a family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial...

Journal of Computational and Applied Mathematics 274 (2015) 88–108

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Journal of Computational and AppliedMathematics

journal homepage: www.elsevier.com/locate/cam

A family of multivariate multiquadric quasi-interpolationoperators with higher degree polynomial reproduction

Ruifeng Wu, Tieru Wu, Huilai Li ∗

School of Mathematics, Jilin University, Changchun 130012, China

h i g h l i g h t s

• A family of multivariate multiquadric quasi-interpolants is proposed.• There is no demand for derivatives of approximated function in quasi-interpolants.• The quasi-interpolants satisfy any degree polynomial reproduction property.• The quasi-interpolants can reach up to a higher approximation order.

a r t i c l e i n f o

Article history:Received 5 August 2013Received in revised form 24 June 2014

Keywords:Quasi-interpolationMultiquadric functionsDimension-splittingLagrange basis functionsPolynomial reproductionConvergence rate

a b s t r a c t

In this paper, by using multivariate divided difference (Rabut, 2001) to approximate thepartial derivative and the idea of the superposition (Waldron, 2009), we modify a multi-quadric quasi-interpolation operator (Ling, 2004) based on adimension-splitting techniquewith the property of linear reproducing to gridded data onmulti-dimensional spaces, suchthat a family of proposedmultivariatemultiquadric quasi-interpolation operatorsΦr+1 hasthe property of r + 1 (r ∈ Z, r > 0) degree polynomial reproducing and converges up toa rate of r + 2. In addition, the proposed quasi-interpolation operator only demands infor-mation of location points rather than the derivatives of the function approximated. More-over, we give the approximation error of our quasi-interpolation operator. Finally, somenumerical experiments are shown to confirm the approximation capacity of our quasi-interpolation operator.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The approximation of themultivariate function from scattered data is an important theme in numerical mathematics. Al-though the classical method to attack the problem is interpolation, the interpolationmatrix quickly becomes ill-conditionedas the number of points increases. There are different ways to overcome this ill-conditioned problem. Compared with in-terpolation, the quasi-interpolation method does not require solution of any linear system, and can reach up to the ex-pected convergence rate. In this paper, we will focus on the quasi-interpolation method. For a set of functional valuesf (qj)nj=1 taken on a set of nodes Ξ = q0, . . . , qn ⊆ Rd and a set of given quasi-interpolation basis functions Ψ (q), the

∗ Corresponding author. Tel.: +86 0431 85166040.E-mail addresses: [email protected] (R. Wu), [email protected] (T. Wu), [email protected] (H. Li).

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R. Wu et al. / Journal of Computational and Applied Mathematics 274 (2015) 88–108 89

quasi-interpolation (Qf )(q) of a d-variate function f takes the standard form via linear combination

(Qf )(q) =

nj=0

f (qj)Ψ (q), q ∈ Rd. (1)

The approximation properties of the quasi-interpolation operator in the case that qj are nodes of a uniform grid can bewell-understood. For example, for a uniform grid with spacing h, the following quasi-interpolation operator

(Qf )(q) =

j∈Zd

f (jh)Ψqh

− j

(2)

proposed by Rabut [1] can be studied via the theory of principal shift-invariant spaces (see, e.g., [2]), which has been devel-oped by de Boor et al. [3]. Here, the quasi-interpolation basis function Ψ is supposed to be a compactly supported or rapidlydecaying function. In [4], Bozzini, Dyn, and Rossini presented a procedure in spaces ofm-harmonic splines inRd which startsfrom a simple generatorΨ0 and recursively defines generatorsΨ1,Ψ2, . . . , Ψm−1 with corresponding quasi-interpolation op-erators defined as (2) reproducing polynomials of degrees 3, 5, . . . , 2m−1 respectively. In [5,6], the Strang–Fix condition isconsidered a necessary and sufficient condition for the polynomial reproduction, convergence and approximation order ofthe quasi-interpolation. Based on the Strang–Fix condition forΨ , the scattered data quasi-interpolation by functions, whichreproduces polynomials, has been proposed by Buhmann et al. [7], Dyn and Ron [8], Wu and Liu [9], andWu and Xiong [10].Wewill discuss themultivariatemultiquadric quasi-interpolation operator which can reproduce higher degree polynomialsin this paper.

With the multiquadric function

ϕi(x) =(x − xi)2 + c2

1/2, x, xi ∈ R, c > 0 (3)

proposed by Hardy [11], Beast and Powell [12] first constructed a univariate quasi-interpolation operator by shifts of themultiquadric basis function of first degree to finite scattered data, which reproduces linear polynomials. However, theoperator requires the derivatives of the function approximated at endpoints, which is not convenient for practical purposes.Thus, Wu and Schaback [13] constructed another univariate multiquadric quasi-interpolation operator with modificationsat endpoints without derivatives.

Ling [14] extended the univariate quasi-interpolation operator (see, e.g., [13]) to multi-dimension by using thedimension-splitting univariate multiquadric basis function approach, which reproduces linear polynomials. Based on [14],Feng and Zhou [15] proposed a multivariate dimension-splitting multiquadric quasi-interpolation operator satisfyingquadratic polynomial reproduction property, and proved that it converged with a rate of O(h3) as c = O(h), h = maxh1,h2, 2h1 = max06i6n−1|xi+1 − xi|, 2h2 = max06j6m−1|yj+1 − yj|. In [16], for any given quasi-interpolation operatorwhich reproduces all polynomials of degree 6 m, Waldron proposed a quasi-interpolation operator which reproduces allpolynomials of degree 6 m + r , where r is a non-negative integer. However, it involves the derivatives of the functionapproximated at each knot.

The aim of our paper is to present a family of multivariate multiquadric quasi-interpolation operators with higher accu-racy for gridded data, which does not require the partial derivatives of the function approximated at each knot. By virtue ofthe constructive idea in [16] and the multivariate divided difference formula in [17], we extend the operator [14] and givea family of multivariate multiquadric quasi-interpolation operators based on the dimension-splitting technique for griddeddata. We show that the new operator could reproduce any degree polynomials and have the relevant order approximationrate under a certain assumption. Our error estimate indicates that the convergence rate depends heavily on the shape pa-rameter c. Thus, our operator could provide the desired smoothness and precision by choosing a suitable value of the shapeparameter c .

The remainder of this paper is organized as follows. In Section 2,we introduce somenecessary definitions. In Section 3,wepropose a family of multivariate multiquadric quasi-interpolation operators satisfying any degree polynomial reproductionproperty. In Section 4, we obtain the error estimate. In Section 5, we give some numerical experiments, and verify the rela-tionship between the approximation order and the degree of polynomial reproducing. In Section 6, we give the conclusions,and put forward the future work.

2. Preliminaries

Definition 1 ([15]). Let k ∈ N, c > 0. The multiquadric functions of 2k − 1 degree (2k order) is defined by

ϕ(q; 2k) =

∥q∥2

+ c2(2k−1)/2

, q = (x1, . . . , xd) ∈ Rd, (4)

where c is called the shape parameter, Rd is the d-dimensional Euclidean space, ∥ · ∥ is the Euclidean norm, and ϕj,2k(q) is ashift of ϕ(q; 2k) centered at qj:

ϕj,2k(q) = ϕ(q − qj; 2k), qj = (xj1, . . . , xjd) ∈ Rd. (5)

90 R. Wu et al. / Journal of Computational and Applied Mathematics 274 (2015) 88–108

Definition 2 ([15]). For a given point set qi = (xi1, . . . , xid)ni=0 ⊂ Ω and a set of real values f (qi)ni=0, Ω is a bounded

domain on Rd, the quasi-interpolation (Lf )(q) of a function f : Rd→ R is defined as follows:

(Lf )(q) =

nj=0

f (qj)αj(q), q ∈ Rd, (6)

where αj(q)nj=0 are quasi-interpolation basis functions.

Definition 3 ([15]). For any real-coefficient polynomial p of degreem, if Lp = p, then the quasi-interpolation operator L issaid to be satisfying the reproduction property of polynomials of degree no more thanm.

Definition 4 ([17]). Let F = f |f : Rd→ R and let A be a discrete subset of Rd, α = (a1, . . . , ad) ∈ Nd

= Zd+, |α| =

a1 + · · · + ad. Suppose that Dα f := Da1 · · ·Dad f =∂ |α|f

∂xa11 ∂x

a22 ···x

add

is the order α derivative, Pr+1 = Pr+1(Rd) is the set of

multivariate polynomials of degree 6 r + 1, where r is a non-negative integer. An operator DαA : F → F is said to be a

Pr+1-exact A-discretization of Dα if and only if(i) There exists a real vector λ = (λa)a∈A such that, for any f ∈ F ,

DαA f (·) =

a∈A

λaf (· + a). (7)

(ii) For any p ∈ Pr+1,

DαAp(·) = Dαp(·). (8)

In such situation, we also say that DαA f is a Pr+1-exact A-discretization of Dα f . Suppose that the points in the set A =

q0, . . . , qN−1 are properly posed for Pr+1(Rd), that is, there does not exist a nonzero polynomial in Pr+1(Rd) which van-ishes on all the points of A, where N =

r + 1 + dd

is the dimension of Pr+1(Rd), and Pr+1(Rd) is the multivariate polynomial

space of all polynomials of degree 6 r + 1. Then, DαA f is determined uniquely.

For the sake of simplification, we set d = 2, then obtain N =r + 1 + 2

2

= (r +3)(r +2)/2. Meanwhile, Pr+1 = Pr+1(R2)

is the bivariate polynomial space of all polynomials of degree 6 r + 1.

Definition 5 ([17]). The set A is said to be Pr+1-unisolvent if and only if for all vectors y = (ya)a∈A, there exists a uniquepolynomial p ∈ Pr+1 such that for any a ∈ A, p(a) = ya.

In the next section, we give explicit forms of DαA f and the associated (λa)a∈A. Since these forms are ratios of determinants,

we need to order the points of A and the basis of Pr+1.

Definition 6 (Order on A [17]). When necessary in what follows, the points of the set A are assumed to be ordered in anarbitrary (but fixed) way; in this case al−1 denotes the (l− 1)th element of A (l = 1, . . . ,N) and Al−1 denotes the set Al−1 =

a0, . . . , al−1.

