View of Bounds for the remainder in the bivariate Shepard interpolation of Lidstone type

Rev. Anal. Num´er. Th´eor. Approx., vol. 34 (2005) no. 1, pp. 47–53 ictp.acad.ro/jnaat

BOUNDS FOR THE REMAINDER IN THE BIVARIATE SHEPARD INTERPOLATION OF LIDSTONE TYPE

TEODORA C ˘ATINAS¸^∗

Abstract. We study the bivariate Shepard-Lidstone interpolation operator and obtain new estimates for the remainder. Some numerical examples are provided.

MSC 2000. 41A63, 41A80.

Keywords. Bivariate Shepard-Lidstone interpolation, remainder.

1. INTRODUCTION

As it is pointed out in [7] and [15], interpolation at nodes having no ex- ploitable pattern is referred to as the case of scattered data and there are two important methods of interpolation in this case: the method of Shepard and the interpolation by radial basis functions.

Consider −∞ < a < b < ∞ and −∞ < c < d < ∞ and let ∆ : a = x₀ < x₁ < . . . < x_N+1 = b and ∆⁰ : c = y₀ < y₁ < . . . < y_M+1 = d denote uniform partitions of the intervals [a, b] and [c, d] with stepsizes h = (b−a)/(N+ 1) andl= (d−c)/(M+ 1),respectively.Further, letρ= ∆×∆⁰ be a rectangular partition of [a, b]×[c, d].For the univariate functionf and the bivariate functiongand each positive integerrwe denote byD^rf = d^rf /dx^r, D^r_xg=∂^rg/∂x^r and D_y^rg=∂^rg/∂y^r.

According to [1] and [2], for a fixed ∆ denote the setL_m(∆) ={h∈C[a, b] : h is a polynomial of degree at most 2m −1 in each subinterval [xi, xi+1], 0≤i≤N}.

Definition 1. [2]For a given functionf ∈C^2m−2[a, b]we say thatL^∆_mf is the Lidstone interpolant of f if L^∆_mf ∈L_m(∆) with

D^2k(L^∆_mf)(x_i) =f^(2k)(x_i), 0≤k≤m−1, 0≤i≤N+ 1.

According to [2], forf ∈C^2m−2[a, b] the Lidstone interpolantL^∆_mf uniquely exists and on the subinterval [xi, xi+1],0≤i≤N, can be explicitly expressed

∗“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, Department of Applied Mathematics, 400084 Cluj-Napoca, Romania, e-mail:

[email protected].

as (1)

(L^∆_mf)|_[xi,xi+1](x) =

m−1

X

k=0

hΛ_k ^xi+1_h^−xf^(2k)(x_i) + Λ_k ^x−x_h ⁱf^(2k)(x_i+1)ⁱh^2k, where Λ_k is the Lidstone polynomial of degree 2k+ 1, k ∈N on the interval [0,1].

We have the interpolation formula

f =L^∆_mf+R^∆_mf, whereR^∆_mf denotes the remainder.

For a fixed rectangular partition ρ= ∆×∆⁰ of [a, b]×[c, d] the set Lm(ρ) is defined as Lm(ρ) =Lm(∆)⊗Lm(∆⁰) (see, e.g., [1] and [2]).

Definition 2. [2] For a given function f ∈ C^{2m−2,2m−2}([a, b]×[c, d]) we say thatL^ρ_mf is the two-dimensional Lidstone interpolant off ifL^ρ_mf ∈Lm(ρ) with

D^2µ_x D_y^2ν(L^ρ_mf)(xi, yj) =f^(2µ,2ν)(xi, yj), 0≤i≤N+ 1, 0≤j≤M + 1, 0≤µ, ν ≤m−1.

According to [2], forf ∈C^{2m−2,2m−2}([a, b]×[c, d]),the Lidstone interpolant L^ρ_mf uniquely exists and can be explicitly expressed as

(2) (L^ρ_mf)(x, y) =

N+1

P

i=0 m−1

P

µ=0 M+1

P

j=0 m−1

P

ν=0

rm,i,µ(x)rm,j,ν(y)f^(2µ,2ν)(xi, yj),

where rm,i,j, 0≤i≤N + 1, 0≤j ≤m−1, are the basic elements of Lm(ρ) satisfying

(3) D^2υrm,i,j(xµ) =δiµδ2υ,j, 0≤µ≤N + 1, 0≤υ≤m−1.

