Estimating error bounds for quaternary subdivision schemes

Estimating error bounds for quaternary subdivision schemes

J. Math. Anal. Appl. 358 (2009) 159–167 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com...

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J. Math. Anal. Appl. 358 (2009) 159–167

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Estimating error bounds for quaternary subdivision schemes ✩ Sadiq Hashmi, Ghulam Mustafa ∗ Department of Mathematics, The Islamia University of Bahawalpur, Pakistan

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 27 February 2008 Available online 3 May 2009 Submitted by M.D. Gunzburger

This paper deals with the error estimation techniques of quaternary subdivision schemes. The estimation is expressed in terms of initial control point sequences and constants. It is independent of subdivision process and parametrization therefore its evaluation is straightforward. © 2009 Published by Elsevier Inc.

Keywords: Subdivision surfaces Control polygon Error bound Subdivision depth Forward differences

1. Introduction Subdivision scheme in the field of Computer Aided Geometric Design is an iterative process that defines a smooth curve or surface as the limit of a sequence of successive refinements applied to an initial control polygon. At each refinement level, new points are added into the existing polygon and the original points remain existed or discarded in all subsequent sequences of control polygons. The number of points inserted at level k + 1 between two consecutive points from level k is called arity of the scheme. In the case when number of points inserted are 2, 3 and 4 the subdivision schemes are called binary, ternary and quaternary respectively. For more details on general n-ary subdivision schemes, we may refer to the thesis of N. Aspert [1]. The most important issues appear in many applications as rendering, intersection, testing and design is how well-refined polygon approximates the limiting curve/surface. The existing methods [2,8,10–12] for computing the error bounds are all based on the parameterizations while methods [13–15] are based on eigenanalysis. [3–6,16] and Huang et al. [7] have estimated error bounds for binary, ternary, tensor product binary volumetric and Catmull–Clark subdivision schemes, in terms of the maximal first/second order differences of the initial control point sequence and constants that depends on the subdivision mask. To best of our knowledge no answer has been found in case of quaternary subdivision schemes. It is forcible and proficient that to increase the arity of the scheme from binary to ternary then to quaternary may result in schemes with higher smoothness and approximation order but smaller support [9]. Quaternary scheme performs quicker topological refinement than binary and ternary one. So we get a quicker generation of curves and surfaces by using quaternary one. It is also noticed that higher arities schemes have slightly lower computational cost than lower arities schemes. These properties of quaternary scheme motivate us to find error bounds in this case. Our error bounds estimation technique is generalization of Mustafa et al. [4,5]. The rest of the paper is arranged in the following way.



*

This work is supported by the Indigenous PhD Scholarship Scheme of Higher Education Commission (HEC), Pakistan. Corresponding author. E-mail addresses: [email protected] (S. Hashmi), [email protected] (G. Mustafa).

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.04.050

©

2009 Published by Elsevier Inc.

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S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

Refinement step of scheme (2.1) Fig. 1. Solid lines show coarse polygon whereas doted lines are refined polygon. It can be obtain by subdividing each line segment into four line segments by using four rules of (2.1).

Error bounds for quaternary subdivision curve and surfaces are presented in Sections 2 and 3 respectively. Future research directions are given in Section 4. To maintain the presentation of our paper as simple as possible for readers we offer typical mathematical proof of preliminary results in Appendices A and B. 2. Error bounds for quaternary curve schemes Given a sequence of control points pki ∈ R N , i ∈ Z, N  2, where the upper index k  0 indicates the subdivision level. A quaternary subdivision process [1] is defined by

pk4i++1α =

m 

aα , j pki+ j ,

j =0

α = 0, 1 , 2 , 3 ,

(2.1)

where m > 0 and m 

a α , j = 1,

α = 0, 1, 2, 3.