Definition 7 (Order on N⋆2 [18]). Let α = (a1, a2), β = (b1, b2) ∈ N⋆2, where N⋆ is the class of non-negative integers. Then,(i) α = β if and only if ak = bk for k = 1, 2.(ii) α ≺ β if either (1) |α| < |β| or (2) |α| = |β| with α strictly preceding β with respect to the lexicographic order.(iii) α ≼ β whenever either α = β or α ≺ β .

Now, we construct the N × N matrix

M(q; A) =

(q0 − q)α0 · · · (q0 − q)αN−1

.... . .

...(qN−1 − q)α0 · · · (qN−1 − q)αN−1

, (9)

where (0, 0) = 0 = α0 ≺ α1 · · · ≺ αN−1.

Example 1. If A = q0, . . . , q5 for qi = (xi, yi) ∈ R2 and if q = (x, y) ∈ R2, then

M(q; A) =

1 Y0 X0 Y 2

0 X0Y0 X20

1 Y1 X1 Y 21 X1Y1 X2

11 Y2 X2 Y 2

2 X2Y2 X22

1 Y3 X3 Y 23 X3Y3 X2

31 Y4 X4 Y 2

4 X4Y4 X24

1 Y5 X5 Y 25 X5Y5 X2

5

,

where Xi = xi − x, Yi = yi − y (i = 0, . . . , 5), and (Xi, Yi)α

= Xa1i Y a2

i for any α = (a1, a2) ∈ N⋆2.


Definition 8 ([18]). (i) For all α, β ∈ N⋆2, we write δβα =

1 if α = β0 otherwise .

(ii) For any α ∈ N⋆2,Mαi (q; A) denotes the matrixM(q; A) but for the row containing qi ∈ A replaced by zeros except at the

intersection with column α where the entry is 1.

Thus, for all q ∈ R2, qi ∈ A and α0 ≼ α ≼ αN−1 we have

Mαi (q; A) =

...

. . ....

(qi−1 − q)α0 · · · (qi−1 − q)αN−1

δα0α · · · δ

αN−1α

(qi+1 − q)α0 · · · (qi+1 − q)αN−1

.... . .

...

. (10)

Example 2. If α = α2 = (1, 0) and if M(q; A) is the matrix given by (9), then

Mα1 (q; A) =

1 Y0 X0 Y 20 X0Y0 X2

0

0 0 1 0 0 0

1 Y2 X2 Y 22 X2Y2 X2

2

1 Y3 X3 Y 23 X3Y3 X2

3

1 Y4 X4 Y 24 X4Y4 X2

4

1 Y5 X5 Y 25 X5Y5 X2

5

.

Definition 9 ([18]). M(A)Mα

i (A)designates M(q; A)

Mα

i (q; A)evaluated at the origin q = 0 = (0, 0) ∈ R2, that is to

say,

M(A) = M ((0, 0); A) and Mαi (A) = Mα

i ((0, 0); A) . (11)

Definition 10. Let A = q0, . . . , qN−1 be a Pr+1-unisolvent set, we denote the Lagrange basis function of the degree r + 1on A by lrj (q) as follows:

lrj (q) =detMr

j,q(A)

detM(A)(j = 0, . . . ,N − 1), (12)

where

Mrj,q(A) =

.... . .

...

qα0j−1 · · · qαN−1

j−1

qα0 · · · qαN−1

qα0j+1 · · · qαN−1

j+1

.... . .

...

, q = (x, y), αN−1 = (r + 1, 0) ∈ N2. (13)

Example 3. If j = 2 and r = 1, then

M12,q(A) =

1 y0 x0 y20 x0y0 x201 y1 x1 y21 x1y1 x211 y x y2 xy x2

1 y3 x3 y23 x3y3 x231 y4 x4 y24 x4y4 x241 y5 x5 y25 x5y5 x25

.

3. Quasi-interpolation operator with any degree polynomial reproduction

In this section, we first propose a family of multivariate multiquadric quasi-interpolation operators Φr+1, then give thepolynomial reproduction property.


In the following, we introduce a well-known univariate quasi-interpolation operator LD.For given data a = x0 < x1 · · · < xn = b, Wu and Schaback [13] proposed the following quasi-interpolation function

by the shifts of the multiquadric function of first degree ϕi(x) =(x − xi)2 + c2

1/2, c > 0 to each knot.

(LDf )(x) =

ni=0

f (xi)αi(x), (14)

where

α0(x) =12

+ϕ1(x) − (x − x0)

2(x1 − x0),

α1(x) =ϕ2(x) − ϕ1(x)2(x2 − x1)

−ϕ1(x) − (x − x0)

2(x1 − x0),

αi(x) =ϕi+1(x) − ϕi(x)2(xi+1 − xi)

−ϕi(x) − ϕi−1(x)2(xi − xi−1)

(2 6 i 6 n − 2),

αn−1(x) =(xn − x) − ϕn−1(x)

2(xn − xn−1)−

ϕn−1(x) − ϕn−2(x)2(xn−1 − xn−2)

,

αn(x) =12

+ϕn−1(x) − (xn − x)

2(xn − xn−1).

They proved that (LDf )(x) possessed the constant and linear polynomial reproduction property.Ling [14] extended the univariate quasi-interpolation formula (14) to multi-dimension using the dimension-splitting

multiquadric basis function technique. For given data (xi, yj, fij), i = 0, . . . , n, j = 0, . . . ,m, the form of dimension-splitting quasi-interpolation for the multiquadric basis function is

(Φ1f )(x, y) =

ni=0

mj=0

fijαi(x)βj(y), (15)

where αi(x)ni=0 are given above. Along that y direction, the basis functions βj(y)mj=0 are defined as follows

β0(y) =12

+ϕ1(y) − (y − y0)

2(y1 − y0),

β1(y) =ϕ2(y) − ϕ1(y)2(y2 − y1)

−ϕ1(y) − (y − y0)

2(y1 − y0),

βj(y) =ϕj+1(y) − ϕj(y)2(yj+1 − yj)

−ϕj(y) − ϕj−1(y)2(yj − yj−1)

(2 6 j 6 m − 2),

βm−1(y) =(ym − y) − ϕm−1(y)

2(ym − ym−1)−

ϕm−1(y) − ϕm−2(y)2(ym−1 − ym−2)

,

βm(y) =12

+ϕm−1(y) − (ym − y)

2(ym − ym−1).

Feng and Zhou [15] proved that Φ1 satisfied the property of linear reproduction.Based on the above operatorΦ1, we give a family of multivariate dimension-splitting quasi-interpolation operatorsΦr+1

for the multiquadric basis function to gridded data as follows by superposition idea in [16]

(Φr+1f )(x, y)

=

ni=0

mj=0

f (xi, yj)αi(x)βj(y) +

ni=0

mj=0

rl=1

rl

r+1l

1l!

ls=0

ls

(x − xi)l−s(y − yj)s

∂ lf∂xl−s∂ys

(xi, yj)

αi(x)βj(y)

=

ni=0

mj=0

r

l=0

r + 1 − lr + 1

ls=0

1(l − s)!s!

(x − xi)l−s(y − yj)s∂ lf

∂xl−s∂ys(xi, yj)

αi(x)βj(y). (16)

The quasi-interpolation operator Φr+1 has the following polynomial reproduction property.

Theorem 1. The quasi-interpolation operator Φr+1 reproduces all polynomials of degree 6 r + 1.

Proof. Let us verify that (Φr+1f )(x, y) = f (x, y), for f (x, y) ∈ Pr+1, then quasi-interpolation (Φr+1f )(x, y) reproduces allpolynomials of degree≤ r +1. Because the univariate quasi-interpolation operator LD in (14) satisfies the constant and the


linear polynomial reproduction property, we haven

i=0

αi(x) = 1,n

i=0

xiαi(x) = x.

Similarlymj=0

βj(y) = 1,mj=0

yjβj(y) = y.

Then when f (x, y) = 1, we have

(Φr+1f )(x, y) =

ni=0

mj=0

αi(x)βj(y) =

n

i=0

αi(x)

mj=0

βj(y)

= 1,

when f (x, y) = x, we have

(Φr+1f )(x, y) =

ni=0

mj=0

1 −

rr + 1

xi +

rr + 1

x

αi(x)βj(y) = x,

when f (x, y) = y, we have

(Φr+1f )(x, y) =

ni=0

mj=0

1 −

rr + 1

yj +

rr + 1

y

αi(x)βj(y) = y,

when f (x, y) = xk (k = 2, . . . , r + 1), we have

(Φr+1f )(x, y) =

ni=0

mj=0

r

l=0

r + 1 − lr + 1

ls=0

1(l − s)!s!



αi(x)βj(y)

=

ni=0

mj=0

r

l=0

r − l + 1r + 1

1l!(x − xi)l

∂ lf∂xl

(xi, yj)

αi(x)βj(y)

=

ni=0

r

l=0

r − l + 1r + 1

1l!(x − xi)l

∂ lf∂xl

(xi, yj)

αi(x)

=

ni=0

k

p=0

kp

k−pl=0

(−1)lk − p

l

r + 1 − l − p

r + 1

xk−pi xp

αi(x).

Becausek−pl=0

(−1)l

k − p

l

r + 1 − l − p

r + 1

= 0, p = 0, . . . , k − 2, k = 2, . . . , r + 1,

we obtain

(Φr+1f )(x, y) =

ni=0

k

p=0

kp

k−pl=0

(−1)lk − p

l

r + 1 − l − p

r + 1

xk−pi xp

αi(x)

=

ni=0

k

k − 1

1l=0

(−1)l1l

r + 1 − l − (k − 1)

r + 1

xixk−1

+

kk

0l=0

(−1)l0l

r + 1 − l − k

r + 1

xk

αi(x)

=

ni=0

kr + 1 − (k − 1)

r + 1−

r + 1 − 1 − (k − 1)r + 1

xixk−1

+r + 1 − kr + 1

xk

αi(x)

= xk.

Similarly, when f (x, y) = yk (k = 2, . . . , r + 1), we have (Φr+1f )(x, y) = yk.


When f (x, y) = xpyq (p, q ∈ N, p + q = k, k = 2, . . . , r + 1), we have

(Φr+1f )(x, y) =

ni=0

mj=0

r

l=0

r + 1 − lr + 1

ls=0

1(l − s)!s!