Lemma 3. [2] If f ∈C^{2m−2,2m−2}([a, b]×[c, d]), then

(L^ρ_mf)(x, y) = (L^∆_mL^∆_m⁰f)(x, y) = (L^∆_m⁰L^∆_mf)(x, y).

Corollary 4. [2] For a function f ∈ C^{2m−2,2m−2}([a, b]× [c, d]), from Lemma 3, we have that

f −L^ρ_mf = (f −L^∆_mf) +L^∆_m(f−L^∆_m⁰f) (4)

= (f −L^∆_mf) + [L^∆_m(f −L^∆_m⁰f)−(f −L^∆_m⁰f)] + (f −L^∆_m⁰f).

With the previous assumptions we denote by L^∆,i_m f the restriction of the Lidstone interpolation polynomialL^∆_mfto the subinterval [xi, xi+1],0≤i≤N, given by (1), and in analogous way we obtain the expression of L^∆_m^0,if, the restriction ofL^∆_m⁰f,to the subinterval [y_i, y_i+1]⊆[c, d],0≤i≤N. We denote

by S_L the univariate combined Shepard-Lidstone operator, introduced by us in [4]:

(SLf)(x) =

N

X

i=0

Ai(x)(L^∆,i_m f)(x), withAi, i= 0, ..., N, given by

(5) Ai(x, y) =

N

Q

j=0 j6=i

r_j^µ(x, y) _N

P

k=0 N

Q

j=0 j6=k

r_j^µ(x, y)

and

(6) ^PN

i=0

A_i = 1.

The univariate Shepard-Lidstone interpolation formula is

(7) f =S_Lf +R_Lf.

We considerf ∈C^{2m−2,2m−2}([a, b]×[c, d]) and the set of Lidstone functio- nals

Λⁱ_Li=ⁿf(xi, yi), f(xi+1, yi+1), . . . , f(2m−2,2m−2)

(xi, yi), f(2m−2,2m−2)(x_i+1, y_i+1)^o,

regarding each subrectangle [x_i, x_i+1]×[y_i, y_i+1],0 ≤ i ≤ N, with Λⁱ_Li = 4m, 0≤i≤N. We denote byL^ρ,i_mf the restriction of the polynomial given by (2) to the subrectangle [xi, xi+1]×[y_i, yi+1],0≤i≤N.This 2m−1 polynomial, in each variable, solves the interpolation problem corresponding to the set Λⁱ_Li, 0≤i≤N and it uniquely exists.

We have

(L^ρ,i_mf)^(2ν,2ν)(xk, yk) =f^(2ν,2ν)(xk, yk), 0≤i≤N; 0≤ν ≤m−1;k=i, i+ 1.

The bivariate Shepard operator of Lidstone type S^Li, introduced by us in [5], is given by

(8) (S^Lif)(x, y) = ^PN

i=0

A_i(x, y)(L^ρ,i_mf)(x, y).

We obtain the bivariate Shepard-Lidstone interpolation formula,

(9) f =S^Lif+R^Lif,

whereS^Lifis given by (8) andR^Lif denotes the remainder of the interpolation formula.

Next, we give an error estimation using the modulus of smoothness of order k.For a functiong defined on [a, b] we have

ω_k(g;δ) = sup{|∆^k_hg(x)| :|h| ≤δ, x, x+kh∈[a, b]},

withδ ∈[0,(b−a)/k] and

∆^k_hg(x) =

k

X

i=0

(−1)^{k+i k}_ig(x+ih).

We consider the norm in the setC(X) of continuous functions defined onX by kfk_C(X)= max

x∈X |f(x)|. We recall first a result from [17]:

Theorem 5. [17] Let L be a bounded operator and let L(P) =P for every P ∈ Pk−1. Then for every bounded function f : [a, b] → R the following inequality is fulfilled

kf −L(f)k_C[a,b]≤(1 +kLk_[a,b])W_kω_kf;^b−a_k , where W_k is Whitney’s constant.

We apply this result for the operators L^∆_m and L^∆_m⁰. For f ∈ C[a, b] and g∈C[c, d] we have

f −L^∆_m(f)

C[a,b]≤(1 +kL^∆_mk_[a,b])W_2mω_2mf;^b−a_2m, (10)

g−L^∆_m⁰(g)_C[c,d]≤(1 +kL^∆_m⁰k_[c,d])W2mω2m

g;^d−c_2m.

Now we give an estimation of the remainderRLf from (7), in terms of the modulus of smoothness.

