesis on the parameter s asking only for s > 3. Moreover, for infinitely differentiable functions their Chebyshev-Fourier coefficients converge to zero geometrically, that is, exponentially with k.

The strategy used in the preceding proposition is different from the one used in [8]; hence by passing the results to the unit circle one can obtain similar results.

Next we study the case of analytic functions on [-1, 1].

If f is an analytic function on [-1, 1], then ℒ2n+1 (f,..)uniformly converges to f on [-1, 1] with a geometric rate of convergence.

Let f be an analytic function on [-1, 1]. Then f can be represented as $f\left(x\right)=\sum _{k=0}^{\infty }{a}_{k}{T}_{k}\left(x\right)$ and it holds that |aK| ≤ Krk for some K > 0 and 0 < r < 1, as can be seen in [4].

We decompose f as follows f = f1, 2n+1 + f2, 2n+1 with ${f}_{1,2n+1}=\sum _{k=0}^{2n+1}{a}_{k}{T}_{k}$ and ${f}_{2,2n+1}=\sum _{k=2n+2}^{\infty }{a}_{k}{T}_{k}$. Since ℒ2n(f1, 2n + 1,..) = f1, 2n+1 we study the behavior of |f2, 2n+1 − ℒ2n(f2, 2n + 1,..)|. Proceeding like in the previous proposition we get $|f2,2n+1|≤∑k=2n+2∞|ak|≤k1−rr2n+2,$(8)

and taking into account Corollary 2.4 it is clear that for h ≥ 2n it holds |ℒ2n(Th,..)| ≤ 4h2 + 14h + 11) = p2(n)r2n+2, Therefore $|ℒ2n+1(f2,2n+1,..)| ≤∑h=2n+2∞|ah||ℒ2n+1(Th,..)|≤∑h=2n+2∞krh(4h2+14h+11)=p2(n)r2n+2,$(9)

where p2(n) is a well determined polynomial of degree 2. Then by using (8) and (9) the result is proved. □

We want to point out that the results in Propositions 2.5 and 2.8 justify the practical use of these interpolants for smooth functions when we have information about the values of the function and its first derivative on the nodal points.

3.1 The case of Chebyshev polynomials of the fourth kind

Let us consider the nodal system ${\left\{{x}_{j}\right\}}_{j=0}^{n-1}={\left\{\mathrm{cos}\frac{2\pi j}{2n-1}\right\}}_{j=0}^{n-1}$ that is, x0 = 1 and x1, …, xn − 1 the zeros of the Chebyshev polynomial of the fourth kind Wn-1.(x) Then, the nodal polynomial is $Mn(x)=wn−1(x)(1−x).$

Our aim is to obtain results on the rate of convergence when we interpolate some types of smooth functions. So we consider f a differentiable function defined on [-1, 1] and we denote by 2n − 1.(f, x) the interpolation polynomial in the space ℙ2n − 1 characterized by satisfying the interpolation conditions $ℋ2n−1(f,xj)=f(xj),ℋ′2n−1(f,xj)=f′(xj) for j=0,⋯,n-1.$(10)

Next we examine some auxiliary polynomials, closely related with the nodal system.

The polynomial ${D}_{0}\left(x\right)=\frac{{\left({M}_{n}\left(x\right)\right)}^{2}}{{\left(2n-1\right)}^{2}\left(x-1\right)}\in {ℙ}_{2n-1}$ satisfies the following properties:

• (i)

D0(xj) = 0 for j = 0; …; n − 1:

• (ii)

${D}_{0}^{\text{'}}\left(1\right)=1,\text{\hspace{0.17em}}{D}_{0}^{\text{'}}\left({x}_{j}\right)=0$ for j = 1; …, n − 1.

• (iii)

If x 𝜖 [-1; 1] then it holds $|{D}_{0}\left(x\right)|\le \frac{4}{{\left(2n-1\right)}^{2}}$.

• (i)

It is immediate.

• (ii)

It can be seen in [8].