αi(x)βj(y)

=

ni=0

mj=0

p

l1=0

ql2=0

pl1

ql2

p−l1s=0

q−l2t=0

(−1)s(−1)tp − l1

s

×

q − l2

t

r + 1 − s − t − l1 − l2

r + 1

xp−l1i xl1yq−l2

j yl2

αi(x)βj(y).

Because ofp−l1s=0

q−l2t=0

(−1)s(−1)tp − l1

s

q − l2

t

r + 1 − s − t − l1 − l2

r + 1= 0, l1 = 0, 1, . . . , p, l2 = 0, 1, . . . , q,

l1 + l2 = 0, 1, . . . , p + q − 2, p + q = 2, . . . , r + 1,

(Φr+1f )(x, y)

=

ni=0

mj=0

p

l1=0

ql2=0

pl1

ql2

p−l1s=0

q−l2t=0

(−1)s(−1)tp − l1

s

×

q − l2

t

r + 1 − s − t − l1 − l2

r + 1

xp−l1i xl1yq−l2

j yl2

αi(x)βj(y)

=

ni=0

mj=0

p

p − 1

qq

1s=0

0t=0

(−1)s(−1)t1s

0t

r + 1 − s − t − (p + q − 1)

r + 1

xixp−1y0j y

q

+

pp

q

q − 1

0s=0

1t=0

(−1)s(−1)t0s

1t

r + 1 − s − t − (p + q − 1)

r + 1

x0i x

pyjyq−1

+

pp

qq

0s=0

0t=0

(−1)s(−1)t0s

0t

r + 1 − s − t − (p + q)

r + 1

x0i x

py0j yq

αi(x)βj(y)

=

ni=0

mj=0

pr + 1 − (p + q − 1)

r + 1−

r + 1 − 1 − (p + q − 1)r + 1

xixp−1yq

+ qr + 1 − (p + q − 1)

r + 1−

r + 1 − 1 − (p + q − 1)r + 1

xpyjyq−1

+

r + 1 − (p + q)

r + 1

xpyq

αi(x)βj(y)

=

ni=0

mj=0

p

r + 1xixp−1yq +

qr + 1

xpyjyq−1+

r + 1 − (p + q)r + 1

xpyq

αi(x)βj(y)

= xpyq, p + q = 2, . . . , r + 1.

Now we have proved that the operator Φr+1 reproduces all polynomials of degree 6 r + 1.

The operator Φr+1 reproduces all polynomials of degree 6 r + 1, but it requires the first to the rth degree derivatives ofthe function f approximated at each knot, which is very difficult to obtain in practice. Therefore, we will use multivariatedivided difference operator Dα

A f defined in Definition 4 to substitute the partial derivatives ∂ |α|f∂xa1 ∂ya2 in the Φr+1f in (16),

where α = (a1, a2) ∈ N2, |α| 6 r .We use Lagrange basis lrj (q)

N−1j=0 (see formula (12)) to replace some arbitrary basis piNi=1 (see [17, Theorem 10]), and

denote by the points in set A (Aq) the gridded data. Then, we give a similar formula (19), and give the modified formula (22)in our paper. In analogy with [17, Theorem 10] we state the following theorem.

Theorem 2. Let A, Aq be Pr+1-unisolvent sets, f ∈ F , α ∈ N2, |α| 6 r, N = (r + 3)(r + 2)/2 and let A, Aq satisfy Definition 6.(i) If A = q0, . . . , qN−1, then

DαA f (0) =

N−1i=0

λqi f (qi) (17)


and

λqi =α!detMα

i (A)

detM(A)(i = 0, . . . ,N − 1), (18)

where Mαi (A) (see formula (10) and (11)) and M(A) (see formula (9) and (11)) are the N × N matrices. Furthermore, we have

DαA f (0) =

N−1i=0

α!detMαi (A)

detM(A)f (qi). (19)

(ii) If Aq = q0 − q, . . . , qN−1 − q, then for any fixed q = (x, y) ∈ R2 and pi = qi − q, we have

DαAq f (q) =

N−1i=0

λpi f (qi) (20)

and

λpi =α!detMα

i (q; A)

detM(q; A)(i = 0, . . . ,N − 1), (21)

where Mαi (q; A) (see formula (10)) and M(q; A) (see formula (9)) are the N × N matrices.

Furthermore, we obtain

DαAq f (q) =

N−1i=0

α!detMαi (q; A)

detM(q; A)f (qi). (22)

Proof. (i) For any A = q0, . . . , qN−1, if we apply (7), then we get

DαA f (0) =

N−1i=0

λqi f (qi)

which is (17). Using the formula (8) and the Lagrange basis functions lrj (q)N−1j=0 of Pr+1 on the above set A, for the fixed j,

we find

DαA l

rj (0) =

N−1i=0

λqi lrj (qi) (j = 0, . . . ,N − 1).

As j = i, we have λqi = DαA l

rj (0) = Dα

A lri (0).

So, applying Definitions 8–10, we obtain

λqi = Dα lri (0) =α!detMα

i (A)

detM(A)(i = 0, . . . ,N − 1)

which is (18). Using (17) and (18), we get (19).(ii) Given Aq = q0 − q, . . . , qN−1 − q, for any fixed q = (x, y) ∈ R2 and pi = qi − q, using A = Aq in (17), we get

DαAq f (0) =

N−1i=0

λpi f (pi).

Applying A = Aq to (18), we immediately get (21).By a straightforward calculation above, we have

DαAq f (q) =

N−1i=0

λpi f (pi + q) =

N−1i=0

λpi f (qi).

Using (21) in (20), we get (22).

According to [17] and Definition 4, we know that

DαAq f (q) ≈

∂ |α|f∂xa1∂ya2

(q), q = (x, y) ∈ R2, (23)


Fig. 1. (a) Choosing point set Aij = q0, . . . , q5 for computing D(1,0)Aij(xi ,yj)

f (xi, yj) and D(0,1)Aij(xi ,yj)

f (xi, yj) as r = 1; (b) Choosing point set Aij = q0, . . . , q9 for

computing D(1,0)Aij(xi ,yj)

f (xi, yj),D(0,1)Aij(xi ,yj)



f (xi, yj), and D(0,2)Aij(xi ,yj)

f (xi, yj) as r = 2.

and when f (x, y) ∈ Pr+1, we obtain

DαAq f (q) =

∂ |α|f∂xa1∂ya2

(q), α = (a1, a2) ∈ N2, |α| 6 r, q = (x, y) ∈ R2. (24)

According to the location of each knot (xi, yj) (i = 0, . . . , n, j = 0, . . . ,m), we introduce the choice of properly posedpoint set Aij of N elements for computing Dα

Aij(xi,yj)f (xi, yj), (xi, yj) ∈ Aij. The principle of choosing lattice Aij is as follows: on

the one hand, lattice Aij is a properly posed point set for Pr+1. On the other hand, lattice Aij is as close to (xi, yj) as possible.By virtue of the above two points, base points in the point set Aij serve as vertices for a triangular array. For the sake ofsimplification, we only set r = 1, 2. Then we choose point set Aij by Fig. 1.

Writing A = Aij, q = (xi, yj) in Theorem 2, we use DαAij(xi,yj)

f (xi, yj) to substitute ∂ |α|f∂xa1 ∂ya2 (xi, yj) in the scheme Φr+1f in

(16), then obtain the following scheme Φr+1f

(Φr+1f )(x, y) =

ni=0

mj=0

r

l=0

r + 1 − lr + 1

1l!0!

(x − xi)lD(l,0)Aij(xi,yj)

f (xi, yj)

+1

(l − 1)!1!(x − xi)l−1(y − yj)D

(l−1,1)Aij(xi,yj)

f (xi, yj)

+ · · · +1

1!(l − 1)!(x − xi)(y − yj)l−1D(1,l−1)

Aij(xi,yj)f (xi, yj) +

10!l!

(y − yj)lD(0,l)Aij(xi,yj)

f (xi, yj)

αi(x)βj(y). (25)

Further, using (25) and Theorem 2, we give the expression of scheme Φr+1f in the Appendix.

Theorem 3. The quasi-interpolation operator Φr+1 reproduces all polynomials of degree 6 r + 1.Proof. When f (x, y) ∈ Pr+1, according to (24), (25) and the proof of Theorem 1, we obtain

(Φr+1f )(x, y) = f (x, y).

4. Convergence rate of the quasi-interpolation operators

In this section, based on the papers [16,17], we obtain convergence rate of the quasi-interpolation operator Φr+1.Let Ω ⊂ R2 be a bounded domain which contains the point set (xi, yj), i = 0, . . . , n, j = 0, . . . ,m. For simplicity, let

x0 < · · · < xn, y0 < · · · < ym. Note thatIρ1(x) = [x − ρ1, x + ρ1], ρ1 > 0,Iρ2(y) = [y − ρ2, y + ρ2], ρ2 > 0,

h1 = infρ1 > 0 : ∀x ∈ [x0, xn], Iρ1(x)

Tx = Ø,

h2 = infρ2 > 0 : ∀x ∈ [y0, ym], Iρ2(y)

Ty = Ø,

Mx = maxx∈[x0,xn]

♯(Ih1

Tx),

My = maxy∈[y0,ym]

♯(Ih2

Ty),

∥Dkf ∥∞ = max|β|=k

supX∈Ω

|(Dβ f )(X)|,

where Tx = x0, . . . , xn, Ty = y0, . . . , ym and ♯(·) denotes the cardinality function. Thus, 2h1 = max16i6n(xi−xi−1), 2h2 =

max16j6m(yj − yj−1), andMxMydefines the maximum number of points of Tx (Ty) contained in the interval Ih1(x)

Ih2(y)

.