(2.2)

j =0

The set of coefficients {aα , j ,

α = 0, 1, 2, 3}mj=0 is called subdivision mask. Given initial values p 0i ∈ RN , i ∈ Z. Then in the

limit k → ∞, the process defines an infinite set of points in R N . The sequence of control points { pki } is related, in a natural

way, with the diadic mesh points t ki = i /4k , i ∈ Z. The process (2.1) then defines a scheme whereby pk4i+1 replaces the value

k+1 k+1 1 k k pki at the mesh point t 4i = t ki while pk4i++11 , pk4i++12 and pk4i++13 are inserted at the new mesh points t 4i +1 = 4 (3t i + t i +1 ), k+1 k+1 1 k 1 k k k t 4i +2 = 2 (t i + t i +1 ) and t 4i +3 = 4 (t i + 3t i +1 ) respectively. Labelling of old and new points is shown in Fig. 1 which illustrates subdivision scheme (2.1) and notation (2.7).

Remark 2.1. By taking the following mask in (2.1) we get a particular four point quaternary approximating subdivision scheme (4PQASS) for 0 < w < 1.5

a0,0 =

7 32 15

7



ω,

64 5

a0,1 =

29

+

13

ω,

64 64 57 7

a0,2 =

5 16



5 64

ω,

a0,3 =

1 64



1

64 7

ω,

49 1 3 − ω, a1,1 = + ω, a1,2 = + ω, a1,3 = − ω, 128 64 128 64 128 64 128 64 a2,0 = a1,3 , a2,1 = a1,2 , a2,2 = a1,1 , a2,3 = a1,0 , a3,0 = a0,3 , a3,1 = a0,2 , a3,2 = a0,1 and a3,3 = a0,0 . a1,0 =

Here we assume

  m      δ1 = max  bβ, j , β = 0, 1, 2, 3 ,   β

(2.3)

j =0

where

⎧ j  ⎪ ⎪ ⎪ ⎪ bβ, j = (aβ,t − aβ+1,t ), ⎪ ⎨ t =0

n −2 ⎪  ⎪ ⎪ ⎪ b 3 , j = a 0, j − bβ, j . ⎪ ⎩

β=0

β = 0, 1 , 2 , (2.4)

S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

Also

 m−1      γ = max  a˜ α , j , α = 0, 1, 2, 3 , α  

161

(2.5)

j =0

where

⎧ m  ⎪ α ⎪ ⎪ ˜ a = aα ,t − , ⎪ α , 0 ⎪ ⎨ n t =1

m  ⎪ ⎪ ⎪ ⎪ a˜ α , j = aα ,t , ⎪ ⎩

α = 0, 1 , 2 , 3 .

(2.6)

j  1,

t = j +1

Further suppose





χ = max p 0i+1 − p 0i .

(2.7)

i

Here we present error bounds between quaternary subdivision curves and their control polygons. Theorem 2.1. Given initial control polygon p 0i = p i , i ∈ Z, let the values pki , k  1, be defined recursively by subdivision process (2.1)

together with (2.2). Suppose P k is the piecewise linear interpolant to the values pki and P ∞ is the limit curve of the process (2.1). If δ1 < 1 then the error bounds between limit curve and its control polygon after k-fold subdivision are

k

P − P ∞



(δ1 )k , γχ ∞ 1 − δ1

(2.8)

where δ1 , γ and χ are defined by (2.3), (2.5) and (2.7) respectively. Proof. Let .∞ denote the maximum norm. Since the maximum difference between P k+1 and P k is attained at a point on the (k + 1)th mesh, we have

k +1

P − P k



where

  max N αk , α = 0, 1, 2, 3 ,





k +1 

1 k k k

Nα =

p 4i +α − 4 (4 − α ) p i + α p i +1 , From (2.1) and (2.2) for

pk4i++1α −

(2.9)

α

1 4

α = 0, 1 , 2 , 3 .