• (iii)

It is a straightforward consequence of the definition of D0 and Wn − 1, (see [13, 14]): $D0(x)=(Mn(x))2(2n−1)2(x−1)=(wn−1(x))2(x−1)(2n−1)2=2sin2(n12arccos x)(x−1)(2n−1)2.$

Next we obtain a new representation of the interpolation polynomials related to the Chebyshev polynomials of the first kind.

Let h be a natural number h =(2n-1)(+1)+k with n a positive integer, ℓ and k nonnegative integers and0 ≤ k < 2n-1. If Th(x) is the Chebyshev polynomial of degree h then ℋ2n-1(Th, x) can be represented as: $ℋ2n−1(Th,x)=akTk(x)+b2n−1−kT2n−1−k(x)+c0D0(x),$(11)

where:

• (i)

If k≠ 0 then ak = 2 + , b2n-1-k =−1−ℓ and c0 = (2 + 3ℓ+ℓ2)(2n-1)2.

• (ii)

If k = 0 then a0 = −2ℓ – ℓ2, b2n-1 =(ℓ+1)2 and c0 = 0.

Actually the coefficients depend on h but we omit it in the notation for the sake of simplicity.

Taking into account that both representations in (11) belong to ℙ2n-1, we only have to prove that 2n-1(Th, x) given in (11) fulfills the corresponding interpolations conditions.

• (i)

If k ≠0, on the one hand we have the following interpolation conditions: $Th(xj)=cos(h arccos xj)=cos(((2n−1)(l+1)+k)2πj2n−1)=cos(2πjk2n−1), for 0≤j≤nTh'(xj)=hsin(h arccos xj)sin arccos xj=hsin(((2n−1)(l+1)+k)2πj2n−1)sin arccos xj=hsin(2πjk2n−1)1−xj2, for 0≤j≤n−1T′h(x0)=T′h(1)=h2.$

On the other hand we have:

Forj ϵ {0,…., n }

For j ∈ {1,…, n} $akTk(xj)+b2n−1−kT2n−1−k(xj)+c0D0(xj)=akcos(k2πj2n−1)+b2n−1−kcos((2n−1−k)2πj2n−1)=(ak+b2n−1−k)cos(2πjk2n−1)=cos(2πjk2n−1).$

For j∈{1,…n−1} $akT′k(xj)+b2n−1−kT′2n−1−k(xj)+c0D′0(xj)=akksin(k arccos xj)sin arccos xj+b2n−1−k(2n−1−k)sin((2n−1−k)arccos xj)sin arccos xj=akksin(k2πj2n−1)1−xj2+b2n−1−k(2n−1−k)sin((2n−1−k)2πj2n−1)1−xj2=sin(2πjk2n−1)1−xj2(akk−b2n−1−k(2n−1−k))=hsin(2πjk2n−1)1−xj2.$

For j = 0 it holds $akT′k(x0)+b2n−1−kT′2n−1−k(x0)+c0D′0(x0)=akk2+b2n−1−k(2n−1−k)2+c0=h2.$

Hence, for k≠ 0, we have proved expression (11).

• (ii)

When k = 0 we have a similar situation. On the one hand we get: $Th(xj)=cos(h arccos xj)=cos((2n−1)(ℓ+1)2πj2n−1)=1, for 0≤j≤n,T′h(xj)=hsin(h arccos xj)sin arccos xj=hsin((2n−1)(ℓ+1)2πj2n−1)sinarccos xj=0, for 1≤j≤n−1,Th'(x0)=Th'(1)=h2.$

On the other hand the following relations hold.

For j ϵ {0,…, n} $a0T0(xj)+b2n−1T2n−1(xj)=a0+b2n−1cos((2n−1)arccos xj)=a0+b2n−1 cos((2n−1)2πj2n−1)=a0+b2n−1=1.$

For j ϵ {1,…, n–1} $a0T′0(xj)+b2n−1T′2n−1(xj)=b2n−1(2n−1)sin((2n−1)arccos xj)sin arccos xj=0.$

For j = 0 $a0T′0(1)+b2n−1T′2n−1(1)=b2n−1(2n−1)2=h2.$

So, for k = 0, the statement has also been proved.