Theorem 4. Let any function f (x) ∈ C r+2(Ω) and its corresponding order of derivatives be bounded onΩ . Then for r ∈ N, r > 2,we have

∥(Φr+1f )(x, y) − f (x, y)∥∞ 6 Ohr+21

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

hr+22

+ O

c2h−1

1

+ O

c2h−1

2

+ O

c2h−2

1 hr+12

+ O

c2h−2

2 hr+11

+ O

c4h−2

1 h−12

+ O

c4h−1

1 h−22

+ O

c4h−1

1 h−12

. (26)

Proof.(Φr+1f )(x, y) − f (x, y) =

(Φr+1f )(x, y) − (Φr+1f )(x, y) + (Φr+1f )(x, y) − f (x, y)

6

(Φr+1f )(x, y) − (Φr+1f )(x, y)+ (Φr+1f )(x, y) − f (x, y)

.Let

N1 =

xn − x02h1

+ 1, Qh1(u) = (u − ρ1, u + ρ1], u ∈ [x0, xn], ρ1 > 0,

N2 =

ym − y02h2

+ 1, Qh2(u) = (u − ρ2, u + ρ2], u ∈ [y0, ym], ρ2 > 0,

and

Tk1 = Qh1(x − 2k1h1)

Qh1(x + 2k1h1), k1 = 0, . . . ,N1,

Tk2 = Qh2(y − 2k2h2)

Qh2(y + 2k2h2), k2 = 0, . . . ,N2N1k1=−N1

Qk1(x+2k1h1) is a covering of [x0, xn]withhalf open intervals. Similarly, the setN2

k2=−N2Qk2(y+2k2h2) is a covering

of [y0, ym]. Therefore, for each i ∈ 0, . . . , n (j ∈ 0, . . . ,m) there exists a unique k1 ∈ 0, . . . ,N1 (k2 ∈ 0, . . . ,N2)such that xi ∈ Tk1

yj ∈ Tk2

. When k1 > 2 (k2 > 2), the inequalities hold as follows

(2k1 − 1)h1 6 |x − xi| 6 (2k1 + 1)h1,

(2(k1 − 1) − 1)h1 6 |x − ξi| 6 (2(k1 + 1) + 1)h1, ∀ξi ∈ (xi−1, xi+1),

(2k2 − 1)h2 6 |y − yj| 6 (2k2 + 1)h2,

(2(k2 − 1) − 1)h2 6 |y − ξj| 6 (2(k2 + 1) + 1)h2, ∀ξj ∈ (yj−1, yj+1).

It follows from the definition ofMxMythat

1 6 ♯(T0

Tx) 6 Mx, 1 6 ♯(Tk1

Tx) 6 2Mx, k1 = 1, . . . ,N1,

1 6 ♯(T0

Ty) 6 My, 1 6 ♯(Tk2

Ty) 6 2My, k2 = 1, . . . ,N2.

Let ϕ−1(x) = |x − x−1|, ϕ0(x) = |x − x0|, ϕn(x) = |x − xn|, ϕn+1 = |x − xn+1|, x−1 < x0, xn < xn+1, x ∈ [x0, xn]. Then,(LDf )(x) in [13] can be expressed in the following form

(LDf )(x) =

ni=0

f (xi)αi(x), (27)

where each αi(x) has the form

αi(x) =ϕi+1(x) − ϕi(x)2(xi+1 − xi)

−ϕi(x) − ϕi−1(x)2(xi − xi−1)

, 0 6 i 6 n,

andwhereαi(x) can be considered as a seconddivideddifference ofϕi(x). By virtue of the relationship between the derivativeand divided difference, we obtain for 2 6 i 6 n − 2

αi(x) =14

c2(x − ξi)2 + c2

32(xi+1 − xi−1) 6

14c2(xi+1 − xi−1)|x − ξi|

3 6 c2h1|x − ξi|3, ξi ∈ (xi−1, xi+1).

When k1 > 2, we obtain

αi(x) 6 c2h−21 (2k1 − 3)−2, 0 6 i 6 n. (28)


Similarly, when k2 > 2, we obtain

βj(y) 6 c2h−22 (2k2 − 3)−2, 0 6 j 6 m. (29)

Let C = ∥Dk+2f ∥∞ = max|β|=k+2

supq∈Ω |(Dβ f )(q)|

and let C0, . . . , Cr+2 be positive constants. According to [16], we know(Φr+1f )(x, y) − f (x, y)

=

1(r + 1)!

ni=0

mj=0

αi(x)βj(y) 1

0t(1 − t)rDr+2

(x−xi,y−yj)ftx + (1 − t)xi, ty + (1 − t)yj

dt

=

1(r + 1)!

ni=0

mj=0

αi(x)βj(y)

(r + 2)!(r + 2)!0!

(x − xi)r+2 1

0t(1 − t)r

∂ r+2f∂xr+2

tx + (1 − t)xi, ty + (1 − t)yj

dt

+(r + 2)!

(r + 1)!1!(x − xi)r+1(y − yj)

1

0t(1 − t)r

∂ r+2f∂xr+1∂y

tx + (1 − t)xi, ty + (1 − t)yj

dt

+ · · · +(r + 2)!1!(r + 1)!

(x − xi)(y − yj)r+1 1

0t(1 − t)r

∂ r+2f∂x∂yr+1

tx + (1 − t)xi, ty + (1 − t)yj

dt

+(r + 2)!0!(r + 2)!

(y − yj)r+2 1

0t(1 − t)r

∂ r+2f∂yr+2

tx + (1 − t)xi, ty + (1 − t)yj

dt

6 C0

n

i=0

αi(x)|x − xi|r+2

+ C1

n

i=0

αi(x)|x − xi|r+1

×

mj=0

βj(y)|y − yj|

+ · · · + Cr+1

n

i=0

αi(x)|x − xi|

×

mj=0

βj(y)|y − yj|r+1

+ Cr+2

mj=0

βj(y)|y − yj|r+2

6 C0

1k1=0

xi∈Tk1

αi(x)|x − xi|r+2+

N1k1=2

xi∈Tk1

αi(x)|x − xi|r+2

+ C1

1k1=0

xi∈Tk1

αi(x)|x − xi|r+1+

N1k1=2

xi∈Tk1

αi(x)|x − xi|r+1

×

1k2=0

yj∈Tk2

βj(y)|y − yj| +

N2k2=2

yj∈Tk2

βj(y)|y − yj|

+ · · · + Cr+1

1k1=0

xi∈Tk1

αi(x)|x − xi| +

N1k1=2

xi∈Tk1

|x − xi|

×

1k2=0

yj∈Tk2

βj(y)|y − yj|r+1+

N2k2=2

yj∈Tk2

βj(y)|y − yj|r+1

+ Cr+2

1k2=0

yj∈Tk2

βj(y)|y − yj|r+2+

N2k2=2

yj∈Tk2

βj(y)|y − yj|r+2

6 C0

4Mx(3h1)

r+2+ 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2((2k1 + 1)h1)

r+2

+ C1

4Mx(3h1)

r+1+ 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2((2k1 + 1)h1)

r+1

×

4My(3h2) + 2My

N2k2=2

c2h−22 (2k2 − 3)−2(2k2 + 1)h2


+ · · · + Cr+1

4Mx(3h1) + 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2(2k1 + 1)h1

×

4My(3h2)

r+1+ 2My

N2k2=2

c2h−22 (2k2 − 3)−2 ((2k2 + 1)h2)

r+1

+ Cr+2

4My(3h2)

r+2+ 2My

N2k2=2

c2h−22 (2k2 − 3)−2((2k2 + 1)h2)

r+2

6 O

hr+21

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

hr+22

+ O

c2h−1

1

+ O

c2h−1

1 h2

+ · · · + O

c2h−1

1 hr2

+ O

c2h−1

2

+ O

c2h−1

2 h1

+ · · · + O

c2h−1

2 hr1

+ O

c2h−2

1 hr+12

+ O

c2h−2

2 hr+11

+ O

c4h−1

1 h−22

+ O

c4h−2

1 h−12

+ O

c4h−1

1 h−12

6 O

hr+21

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

hr+22

+ O

c2h−1

1

+ O

c2h−1

2

+ O

c2h−2

1 hr+12

+ O

c2h−2

2 hr+11

+ O

c4h−2

1 h−12

+ O

c4h−1

1 h−22

+ O

c4h−1

1 h−12

. (30)

Let K1 =h1maxh1min

> 1, K2 =h2maxh2min

> 1, K =maxh1max,h2maxminh1min,h2min

> 1, where h1max, h1min define the maximum and the minimumdistance between adjacent knots respectively along the x direction, and h2max, h2min are defined along the y direction in asimilar way. According to [17, sec.2], we have

pi∈Aq

|β|=r+2

|λpi ||aβ

|

β!6

pi∈Aq

|λpi |

|β|=r+2

1/β!

maxpi∈Aqj=1,2

|pij|

6 CαK r+1hr+2−|α|

3 , (i = 0, . . . ,N − 1), (31)

where h3 = maxh1, h2, α ∈ N2, the fixed q = (xi, yj), Aq = q0 − q, . . . , qN−1 − q, pi = qi − q and Cα is a positiveconstant dependent on variables α and the point set Aq. Therefore, we haveD(a1,a2)

Aij(xi,yj)f (xi, yj) −

∂ (a1+a2)f∂xa1ya2

(xi, yj) 6 CC(a1,a2)K

r+1h3r+2−(a1+a2), (32)

where α = (a1, a2) ∈ N2, a1, a2 ∈ N, a1 + a2 = l, l = 2, . . . , r . We also obtainD(l,0)Aij(xi,yj)

f (xi, yj) −∂ lf∂xl

(xi, yj) 6 CC(l,0)K r+1

1 hr+2−l1 ,D(0,l)

Aij(xi,yj)f (xi, yj) −

∂ lf∂yl

(xi, yj) 6 CC(0,l)K r+1

2 hr+2−l2 , (33)

where α = (0, l) ((l, 0)) ∈ N2, l = 1, . . . , r . In the following, we still denote these constants in the upper bound by thesame symbol Cα , but their values may be different at different places. We have(Φr+1f )(x, y) − (Φr+1f )(x, y)

6

ni=0

mj=0

r

l=1

r + 1 − lr + 1

1l!0!

(x − xi)lD(l,0)Aij(xi,yj)

f (xi, yj) −∂ lf∂xl

(xi, yj)

+1

(l − 1)!1!(x − xi)l−1(y − yj)

D(l−1,1)Aij(xi,yj)

f (xi, yj) −∂ lf

∂xl−1∂y(xi, yj)

+ · · · +

11!(l − 1)!

