View of Combined Shepard operators with Chebyshev nodes

(1)

Rev. Anal. Num´er. Th´eor. Approx., vol. 33 (2004) no. 1, pp. 73–78 ictp.acad.ro/jnaat

COMBINED SHEPARD OPERATORS WITH CHEBYSHEV NODES

CRISTINA O. OS¸AN^∗and RADU T. TRˆIMBIT¸ AS¸^†

Abstract. In this paper we study combined Shepard-Lagrange univariate interpolation operator

S^L,m_n,µ(Y;f, x) :=S_n,µ^L,m(f, x) =

n+1

P

k=0

|x−yn,k|^−µ(Lmf)(x, yn,k)

n+1

P

k=0

|x−yn,k|^−µ

,

where (yn,k) are the interpolation nodes and (Lmf)(x;yn,k) is the Lagrange interpolation polynomial with nodesyn,k,yn,k+1,yn,k+2,. . . ,yn,k+m, when the interpolation nodes (yn,k)_k=1,nare the zeros of first kind Chebyshev polynomial completed with yn,0 = −1 and yn,n+1 = 1. We give a direct proof for error estimation and some numerical examples.

MSC 2000. 65D05, 41A05.

Keywords. Shepard interpolation, Chebyshev nodes.

1. INTRODUCTION

Let Y = yn,i∈[−1,1]; i= 0, n+ 1; n∈N be an infinite matrix where each row is a set of distinct points in [−1,1]. Forf ∈C^m([−1,1]) the Shepard- Lagrange operator is defined by

(1) S_n,µ^L,m(Y;f, x) :=S_n,µ^L,m(f, x) =

n+1

P

k=0

|x−y_n,k|^−µ(L_mf)(x, y_n,k)

n+1

P

k=0

|x−y_n,k|^−µ

,

wherem∈N,m < nis prescribed.

The Shepard-Lagrange operator was treated in [1] and [6]. Its most impor- tant properties are: it preserves polynomials of degree m, i.e.,

S_n,µ^L,m(ej;x) =ej(x), j= 0, m, whereej(x) =xⁱ. Also,

S_n,µ^L,m(f;y_n,k) =f(y_n,k), k= 1, n.

∗“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, str. M.

Kogˇalniceanu 1, 400084 Cluj-Napoca, Romania, e-mail: [email protected].

†“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, str. M.

Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania, e-mail: [email protected].

(2)

In this paper we give an error estimation analogous to that given in [6], but the present proof is direct and exploits the properties of Cebyshev nodes and Lagrange interpolation polynomials with such nodes.

2. ERROR ESTIMATION

If f ∈ C^p([a, b]), p ∈ N, p < m, and (L_mf) is the m-th degree Lagrange interpolation polynomial with nodes x0, x1, . . . , xm ∈ [a, b], the following error estimation holds (see [3])

(2) kf −L_mfk_∞≤C_p(b−a)^p ^log_mp^mω f^(p);_m−p^b−a, whereC_p >0.

Let (yn,k)_k=0,n+1 be the set of the roots of first kindn-th degree Chebyshev polynomial, completed with yn,0 = −1 and yn,n+1 = 1. For k = 1, n, y_n,k = cosθ_n,k, where θ_n,k = (2k−1)π/(2n).

Remark 1. We have

|θ_n,k−θ_n,k+1|= ^π_n.

The following theorem holds:

Theorem 1. If f ∈ C^p([−1,1]) and µ > p+ 1, we have (3) f(x)−S_n,µ^L,m(f;x)≤ ^(1−x_nµ−1²⁾^p

Z 1 n⁻¹

ω(f^(p);t√ 1−x²) t^µ−p dt.

Proof. Lety_n,d be the closest point to x. We share the nod set (y_n,k) into three classes: k= 0, d−m;k=d−m+ 1, d;k=d+ 1, n+ 1.

Equations (1) and (2) imply

f(x)−S_n,µ^L,m(f;x)≤

≤

n+1

X

k=0

x−y_n,d x−y_n,k

µ|f(x)−(Lmf)(x, yn,k)|

≤C_p^log_mp^m

"_d−m X

k=0

x−yn,d

x−y_n,k

µ|x−y_n,k|^pω f^(p);^|x−y_m−p^n,k^|

+

d

X

k=d−m+1

µ|y_n,k+m−y_n,k|^pω f^(p);^|y^n,k+m_m−p^−y^n,k^|+

+

n+1

X

k=d+1

x−yn,d

x−y_n,k

µ|x−y_n,k+m|^pω f^(p);^|x−y_m−p^n,k+m^|





:=Cplogm

m^p (S1+S2+S3).

(3)

We also have

|x−y_n,d| ≤^π_n^p1−x², (4)

|x−y_n,k| ≥^π_n|k−d|^p1−x², (5)

|x−yn,k+m| ≤ |x−yn,k|+π^m_n. (6)

We shall also use the following properties for modulus of continuity δ2 ≥δ₁ ⇒ ^ω(f;δ_δ ²⁾

2 ≤ ^ω(f;δ_δ ¹⁾

1 , (7)

ω(f;λδ)≤(1 +λ)ω(f;δ) if λ∈R+, (8)

δ₁ ≤δ₂ ⇒ω(f;δ₁)≤ω(f;δ₂), (9)

ω(f;δ₁+δ₂)≤ω(f;δ₁) +ω(f;δ₂).