Let h be h = (2n – 1)(ℓ + 1)+ k with n a positive integer, ℓ and k nonnegative integers and 0 ≤ k < 2n – 1. Then we have:

• (i)

If k ≠ 0 then 𝓗2n – 1(Th, x)| ≤ 4ℓ2 + 14ℓ + 11 ∀x ∊ [–1, 1].

• (ii)

If k = 0 then 𝓗2n – 1(Th, x)| ≤ 2ℓ2 + 4ℓ + 1 ∀x ∊ [–1, 1].

Both statements are straightforward consequences of the previous representations.

Now we are in a position for proving our main results concerning some kind of smooth functions.

Let f be a function defined for x ∈ [-1, 1] by $f\left(x\right)=\sum _{k=0}^{\infty }{a}_{k}{T}_{k}\left(x\right)$ with $|{a}_{k}|\le K\frac{1}{{k}^{s}}$ for k ≠ 0 and s ≥ 4. Then 𝓗2n–1(f,.) uniformly converges to f on [-1, 1] and the rate of convergence is $\mathcal{O}\left(\frac{1}{{n}^{s-1}}\right)$.

By using the same technique as in Proposition 2.5, we can write f = f1, 2n–1 + f2, 2n–1 with ${f}_{1,\text{\hspace{0.17em}}2n-1}=\sum _{k=0}^{2n-1}{a}_{k}{T}_{k}$ and ${f}_{2,\text{\hspace{0.17em}}2n-1}=\sum _{k=2n}^{\infty }{a}_{k}{T}_{k}$.

Since ℋ2n–1(f1,2n–1,.) = f1,2n–1, we study the behavior of ℋ2n–1(f2,2n–1,.) – f2,2n–1. If we denote by Hn, s the generalized harmonic number defined by ${H}_{n,\text{\hspace{0.17em}}s}=\sum _{k=1}^{n}\frac{1}{{k}^{s}}$ and $H\left(s\right)=\sum _{k=1}^{\infty }\frac{1}{{k}^{s}}$ then $|f2,2n−1(x)|≤∑k=2n∞|ak|≤∑k=2n∞k1ks=k(H(s)−H2n−1,s)≤k(s−1)(2n−1)s−1.$(12)

Proceeding in a similar way $H2n−1f2,2n−1,.≤∑h=2n∞ahH2n−1Th,.≤∑ℓ=0∞∑k=02n−2a2n−1+ℓ2n−1+kH2n−1T2n−1+ℓ2n−1+k,.≤∑ℓ=0∞∑k=12n−2a2n−1+ℓ2n−1+kH2n−1T2n−1+ℓ2n−1+k,.+⏟∗∑ℓ=0∞|a2n−1+ℓ(2n−1)||H2n−1(T2n−1+ℓ(2n−1).,)|⏟∗∗$(13)

and taking into account the previous corollary we obtain $∗≤∑ℓ=0∞∑k=12n−2k2n−1ℓ+1+ks4ℓ2+14ℓ+11≤∑ℓ=0∞2n−1k2n−1ℓ+1s4ℓ2+14ℓ+11≤k2n−1s−1∑ℓ=0∞4ℓ2+14ℓ+11ℓ+1s≤11k2n−1s−1∑ℓ=0∞1ℓ+1s−2$(14)

and $∗∗≤∑ℓ=0∞k2n−1ℓ+1s2ℓ2+4l+1≤k2n−1s∑ℓ=0∞2ℓ2+4ℓ+1ℓ+1s≤2k2n−1s∑ℓ=0∞1ℓ+1s−2.$(15)

Therefore, taking into account $|f(x)−ℋ2n−1(f,x)|=|f2,2n−1(x)−ℋ2n−1(f2,2n−1,x)|≤|f2,2n−1(x)|+|ℋ2n−1(f2,2n−1,x)|,$

if we use (12), (13), (14) and (15), the result is obtained.

Next we study the case of analytic functions on [-1, 1].