(x − xi)(y − yj)l−1D(1,l−1)Aij(xi,yj)

f (xi, yj) −∂ lf

∂x∂yl−1(xi, yj)

+10!l!

(y − yj)lD(0,l)Aij(xi,yj)

f (xi, yj) −∂ lf∂yl

(xi, yj)

αi(x)βj(y)


6r

r + 1C

1

1!0!C(1,0)K r+1

1 hr+11

ni=0

|x − xi|αi(x) +1

0!1!C(0,1)K r+1

2 hr+12

mj=0

|y − yj|βj(y)

+r − 1r + 1

C

1

2!0!C(2,0)K r+1

1 hr1

ni=0

|x − xi|2αi(x) +1

1!1!C(1,1)K r+1hr

3

ni=0

|x − xi|αi(x)mj=0

|y − yj|βj(y)

+1

0!2!C(0,2)K r+1

2 hr2

mj=0

|y − yj|2βj(y)

+ · · · +1

r + 1C

1

r!0!C(r,0)K r+1

1 h21

ni=0

|x − xi|rαi(x)

+1

(l − 1)!1!C(r−1,1)K r+1h2

3

ni=0

|x − xi|r−1αi(x)mj=0

|y − yj|βj(y)

+ · · · +1

0!r!C(0,r)K r+1

2 h22

mj=0

|y − yj|rβj(y)

6r

r + 1C

11!0!

C(1,0)K r+11 hr+1

1

1k1=0

xi∈Tk1

αi(x)|x − xi| +

N1k1=2

xi∈Tk1

αi(x)|x − xi|

+

10!1!

C(0,1)K r+12 hr+1

2

1k2=0

yj∈Tk2

βj(y)|y − yj| +

N2k2=2

yj∈Tk2

βj(y)|y − yj|

+

r − 1r + 1

C

12!0!

C(2,0)K r+11 hr

1

1k1=0

xi∈Tk1

αi(x)|x − xi|2 +

N1k1=2

xi∈Tk1

αi(x)|x − xi|2

+

11!1!

C(1,1)K r+1hr3

1k1=0

xi∈Tk1

αi(x)|x − xi| +

N1k1=2

xi∈Tk1

αi(x)|x − xi|

×

1k2=0

yj∈Tk2

βj(y)|y − yj| +

N2k2=2

yj∈Tk2

βj(y)|y − yj|

+

10!2!

C(0,2)K r+12 hr

2

1k2=0

yj∈Tk2

βj(y)|y − yj|2 +

N2k2=2

yj∈Tk2

βj(y)|y − yj|2

+ · · · +

1r + 1

C

1r!0!

C(r,0)K r+11 h2

1

1k1=0

xi∈Tk1

αi(x)|x − xi|r +

N1k1=2

xi∈Tk1

αi(x)|x − xi|r

+

1(r − 1)!1!

C(r−1,1)K r+1h23

1k1=0

xi∈Tk1

αi(x)|x − xi|r−1+

N1k1=2

xi∈Tk1

αi(x)|x − xi|r−1

×

1k2=0

yj∈Tk2

βj(y)|y − yj| +

N2k2=2

yj∈Tk2

βj(y)|y − yj|

+ · · · +

10!r!

C(0,r)K r+12 h2

2

1k2=0

yj∈Tk2

βj(y)|y − yj|r +

N2k2=2

yj∈Tk2

βj(y)|y − yj|r

6

rr + 1

C

1

1!0!C(1,0)K r+1

1 hr+11

4Mx(3h1) + 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2(2k1 + 1)h1


+1

0!1!C(0,1)K r+1

2 hr+12

4My(3h2) + 2My

N2k2=2

c2h−22 (2k2 − 3)−2(2k2 + 1)h2

+r − 1r + 1

C

1

2!0!C(2,0)K r+1

1 hr1

4Mx(3h1)

2+ 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2 ((2k1 + 1)h1)

2

+1

1!1!C(1,1)K r+1hr

3

4Mx(3h1) + 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2(2k1 + 1)h1

×

4My(3h2) + 2My

N2k2=2

c2h−22 (2k2 − 3)−2(2k2 + 1)h2

+1

0!2!K r+12 hr

2

4My(3h2)

2+ 2My

N2k2=2

c2h−22 (2k2 − 3)−2 ((2k2 + 1)h2)

2

+ · · · +1

r + 1C

1

r!0!C(r,0)K r+1

1 h21

4Mx(3h1)

r+ 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2 ((2k1 + 1)h1)

r

+1

(r − 1)!1!C(r−1,1)K r+1h2

3

4Mx(3h1)

r−1+ 2Mx

N1k1=2

c2h−21 (2k1 − 3)−2((2k1 + 1)h1)

r−1

×

4My(3h2) + 2My

N2k2=2

c2h−22 (2k2 − 3)−2(2k2 + 1)h2

+ · · · +1

0!r!C(0,r)K r+1

2 h22

4My(3h2)

r+ 2My

N2k2=2

c2h−22 (2k2 − 3)−2 ((2k2 + 1)h2)

r

6 Ohr+21

+ O

c2hr−1

1

+ · · · + O

c2h1

+ O

hr+22

+ O

c2hr−1

2

+ · · · + O

c2h2

+ O

hr−11 (h2h2

3) + hr−21 (h2h3

3 + h22h

23) + · · · + h1(h2hr

3 + h22h

r−13 + · · · + hr−1

2 h23)

+ Oc2h−1

1 (hr−22 (h2

3) + hr−32 (h3

3 + h23) + · · · + h2(hr−1

3 + hr−23 + · · · + h2

3))

+ Oc2h−1

2 (hr−21 (h2

3) + hr−31 (h3

3 + h23) + · · · + h1(hr−1

3 + hr−23 + · · · + h2

3))

+ Oc2h−2

1 (hr−12 h2

3 + hr−22 h3

3 + · · · + h2hr3)

+ Oc2h−2

2 (hr−11 h2

3 + hr−21 h3

3 + · · · + h1hr3)

+ Oc4h−1

1 h−22 (hr−1

3 + hr−23 + · · · + h2

3)

+ Oc4h−2

1 h−12 (hr−1

3 + hr−23 + · · · + h2

3)

+ Oc4h−1

1 h−12 (hr−2

3 + hr−33 + · · · + h2

3)

+ Oc4h−2

1 h−22 hr

3

6 O

hr+21

+ O

c2h1

+ O

hr+22

+ O

c2h2

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

c2h−1

1 h2h23

+ O

c2h−1

2 h1h23

+ O

c2h−2

1 hr+13

+ O

c2h−2

2 hr+13

+ O

c4h−1

1 h−22 h2

3

+ O

c4h−2

1 h−12 h2

3

+ O

c4h−1

1 h−12 h2

3

+ O

c4h−2

1 h−22 hr

3

6 O

hr+21

+ O

hr+22

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

c2h1

+ O

c2h2

+ O

c2h1h2

+ O

c2h−1

1 h32

+ O

c2h−1

2 h31

+ O

c2hr−1

1

+ O

c2hr+1

2 h−21

+ O

c2hr−1

2

+ O

c2hr+1

1 h−22

+ O

c4h1h−2

2

+ O

c4h−1

1

+ O

c4h−2

1 h2

+ O

c4h−1

2


+ Oc4h1h−1

2

+ O

c4h−1

1 h2

+ O

c4hr−2

1 h−22

+ O

c4h−2

1 hr−22

6 O

hr+21

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

hr+22

+ O

c2h−1

1

+ O

c2h−1

2

+ O

c2h−2

1 hr+12

+ O

c2h−2

2 hr+11

+ O

c4h−2

1 h−12

+ O

c4h−1

1 h−22

+ O

c4h−1

1 h−12

. (34)

Combining (30) with (34), we get Theorem 4.

Corollary 1. Let any function f (x, y) ∈ C2(Ω) and its corresponding order of derivatives be bounded on Ω . Then for r ∈ N, r =

0, the quasi-interpolation operator Φ1 satisfies

∥(Φ1f (x, y)) − f (x, y)∥∞ 6 Oh21

+ O

h22

+ O (h1h2) + O

c2h−1

1

+ O

c2h−1

2

+ O

c4h−2

1 h−22

. (35)

Also, the approximation capacity of the quasi-interpolation operator Φ2 is provided in the following Corollary.

Corollary 2. Let any function f (x, y) ∈ C3(Ω) and its corresponding order of derivatives be bounded on Ω . Then for r = 1, wehave

∥Φ2f (x, y) − f (x, y)∥∞ 6 O(h31) + O

h32

+ O

h21h2+ O

h1h2

2

+ O

c2h−1

1

+ O

c2h−1

2

+ O

c2h2

1h−22

+ O

c2h2

2h−21

+ O

c4h−1

1 h−22

+ O

c4h−2

1 h−12

. (36)

Remark 1. Putting r = 0 and r = 1 in Φr+1, we can get Ling’s operatorΦ1 [14] and similar to Feng–Zhou’s operatorΦ2 [15]respectively. Meanwhile, we give a new error estimate (see Corollaries 1 and 2).

Remark 2. For simplicity, we only extend the univariate scheme to 2-dimension using the dimension-splitting technique.We can construct the analogue quasi-interpolation operator for higher dimensions.

Now let h = maxh1, h2, we simplify Theorem 4, then we get Theorem 5.

Theorem 5. If f (x, y) satisfies the conditions of Theorem 4, then there exists constant p > 1 such that

∥(Φr+1f )(x, y) − f (x, y)∥∞ 6 Ohr+2

+ Oc2h−p

+ Oc2hr+1−2p

+ Oc4h−3p

+ Oc4h−2p , h → 0. (37)

Proof. For constant p > 1 s.t. h1 = hp2 or h2 = hp

1, we find

maxh−11 , h−1

2 = h−p.

Therefore, we have for the formula (26)

∥(Φr+1f )(x, y) − f (x, y)∥∞ 6 Ohr+21

+ O

hr+11 h2

+ O

hr1h

22

+ · · · + O

h1hr+1

2

+ O

hr+22

+ O

c2h−1

1

+ O

c2h−1

2

+ O

c2h−2

1 hr+12

+ O

c2h−2

2 hr+11

+ O

c4h−2

1 h−12

+ O

c4h−1

1 h−22

+ O

c4h−1

1 h−12

6 O

hr+2

+ Oc2h−p

+ Oc2hr+1−2p

+ Oc4h−3p

+ Oc4h−2p , h → 0.