Theorem 6. If f ∈C^2m−2[a, b], then (11) kR_Lfk_C[a,b]≤(1 +L^∆_m

[a,b])W2mω2m

f;^b−a_2m. Proof. We have

(RLf)(x) =f(x)−^PN

i=0

Ai(x)(L^∆,i_m f)(x)

= ^PN

i=0

A_i(x)f(x)−^PN

i=0

A_i(x)(L^∆,i_m f)(x)

= ^PN

i=0

A_i(x)[f(x)−(L^∆,i_m f)(x)], and taking into account (10) and that

N

P

i=0

|A_i(x)|= 1,

relation (11) follows.

The next result provides an estimation of the error in formula (9).

Theorem 7. If f ∈C^{2m−2,2m−2}([a, b]×[c, d]), then

R^Lif

C[a,b]≤ 1 +L^∆_m

C[a,b]

W_2m max

y∈[c,d]ω_2mf(·, y);^b−a_2m + 1 +L^∆_m

C[a,b]

W2m max

y∈[c,d]ω2m (f −L^∆_m⁰f)(·, y);^b−a_2m + 1 +kL^∆_m⁰k_C[c,d]W2m max

x∈[a,b]ω2m

f(x,·);^d−c_2m,

where Wk is Whitney’s constant.

Proof. Taking into account (8) and (6) we get (R^Lif)(x, y) =f(x, y)−(S^Lif)(x, y)

=f(x, y)− ^PN

i=0

A_i(x, y)(L^ρ,i_mf)(x, y)

=

N

P

i=0

Ai(x, y)f(x, y)− ^PN

i=0

Ai(x, y)(L^ρ,i_mf)(x, y)

=

N

P

i=0

Ai(x, y)[f(x, y)−(L^ρ,i_mf)(x, y)].

Next applying the results (4) given by Corollary 4 and (6) it follows that (R^Lif)(x, y) =

N

P

i=0

Ai(x, y){(f−L^∆,i_m f)(x, y)

+ [L^∆,i_m (f−L^∆_m^0,if)(x, y)−(f−L^∆_m^0,if)(x, y)]

+ (f−L^∆_m^0,if)(x, y)}

=^PN

i=0

A_i(x, y)(f −L^∆,i_m f)(x, y) + ^PN

i=0

A_i(x, y)[L^∆,i_m (f −L^∆_m^0,if)(x, y)−(f−L^∆_m^0,if)(x, y)]

+

N

P

i=0

Ai(x, y)(f −L^∆_m^0,if)(x, y)

=

"

f(x, y)^PN

i=0

A_i(x, y)− ^PN

i=0

A_i(x, y)(L^∆,i_m f)(x, y)

#

− ^PN

i=0

Ai(x, y)[(f−L^∆_m^0,if)(x, y)−L^∆,i_m (f−L^∆_m^0,if)(x, y)]

+

"

f(x, y)^PN

i=0

A_i(x, y)− ^PN

i=0

A_i(x, y)(L^∆_m^0,if)(x, y)

#

and, finally, (R^Lif)(x, y) =

"

f(x, y)− ^PN

i=0

Ai(x, y)(L^∆,i_m f)(x, y)

#

− ^PN

i=0

Ai(x, y)[(f−L^∆_m^0,if)(x, y)−L^∆,i_m (f−L^∆_m^0,if)(x, y)]

+

"

f(x, y)− ^PN

i=0

Ai(x, y)(L^∆_m^0,if)(x, y)

# .

Applying Theorem 6 three times the conclusion follows.

Example 1. Letf : [−2,2]×[−2,2]→R, f(x, y) =xe^−(x2+y2)

and consider the nodes z₁ = (−1,−1), z₂ = (−0.5,−0.5), z₃ = (−0.3,−0.1), z₄ = (0,0), z₅ = (0.5,0.8), z₆ = (1,1). In Figure 1 we plot the graphics of f and S^Lif for µ = 1. In Figure 2 we plot the error (in absolute value) for Shepard interpolation regarding these data, and also, the error for Shepard interpolation of Lidstone type; we notice that in both cases, the maximum value is 0.5.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1 0 1

−0.52

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

(a) Graph off.

−1 −0.5 0 0.5 1

−1

−0.5 0.5 0

1

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3

(b) The interpolant S_Li⁽²⁾f, µ = 1.

Fig. 1. Graph off andS_Li⁽²⁾f.

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−1

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0.5 1

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(a) Error for the Shepard inter- polation.

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0.5 1

−1

−0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5

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Received by the editors: March 10, 2004.