Let f be an analytic function on [-1, 1]. Then ℋ2n-1(f,.) uniformly converges to f on [-1, 1] with a geometric rate of convergence.

Proceeding like in Proposition 2.8, f can be represented as $f\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sum _{k=0}^{\infty }{a}_{k}{T}_{k}$, with |ak| ≤ Krk for some K > 0 and 0 < r < 1. We decompose f as follows f = f1,2n-1 + f2,2n-1 with ${f}_{1,2n-1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sum _{k=0}^{2n-1}{a}_{k}{T}_{k}$ and ${f}_{2,2n-1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sum _{k=2n}^{\infty }{a}_{k}{T}_{k}$. Since ℋ2n-1(f1,2n-1,.) = f1,2n-1 we study the behavior of |f2,2n-1 − 𝓗2n-1(f2,2n-1.,)|. Thus, we have $|f2,2n−1|=|∑k=2n∞akTk|≤∑k=2n∞|ak|≤k1−rr2n,$(16)

and $|ℋ2n−1(f2,2n−1,.)|=|∑k=2n∞akℋ2n−1(Tk,.)|≤∑k=2n∞|ak||ℋ2n−1(Tk,.)|≤ ∑k=2n∞krk(4k2+14k+11)=p2(n)r2n,$(17)

where p2(n)denotes a polynomial of degree 2.

Hence using (16) and (17) the result is proved.

3.2 The case of Chebyshev polynomials of the third kind

Let us consider the nodal system ${\left\{{x}_{j}\right\}}_{j=1}^{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left\{\mathrm{cos}\text{\hspace{0.17em}}\frac{\left(2j-1\right)\pi }{2n-1}\right\}}_{j=1}^{n}$, that is, x1,...,xn-1 are the zeros of the Chebyshev polynomial of the third kind Vn-1(x) and xn = -1. Then the nodal polynomial is Vn-1(x)(1+x).

If f is a differentiable function defined on [-1, 1], we denote by 𝒦2n-1(f, x) the interpolation polynomial in the space ℙ2n-1 characterized by satisfying the interpolation conditions $K2n−1(f,xj) = (fxj), K2n−1′(f,xj) = f′(xj) for j = 1,⋯ , n.$

It is clear that all the asserts related to the extended nodal system corresponding to Wn-1 can be reproduced with the extended nodal system corresponding to Vn-1. The Hermite interpolation polynomial corresponding to a smooth function f(x) with the extended nodal system of Vn-1 is the Hermite interpolation polynomial corresponding to a smooth function g(x) = f(-x) on the extended nodal system corresponding to Wn-1 and vice versa.

Indeed, it is easy to obtain that if h is a natural number h = (2n-1)(l+1)+k with n a positive integer, l and k nonnegative integers and 0 ≤ k < 2n-1, then 𝒦2n-1(Th,x) can be represented as: $K2n−1(Th,x)=ℋ2n−1(Th(−x),x)=ak(−1)kTk(x)+b 2n−1−k(−1)2n−1−kT2n−1−k(x)+c0D0(−x),$

where the sequences of coefficients are given in Proposition 3.2.

Moreover, proceeding like in the previous subsection one can obtain similar results as those in Propositions 3.4 and 3.5 in a straight way as follows.

Let f be a function defined for x ∈[-1, 1] by $f\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sum _{k=0}^{\infty }{a}_{k}{T}_{k}\left(x\right)$ with $|{a}_{k}|\le k\frac{1}{{k}^{s}}$ for k ≠ 0 and s ≥ 4. Then 𝒦2n-1(f,.) uniformly converges to f on [-1, 1] and the rate of convergence is $\mathcal{O}\left(\frac{1}{{n}^{s-1}}\right)$.

Let f be an analytic function on [-1, 1]. Then 𝒦2n-1(f,.) uniformly converges to f on [-1, 1] with a geometric rate of convergence.

Acknowledgement

The research was supported by Ministerio de Educación y Ciencia of Spain under grant number MTM2011-22713.

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Accepted: 2016-02-12

Published Online: 2016-03-29

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455,

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