Remark 3. The approximation order of the quasi-interpolation operator Φr+1 in Theorem 5 can reach up to r + 2 providedc2h−p

= O(hr+2) or c2hr+1−2p= O(hr+2).

5. Numerical examples

In this section, we observe the polynomial reproducing property and approximation capacity of the quasi-interpolationoperator Φr+1, and verify that the relationship between the approximation order and the degree of polynomial repro-ducing is very close. On the other hand, we observe the approximation capacity and the computational cost of the quasi-interpolation operator Φr+1, and believe that our operator is desirable. Quasi-interpolation operators Φ3 and Φ4 involvedin the following are Φr+1 as r = 2 and 3, respectively. We consider here the Ling’s quasi-interpolation operator Φ1 definedas (15) and Feng–Zhou’s quasi-interpolation operator Φ2 introduced in [15].

In the following, for sake of simplification, we suppose that sampled points (xi, yj)06i6n;06j6m satisfy xi+1 − xi = h1,yj+1 − yj = h2, where h1 and h2 are the step-size along x and y directions respectively. All errors are computed using thesame grid of the above sampled points used for computing the function values that are quasi-interpolated.


Table 1We consider the test function f (x, y) = x3 + y3 , h1 = h2 = h.

c 0.01 0.02 0.05 0.1 0.2h 0.1 0.1 0.1 0.1 0.1

∥(Φ1f )(x, y) − f (x, y)∥∞ 7.534 × 10−3 9.528 × 10−3 1.030 × 10−2 5.346 × 10−2 1.253 × 10−1

∥(Φ2f )(x, y) − f (x, y)∥∞ 3.673 × 10−3 3.972 × 10−3 4.875 × 10−3 9.727 × 10−3 1.285 × 10−2

∥(Φ3f )(x, y) − f (x, y)∥∞ 3.828 × 10−15 4.002 × 10−15 4.328 × 10−15 7.929 × 10−15 9.311× 10−15

∥(Φ4f )(x, y) − f (x, y)∥∞ 3.214 × 10−16 3.455 × 10−16 7.426 × 10−16 5.374 × 10−15 7.632× 10−15

Table 2We consider the test function f (x, y) = x4 + y4 , h1 = h2 = h.

c 0.01 0.02 0.05 0.1 0.2h 0.1 0.1 0.1 0.1 0.1

∥(Φ1f )(x, y) − f (x, y)∥∞ 1.216 × 10−2 1.528 × 10−2 1.879 × 10−2 9.528 × 10−2 1.982 × 10−1

∥(Φ2f )(x, y) − f (x, y)∥∞ 3.499 × 10−3 3.961 × 10−3 4.854 × 10−3 1.183 × 10−2 2.377 × 10−2

∥(Φ3f )(x, y) − f (x, y)∥∞ 3.750 × 10−4 7.907 × 10−4 1.489 × 10−3 4.508 × 10−3 9.727 × 10−3

∥(Φ4f )(x, y) − f (x, y)∥∞ 4.132 × 10−16 4.613 × 10−16 8.032 × 10−16 9.324 × 10−15 9.852× 10−15

Table 3The test function f (x, y) = x3 + y3, h1 = h2 .

c 0.1 0.1 0.05 0.05 0.01h1 0.1 0.2 0.1 0.2 0.2h2 0.2 0.1 0.2 0.1 0.1

∥(Φ1f )(x, y) − f (x, y)∥∞ 4.052 × 10−2 4.053 × 10−2 3.082 × 10−2 3.127 × 10−2 2.421 × 10−2

∥(Φ2f )(x, y) − f (x, y)∥∞ 2.234 × 10−2 2.237 × 10−2 1.294 × 10−2 1.299 × 10−2 7.834 × 10−3

∥(Φ3f )(x, y) − f (x, y)∥∞ 4.426 × 10−16 4.447 × 10−16 4.089 × 10−16 4.205 × 10−16 8.727× 10−17

∥(Φ4f )(x, y) − f (x, y)∥∞ 3.812 × 10−16 3.920 × 10−16 2.933 × 10−16 2.945 × 10−16 8.220× 10−17

Table 4The test function f (x, y) = x4 + y4, h1 = h2 .

c 0.1 0.1 0.01 0.01 0.001h1 0.1 0.2 0.1 0.2 0.1h2 0.2 0.1 0.2 0.1 0.2

∥(Φ1f )(x, y) − f (x, y)∥∞ 4.065 × 10−2 4.706 × 10−2 3.703 × 10−2 3.528 × 10−2 3.183 × 10−2

∥(Φ2f )(x, y) − f (x, y)∥∞ 2.234 × 10−2 2.216 × 10−2 1.575 × 10−2 1.520 × 10−2 8.091 × 10−3

∥(Φ3f )(x, y) − f (x, y)∥∞ 1.290 × 10−2 1.292 × 10−2 9.202 × 10−3 9.307 × 10−3 2.408 × 10−3

∥(Φ4f )(x, y) − f (x, y)∥∞ 5.826 × 10−16 5.913 × 10−16 5.028 × 10−16 5.092 × 10−16 9.347× 10−17

5.1. Verification of the polynomial reproduction property

First of all, let f (x, y) = x3+y3, x4+y4 defined on [0, 1]2 be the test functions, thenwe choose different shape parametersc , h1, and h2.

When h1 = h2 and h1 = h2, we compare the convergence capacity of the quasi-interpolation operators Φ3 and Φ4 withthat of Φ1 and Φ2, and verify that the operators Φ3 and Φ4 reproduce the cubic polynomial in Tables 1 and 3. From Tables 2and 4, we verify that the operator Φ4 meets the quartic polynomial reproducing property. However, the operator Φ3 cannotreproduce the quartic polynomial. Obviously, the operator Φr+1 can reproduce all polynomials of degree 6 r + 1.

5.2. Verification of approximation capacity

Let f1(x, y) = x5 + y5 and f2(x, y) =1.25+cos(5.4y)6+6(3x−1)2

defined on [0, 1]2 be the test functions, then we choose different shapeparameters c , h1, h2 to compute the ∥(Φ3f )(x, y) − f (x, y)∥∞ and ∥(Φ4f )(x, y) − f (x, y)∥∞. From Tables 5–10, we verifythat the maximum convergence rates of our quasi-interpolation operators Φ3 and Φ4 are 4 and 5, respectively. Let shapeparameter c become bigger and let h1 and h2 be constant, then the convergence rates of the operatorsΦ3 andΦ4 reduce. Letshape parameters c , h1, and h2 become smaller, then the approximation effects will be more desirable. So, we conclude thatthe approximation capacity of quasi-interpolation operators Φ3 and Φ4 is dependent on the shape parameters c , h1, and h2.

FromTable 5, we see that the convergence rates rh ofΦ3f1 andΦ3f2 can attain 3 as h1 = h2 = h = 0.2, 0.1, 0.05, 0.02, p =

1, c2h−p= O(hr+2), r = 2. Furthermore, the convergence rates rh of Φ3f1 and Φ3f2 can attain 4 as h = 0.01, c = 0.000001,

c < h2.


Table 5The convergence rate of the quasi-interpolation operator Φr+1 as h1 = h2 = h, p = 1, c2h−p

= O(hr+2), r = 2.

c 0.04 0.01 0.0025 0.0004 0.000001h 0.2 0.1 0.05 0.02 0.01

∥(Φ3f1)(x, y) − f1(x, y)∥∞ 6.511 × 10−3 5.943 × 10−4 3.276 × 10−5 1.641 × 10−6 9.638 × 10−9

rh 3.128 3.226 3.447 3.405 4.008

∥(Φ3f2)(x, y) − f2(x, y)∥∞ 6.844 × 10−3 7.674 × 10−4 4.679 × 10−5 2.361 × 10−6 9.954 × 10−9

rh 3.097 3.115 3.328 3.312 4.001

Table 6The convergence rate of the quasi-interpolation operator Φr+1 as h1 = h2 = h, p = 1, c2h−p

= O(hr+2), r = 3.

c 0.01 0.003 0.0005 0.00005 0.0000008h 0.2 0.1 0.05 0.02 0.01

∥(Φ4f1)(x, y) − f1(x, y)∥∞ 6.796 × 10−4 2.780 × 10−5 1.561 × 10−6 1.097 × 10−8 9.036× 10−11

rh 4.532 4.556 4.463 4.685 5.022

∥(Φ4f2)(x, y) − f2(x, y)∥∞ 8.126 × 10−4 3.873 × 10−5 1.858 × 10−6 1.585 × 10−8 9.908× 10−11

rh 4.421 4.412 4.405 4.591 5.002

Table 7The convergence rate of the quasi-interpolation operator Φr+1 as h1 = h2, h = maxh1, h2, p = (ln h − ln 2)/ ln h, c2h−p

= O(hr+2), r = 2.

c 0.04 0.01 0.0016 0.0009 0.0000004h 0.2 0.1 0.05 0.025 0.02h1 0.1 0.05 0.025 0.0125 0.01h2 0.2 0.1 0.05 0.025 0.02

∥(Φ3f1)(x, y) − f1(x, y)∥∞ 6.745 × 10−3 6.982 × 10−4 2.593 × 10−5 1.853 × 10−6 1.069 × 10−7

rh 3.106 3.156 3.525 3.764 4.103

∥(Φ3f2)(x, y) − f2(x, y)∥∞ 7.194 × 10−3 8.356 × 10−4 3.992 × 10−5 4.660 × 10−6 1.078 × 10−7

rh 3.066 3.078 3.381 3.501 4.101

Table 8The convergence rate of the quasi-interpolation operator Φr+1 as h1 = h2, h = maxh1, h2, p = (ln h − ln 2)/ ln h, c2h−p

= O(hr+2), r = 3.

c 0.01 0.003 0.0005 0.0001 0.00000001h 0.2 0.1 0.05 0.025 0.02h1 0.1 0.05 0.025 0.0125 0.01h2 0.2 0.1 0.05 0.025 0.02