(10)

Now, it follows the estimations for S₁,S₂, and S₃. S₁=

d−m

X

k=0

µ|x−y_n,k|^pω f^(p);^|x−y_m−p^n,k^|.

From (7) forδ1:= ^π_n|k−d|√

1−x² and δ2 :=|x−yn,k|we have

ω(^f^(p);|^x−yn,k|)

|^x−yn,k| ≤ ^2n(π+1)

π|k−d|√ 1−x². We obtain

S₁≤c₁

√1−x² n

p

·

d−m

X

k=0 1

|k−d|^µ−p ω f^(p);^|k−d|_n ^p1−x² with

c₁ =2π^p(π+ 1)1 +_m−p¹ , S₂ =

d

X

k=d−m+1

x−yn,d

x−y_n,k

µ|y_n,k+m−y_n,k|^pω f^(p);^|y^n,k+m_m−p^−y^n,k^|.

Since |y_n,k+m−yn,k| ≤ ^mπ_n √

1−x² we have

ω f^(p);^|y^n,k+m_m−p^−y^n,k^|≤ 1 +_m−p¹ (mπ+ 1)ω f^(p);

√1−x² n

. We obtain

S2 ≤c2

√1−x² n

p

ω f^(p);

√1−x² n

with

c₂ = (mπ)^p(mπ+ 1)^p(m−1) 1 +_m−p¹ ,

S₃ =

n+1

X

k=d+1

x−yn,d

x−y_n,k

µ|x−y_n,k+m|^pω f^(p);^|x−y_m−p^n,k+m^|.

(4)

ForS3 we remark that

|x−y_n,k+m|^p≤(|x−y_n,k|+|y_n,k+m−y_n,k|)^p

=

p

X

r=0 p r

|x−yn,k|^p−r|y_n,k+m−yn,k|^r. Now, we have

S3 ≤ |x−y_n,d|^µ

n+1

X

k=d+1 p

X

r=0 p r

|yn,k−y_n,k+m|^r

|^x−yn,k|^µ−p+r ω f^(p);^|x−y_m−p^n,k+m^|

≤π^p

√1−x² n

p ^p X

r=0 p r

m^r _n+1

P

k=d+1 1

|k−d|^µ−p ω f^(p);^|x−y_m−p^n,k+m^|

= 1+_m−p¹ π^p(m+1)^p

√1−x² n

p n+1

X

k=d+1 1

From this observation and (10), (7) we obtain ω f^(p);|x−yn,k+m|≤

≤M(mπ+ 1)ω f^(p);

√1−x² n

+ 2(π+ 1)ω f^(p);^|k−d|_n ^p1−x². In conclusion, we obtain forS₃ the following inequality

S3≤

√1−x² n

p

c3ω f^(p);

√1−x² n

+c4 n+1

X

k=d+1

ω f^(p);^|k−d|_n √ 1−x²

|k−d|^µ−p

with the constants

c₃=π^p(m+ 1)^pM(mπ+ 1) 1 + _m−p¹ , c4= 2π^p(m+ 1)^p(π+ 1) 1 + _m−p¹ . Since

ω f^(p);

√1−x² n

≤ _nµ−p−1¹

Z ₁

n⁻¹

ω f^(p);t√

1−x² t^µ−p dt and

S1+S2+S3 ≤

≤

√1−x² n

ph

C(c₂, c₃)·ω f^(p);

√1−x² n

+C(c₁, c₄)·ω f^(p);^|k−d|_n ^p1−x²ⁱ

(3) follows.

(5)

3. EXAMPLES AND GRAPHS

Let us consider the functionf : [−1,1]→R, f(x) = sinπx.

Its graph appears in figure 1, together with the Shepard-Lagrange approximation functions forµ∈ {2,4},n= 16 andm∈ {1,2}.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5

(a) Graph off.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5

(b) Shepard-Lagrange, n = 16, µ= 2,m= 1.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5

(c) Shepard-Lagrange, n = 16, µ= 2,m= 2.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5

(d) Shepard-Lagrange, n = 16, µ= 4,m= 1.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5 0 0.5 1 1.5

(e) Shepard-Lagrange, n = 16, µ= 4,m= 2.

Fig. 1. Graph off and various Shepard-Lagrange interpolants.

(6)

REFERENCES

[1] Gh. Comanand R. Trˆımbit¸as¸,Combined Shepard univariate operators, East Journal on Approximations,7, no. 4, pp. 471–483, 2001.

[2] G. CriscuoloandG. Mastroianni,Estimates of the Shepard interpolatory procedure, Acta. Math. Hung.,61, nos. 1–2, pp. 79–91, 1993.

[3] M. Crouzeix and A. L. Mignot, Analyse numérique des équations différentielles, 2êédition, Masson, Paris, 1989.

[4] B. Della Vecchia and G. Mastroianni, Pointwise estimates of rational operators based on general distribution of knots, Facta Universitatis (Niˇs), Ser. Math. Inform.,6, pp. 63–78, 1991.

[5] B. Della Vecchia and G. Mastroianni, Pointwise simultaneous approximation by rational operators, J. Approx. Theory,65, pp. 140–154, 1991.

[6] R. Trˆımbit¸as¸, Univariate Shepard-Lagrange interpolation, Kragujevac J. Math., 24, pp. 85–94, 2002.

[7] D. Shepard,A two dimensional interpolation function for irregularly spaced data, Proc.

23rd Nat. Conf. ACM, pp. 517–523, 1968.

Received by the editors: August 5, 2003.