∥(Φ4f1)(x, y) − f1(x, y)∥∞ 7.413 × 10−4 2.618 × 10−5 1.643 × 10−6 1.978 × 10−7 2.057 × 10−9

rh 4.478 4.582 4.446 4.402 5.113

∥(Φ4f2)(x, y) − f2(x, y)∥∞ 8.391 × 10−4 3.690 × 10−5 1.891 × 10−6 2.092 × 10−7 2.114 × 10−9

rh 4.401 4.433 4.399 4.386 5.106

From Table 6, we see that the convergence rates rh ofΦ4f1 andΦ4f2 can attain 4 as h1 = h2 = h = 0.2, 0.1, 0.05, 0.02, p =

1, c2h−p= O(hr+2), r = 3. Furthermore, the convergence rates rh ofΦ4f1 andΦ4f2 can attain 5 as h = 0.01, c = 0.0000008,

c < h5/2.FromTable 7,we see that the convergence rates rh ofΦ3f1 andΦ3f2 can attain 3 as h1 = 0.1, 0.05, 0.025, 0.0125, h2 = 2h1,

h = maxh1, h2, 1 < p = (ln h − ln 2)/ ln h < 2, c2h−p= O(hr+2), r = 2. Furthermore, the convergence rates rh of Φ3f1

and Φ3f2 can attain 4 as h1 = 0.01, h2 = 0.02, h = 0.02, c = 0.0000004, c < h2.FromTable 8,we see that the convergence rates rh ofΦ4f1 andΦ4f2 can attain 4 as h1 = 0.1, 0.05, 0.025, 0.0125, h2 = 2h1,

h = maxh1, h2, 1 < p = (ln h − ln 2)/ ln h < 2, c2h−p= O(hr+2), r = 3. Furthermore, the convergence rates rh of Φ4f1

and Φ4f2 can attain 5 as h1 = 0.01, h2 = 0.02, h = 0.02, c = 0.00000001, c < h5/2.From Table 9, we see that the convergence rates rh ofΦ3f1 andΦ3f2 can attain 3 as h1 = h3

2, h2 = 0.2, 0.1, 0.05, 0.025, h =

maxh1, h2, p = 3, c2h−p= c2hr+1−2p

= O(hr+2), r = 2. Furthermore, the convergence rates rh of Φ3f1 and Φ3f2 can attain4 as h1 = 8.0 × 10−6, h2 = 0.02, h = 0.02, c = 0.0000004, c < h5/2.

FromTable 10,we see that the convergence rates rh ofΦ4f1 andΦ4f2 can attain 4 as h1 = h32, h2 = 0.2, 0.1, 0.05, 0.025, h =

maxh1, h2, p = 3, c2h−p= O(hr+2), r = 3. Furthermore, the convergence rates rh of Φ4f1 and Φ4f2 can attain 5 as h1 =

8.0 × 10−6, h2 = 0.02, h = 0.02, c = 0.00000008, c < h5/2.Thus, the convergence rate of the operatorΦr+1 can reach up to r+2 provided c2h−p

= O(hr+2) or c2hr+1−2p= O(hr+2).

Moreover, we conclude that there is a very close relation between the convergence rate of our quasi-interpolation operatorand the degree of polynomial reproducing.


Table 9The convergence rate of quasi-interpolation operator Φr+1 as h1 = h2, h = maxh1, h2, p = 3, c2h−p

= c2hr+1−2p= O(hr+2), r = 2..

c 0.01 0.003 0.0005 0.0001 0.0000001h 0.2 0.1 0.05 0.025 0.02h1 8.0 × 10−3 1.0 × 10−3 1.25 × 10−4 1.5625 × 10−5 8.0 × 10−6

h2 0.2 0.1 0.05 0.025 0.02

∥(Φ3f1)(x, y) − f1(x, y)∥∞ 2.347 × 10−3 1.774 × 10−4 1.685 × 10−5 3.270 × 10−6 1.474 × 10−7

rh 3.762 3.751 3.669 3.603 4.021

∥(Φ3f2)(x, y) − f2(x, y)∥∞ 2.674 × 10−3 2.128 × 10−4 1.767 × 10−5 3.282 × 10−6 1.412 × 10−7

rh 3.681 3.672 3.653 3.601 4.032

Table 10The convergence rate of quasi-interpolation operator Φr+1 as h1 = h2, h = maxh1, h2, p = 3, c2h−p

= O(hr+2), r = 3..

c 0.008 0.001 0.0001 0.00002 0.00000008h 0.2 0.1 0.05 0.025 0.02h1 8.0 × 10−3 1.0 × 10−3 1.25 × 10−4 1.5625 × 10−5 8.0 × 10−6

h2 0.2 0.1 0.05 0.025 0.02

∥(Φ4f1)(x, y) − f1(x, y)∥∞ 5.339 × 10−4 1.901 × 10−5 7.585 × 10−7 6.133 × 10−8 2.982 × 10−9

rh 4.682 4,721 4.704 4.736 5.018

∥(Φ4f2)(x, y) − f2(x, y)∥∞ 5.870 × 10−4 2.065 × 10−5 8.199 × 10−7 6.909 × 10−8 2.726 × 10−9

rh 4.623 4.685 4.678 4.702 5.041

Table 11Approximation capacity of each quasi-interpolation operator approximating f (x, y) =

64 − 81((x − 0.5)2 + (y − 0.5)2)/9 − 0.5, h1 = h2 = h.

c 0.1 0.01 0.001 0.1 0.01 0.001h 0.2 0.2 0.2 0.04 0.04 0.04

∥(Φ1f )(x, y) − f (x, y)∥∞ 2.991 × 10−1 7.262 × 10−2 5.344 × 10−2 6.825 × 10−2 4.821 × 10−2 1.830 × 10−2

∥(Φ2f )(x, y) − f (x, y)∥∞ 5.981 × 10−2 1.573 × 10−2 1.142 × 10−2 2.593 × 10−3 2.124 × 10−3 7.780 × 10−4

∥(Φ3f )(x, y) − f (x, y)∥∞ 1.383 × 10−2 3.692 × 10−3 2.678 × 10−3 1.386 × 10−4 7.839 × 10−5 3.004 × 10−5

∥(Φ4f )(x, y) − f (x, y)∥∞ 2.552 × 10−3 7.631 × 10−4 5.356 × 10−4 4.719 × 10−6 3.301 × 10−6 1.036 × 10−6

Table 12Approximation capacity of each quasi-interpolation operator approximating f (x, y) =

64 − 81((x − 0.5)2 + (y − 0.5)2)/9 − 0.5, h1 = h2 .

c 0.1 0.01 0.001 0.1 0.01 0.001h1 0.2 0.2 0.2 0.04 0.04 0.04h2 0.04 0.04 0.04 0.2 0.2 0.2

∥(Φ1f )(x, y) − f (x, y)∥∞ 2.755 × 10−1 6.462 × 10−2 4.931 × 10−2 2.775 × 10−1 2.342 × 10−1 1.426 × 10−1

∥(Φ2f )(x, y) − f (x, y)∥∞ 5.590 × 10−2 1.475 × 10−2 1.086 × 10−2 5.608 × 10−2 4.923 × 10−2 3.389 × 10−2

∥(Φ3f )(x, y) − f (x, y)∥∞ 1.255 × 10−2 2.997 × 10−3 2.593 × 10−3 1.276 × 10−2 9.202 × 10−3 5.836 × 10−3

∥(Φ4f )(x, y) − f (x, y)∥∞ 2.280 × 10−3 5.994 × 10−4 4.331 × 10−4 2.393 × 10−3 1.873 × 10−3 1.152 × 10−3

Table 13Computational time(s) of each quasi-interpolation operator approximating f (x, y) =

64 − 81((x − 0.5)2 + (y − 0.5)2)/9 − 0.5, h1 = h2 = h.

c 0.001 0.001 0.001 0.001 0.001 0.001h 0.2 0.1 0.05 0.025 0.0125 0.00625

(Φ1f )(x, y) 1.421 × 10−3 1.663 × 10−3 2.514 × 10−3 6.637 × 10−3 4.612 × 10−3 1.513 × 10−2

(Φ2f )(x, y) 1.753 × 10−3 1.740 × 10−3 3.643 × 10−3 7.058 × 10−3 3.235 × 10−3 1.029 × 10−2

(Φ3f )(x, y) 3.201 × 10−3 4.625 × 10−3 8.549 × 10−3 1.073 × 10−2 3.557 × 10−2 8.192 × 10−2

(Φ4f )(x, y) 8.012 × 10−3 9.213 × 10−3 8.057 × 10−2 1.424 × 10−2 6.194 × 10−2 1.458 × 10−1

5.3. Comparison of computational cost

To verify the feasibility of our method, we suppose that f (x, y) =64 − 81((x − 0.5)2 + (y − 0.5)2)/9 − 0.5 defined

on [0, 1]2 is the test function. Then we choose the same shape parameters c , h1 = h2 = h, and h1 = h2 to compare theapproximation capacity of our quasi-interpolation operators Φ3 and Φ4 with that of Φ1 and Φ2. The comparison results areshown in Tables 11 and 12. Meanwhile, we choose the above approximation function f (x), shape parameters c , h1 and h2 tocompare the computational cost of our quasi-interpolation operatorsΦ3 andΦ4 with that ofΦ1 andΦ2. The laptopwhichweuse to compute the error has the following properties: processor type is Intel Core 2 Duo T7250, CPU frequency is 2.00 GHz,memory capacity is 2.00 GB. In Tables 13 and 14, we observe the computational time(s) of each quasi-interpolation operator.


Table 14Computational time(s) of each quasi-interpolation operator approximating f (x, y) =

64 − 81((x − 0.5)2 + (y − 0.5)2)/9 − 0.5, h1 = h2 .

c 0.001 0.001 0.001 0.001 0.001 0.001h1 0.2 0.2 0.1 0.1 0.05 0.05h2 0.1 0.05 0.2 0.05 0.2 0.1

(Φ1f )(x, y) 2.246 × 10−3 1.896 × 10−3 2.209 × 10−3 2.495 × 10−3 3.021 × 10−3 5.197 × 10−3

(Φ2f )(x, y) 2.674 × 10−3 3.012 × 10−3 3.059 × 10−3 3.268 × 10−3 4.135 × 10−3 7.031 × 10−3

(Φ3f )(x, y) 4.325 × 10−3 4.661 × 10−3 7.956 × 10−3 8.315 × 10−3 9.226 × 10−3 1.035 × 10−2

(Φ4f )(x, y) 8.327 × 10−3 9.235 × 10−3 1.134 × 10−2 1.603 × 10−2 2.715 × 10−2 5.034 × 10−2

From Tables 11 and 12, we find that for the same c , h1 = h2 = h, and h1 = h2, our quasi-interpolation operators Φ3 andΦ4 provide a high degree of accuracy.

From Tables 13 and 14, we find that for the same c , h1 = h2 = h, and h1 = h2, our quasi-interpolation operators Φ3 andΦ4 cost a little longer time than Φ1 and Φ2. Therefore, we believe that it is really a good idea to move up to the high-orderscheme.

6. Conclusions

In this paper, by combining Ling’s multivariate quasi-interpolation operator Φ1 for dimension-splitting multiquadricbasis function with the modified Taylor polynomials of degree r , we first construct a family of multivariate multiquadricquasi-interpolation operators Φr+1. The operator Φr+1 with special r reproduces polynomials of degree at most r + 1, andreaches up to higher convergence rate. Finally, we propose the quasi-interpolation operatorΦr+1 which does not require thederivatives of function approximated at each knot, and possesses the same convergence rate with the operator Φr+1. Thus,the latter ismore reasonable for people in various applications. The numerical examples show that the convergence capacityof our operator is closely related to the degree of polynomial reproducing. However, we just put forward the dimension-splitting multiquadric quasi-interpolation operator for gridded data.

In our future work, we intend to use the operator Φr+1 to solve partial differential equations, and construct the multi-variate multiquadric quasi-interpolation operator for completely scattered data.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant Nos. 61373003 and 11271041) and theNatural Science Foundation of Jilin Province, China (Grant No. 2013101062JC).

Appendix. The expression of our multiquadric quasi-interpolation operator Φ2

Based on the quasi-interpolation scheme Φ1 in [14] reproducing linear polynomials, we give a new quasi-interpolationoperator Φ2 in (16) provided r = 1.

(Φ2f )(x, y) =

ni=0

mj=0

f (xi, yj)αi(x)βj(y) +

ni=0

mj=0

12

∂ f∂x

(xi, yj) +12

∂ f∂y

(xi, yj)

αi(x)βj(y)

=

ni=0

mj=0

f (xi, yj) +

12

∂ f∂x

(xi, yj) +12

∂ f∂y

(xi, yj)

αi(x)βj(y).

We substitute ∂ f∂x (xi, yj),

∂ f∂y (xi, yj) (i = 0, . . . , n, j = 0, . . . ,m) in (Φ2f )(x, y) by D(1,0)

6 f (xi, yj), D(0,1)6 f (xi, yj) (i = 0, . . . ,

n, j = 0, . . . ,m) respectively. Thus, by Theorem 2 we get the modification (Φ2f )(x, y) of (Φ2f )(x, y) as follow:

Φ2f (x, y) =

ni=0

mj=0

f (xi, yj)αi(x)βj(y)

+

0i=0

m−2j=0

12D(1,0)6 f (xi, yj) +

12D(0,1)6 f (xi, yj)

αi(x)βj(y)

+

0i=0

mj=m−1

12D(1,0)6 f (xi, yj) +

12D(0,1)6 f (xi, yj)

αi(x)βj(y)

+

n−1i=1

m−2j=0

12D(1,0)6 f (xi, yj) +

12D(0,1)6 f (xi, yj)

αi(x)βj(y)


+

n−1i=1

mj=m−1

12D(1,0)6 f (xi, yj) +

12D(0,1)6 f (xi, yj)

αi(x)βj(y)

+

ni=n

1j=0

12D(1,0)6 f (xi, yj) +

12D(0,1)6 f (xi, yj)

αi(x)βj(y)

+

ni=n

mj=2

12D(1,0)6 f (xi, yj) +

12D(0,1)6 f (xi, yj)

αi(x)βj(y),

where

D(1,0)6 f (xi, yj) = −

xi+1 + xi+2 − 2xi(xi+2 − xi)(xi+1 − xi)

f (xi, yj) −xi+2 − xi

(xi − xi+1)(xi+2 − xi+1)f (xi+1, fj) + 0f (xi+1, yj+1)

−xi+1 − xi

(xi − xi+2)(xi+1 − xi+2)f (xi+2, yj) + 0f (xi+2, yj+1) + 0f (xi+2, yj+2) (i = 0, 0 6 j 6 m − 2),

D(1,0)6 f (xi, yj) = −

xi+1 − xi(xi+1 − xi−1)(xi − xi−1)

f (xi−1, yj) −xi−1 + xi+1 − 2xi

(xi−1 − xi)(xi+1 − xi)f (xi, yj) + 0f (xi, yj+1)

−xi−1 − xi

(xi−1 − xi+1)(xi − xi+1)f (xi+1, yj)

+ 0f (xi+1, yj+1) + 0f (xi+1, yj+2) (1 6 i 6 n − 1, 0 6 j 6 m − 2),

D(1,0)6 f (xi, yj) = −

xi−1 − xi(xi − xi−2)(xi−1 − xi−2)

f (xi−2, yj) −xi−2 − xi

(xi−2 − xi−1)(xi − xi−1)f (xi−1, yj) + 0f (xi−1, yj+1)

−xi−1 + xi−2 − 2xi

(xi−2 − xi)(xi−1 − xi)f (xi, yj) + 0f (xi, yj+1) + 0f (xi, yj+2) (i = n, 0 6 j 6 1),

D(1,0)6 f (xi, yj) = −

xi−1 + xi−2 − 2xi(xi−2 − xi)(xi−1 − xi)

f (xi, yj) −xi−2 − xi

(xi−2 − xi−1)(xi − xi−1)f (xi−1, yj) + 0f (xi−1, yj−1)

−xi−1 − xi

(xi − xi−2)(xi−1 − xi−2)f (xi−2, yj) + 0f (xi−2, yj−1) + 0f (xi−2, yj−2) (i = n, 2 6 j 6 m),

D(1,0)6 f (xi, yj) = −

xi−1 − xi(xi−1 − xi+1)(xi − xi+1)

f (xi+1, yj) −xi−1 + xi+1 − 2xi

(xi+1 − xi)(xi−1 − xi)f (xi, yj) + 0f (xi, yj−1)

−xi+1 − xi

(xi+1 − xi−1)(xi − xi−1)f (xi−1, yj)

+ 0f (xi−1, yj−1) + 0f (xi−1, yj−2) (1 6 i 6 n − 1, m − 1 6 j 6 m),

D(1,0)6 f (xi, yj) = −

xi − xi+1

(xi − xi+2)(xi+1 − xi+2)f (xi+2, yj) −

xi+2 − xi(xi+2 − xi+1)(xi − xi+1)

f (xi+1, yj) + 0f (xi+1, yj−1)

−xi+1 + xi+2 − 2xi

(xi+2 − xi)(xi+1 − xi)f (xi, yj) + 0f (xi, yj−1) + 0f (xi, yj−2) (i = 0, m − 1 6 j 6 m),

D(0,1)6 f (xi, yj) = 0f (xi+2, yj+2) + 0f (xi+1, yj+2) + 0f (xi+1, yj+1) +

yj+1 − yj(yj+1 − yj+2)(yj − yj+2)

f (xi, yj+2)

−yj+2 − yj

(yj+2 − yj+1)(yj − yj+1)f (xi, yj+1) −

yj+1 + yj+2 − 2yj(yj+2 − yj)(yj+1 − yj)

f (xi, yj) (i = 0, 0 6 j 6 m − 2),

D(0,1)6 f (xi, yj) = 0f (xi−1, yj) −

1yj+1 − yj

f (xi, yj) −1

yj − yj+1f (xi, yj+1) +

1yj+2 − yj

f (xi+1, yj)

+1

yj+2 − yj+1f (xi+1, yj+1) −

yj+1 − yj(yj − yj+2)(yj+1 − yj+2)

f (xi+1, yj+2) (1 6 i 6 n − 1, 0 6 j 6 m − 2),

D(0,1)6 f (xi, yj) = 0f (xi−2, yj) + 0f (xi−1, yj) + 0f (xi−1, yj+1) +

yj+1 + yj+2 − 2yj(yj+1 − yj)(yj+2 − yj)

f (xi, yj)

−yj+2 − yj

(yj − yj+1)(yj+2 − yj+1)f (xi, yj+1) −

yj+1 − yj(yj − yj+2)(yj+1 − yj+2)

f (xi, yj+2) (i = n, 0 6 j 6 1),

D(0,1)6 f (xi, yj) = 0f (xi−2, yj−2) + 0f (xi−1, yj−2) + 0f (xi−1, yj−1) +

yj−2 − yj(yj−1 − yj−2)(yj − yj−2)

f (xi, yj−2)

−yj−2 − yj

(yj−2 − yj−1)(yj − yj−1)f (xi, yj−1) −

yj−2 + yj−1 − 2yj(yj−2 − yj)(yj−1 − yj)

f (xi, yj) (i = n, 2 6 j 6 m),


D(0,1)6 f (xi, yj) = 0f (xi+1, yj) −

1yj−1 − yj

f (xi, yj) −1

yj − yj−1f (xi, yj−1) +

1yj−2 − yj

f (xi−1, yj)

+1

yj−2 − yj−1f (xi−1, yj−1) −

yj−1 − yj(yj − yj−2)(yj−1 − yj−2)

f (xi−1, yj−2) (1 6 i 6 n − 1, m − 1 6 j 6 m),

D(0,1)6 f (xi, yj) = 0f (xi+2, yj) + 0f (xi+1, yj) + 0f (xi+1, yj−1) +

yj−2 + yj−1 − 2yj(yj−2 − yj)(yj−1 − yj)

f (xi, yj)

−yj−2 − yj

(yj−2 − yj−1)(yj − yj−1)f (xi, yj−1) −

yj−1 − yj(yj−1 − yj−2)(yj − yj−2)

f (xi, yj−2) (i = 0, m − 1 6 j 6 m).

Similarly, we can get the expression of the quasi-interpolation operator Φr+1 provided r ∈ N, r > 2.

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