(2.10)

α = 0, 1, 2, 3 we obtain

−1  m   (4 − α ) pki + α pki+1 = a˜ α , j pki+ j +α +1 − pki+ j +α ,

(2.11)

j =0

where a˜ α , j is defined by (2.6). From (2.1), (2.2) and induction on m for β = 0, 1, 2, 3 we get

pk4i +β+1 − pk4i +β =

m  j =0





−1 bβ, j pki + − pki +−1j , j +1

(2.12)

where bβ, j is defined by (2.4). It follows from (2.10) and (2.11) that

m−1   



  k Nα  max a˜ α , j  max pki+1 − pki . α   i

(2.13)

j =0

Using (2.12) recursively gives

max pk i

i +1



pki



m k  



  bβ, j  max p 0i +1 − p 0i .  max  β  i 

(2.14)

j =0

By (2.9), (2.13) and (2.14) we have

k +1

P − P k



where δ1 , γ and proof. 2

 γ χ (δ1 )k ,

(2.15)

χ are defined by (2.3), (2.5) and (2.7) respectively. Triangle inequality yields (2.8). This completes the

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S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

Corollary 2.2. Given p 0i = p i , i ∈ Z, let the values pki , k  0, be defined recursively by 4PQASS (defined in Remark 2.1). Suppose P k is

the piecewise linear interpolant to the values pki and P ∞ is the limit curve of 4PQASS. Then by Theorem 2.1

k

P − P ∞



γχ ∞

(δ1 )k , 1 − δ1

where δ1 = 1/4, γ = 1.125 and the best possible choice of χ = 0.1. Remark 2.2. Theorem 1 in [4] and Theorem 2.1 in [5] are predominantly designed to estimate the error bounds of binary (i.e. each edge of control polygon divided into two sub-edges) and ternary (i.e. each edge divided into three sub-edges) curve schemes respectively. But in quaternary schemes each edge of control polygon is divided into four sub-edges therefore it is not feasible to evaluate error bounds of quaternary curve schemes (particularly 4PQASS as well) by [4, Theorem 1] and [5, Theorem 2.1]. In view of the above, contribution of our Theorem 2.1 will be very accommodating and defensible. 3. Error bounds for quaternary surface schemes In this section, first we define quaternary scheme and fix some notation for clear reading. Then we will prove some preliminary results on error bounds for quaternary subdivision surfaces. 3.1. Definition and notations Given a sequence of control points pki, j ∈ R N , i , j ∈ Z, N  2, where the upper index k  0 indicates the subdivision level. A tensor product of quaternary subdivision process (2.1) is defined by

pk4i++1α ,4 j +β =

m m   r =0 s =0

aα ,r aβ,s pki+r , j +s ,

α , β = 0, 1, 2, 3,

(3.1)

where aα ,r satisfies (2.2). Given initial values p 0i , j ∈ R N , i , j ∈ Z, then in the limit k → ∞, the process defines an infinite set of points in RN . The sequence of values { pki, j } is related, in a natural way, with the diadic mesh points ( ik , k ), 4 4 i , j ∈ Z. The process (3.1) then defines a scheme whereby pk4i+,41 j , pk4i++14,4 j , pk4i+,41 j +4 and pk4i++14,4 j +4 replace the values j j j +1 j +1 pki, j , pki+1, j , pki, j +1 and pki+1, j +1 at the mesh points ( ik , k ), ( i +k1 , k ), ( ik , k ) and ( i +k1 , k ) respectively. The values j

4

4

4

4

4

4

4

4

pk4i++11,4 j , pk4i++12,4 j , pk4i++13,4 j , pk4i+,41 j +1 , pk4i++11,4 j +1 , pk4i++12,4 j +1 , pk4i++13,4 j +1 , pk4i+,41 j +2 , pk4i++11,4 j +2 , pk4i++12,4 j +2 , pk4i++13,4 j +2 , pk4i+,41 j +3 ,

1 +2 +3 pk4i++11,4 j +3 , pk4i++12,4 j +3 and pk4i++13,4 j +3 are inserted at the new mesh points ( 3ik+ , ), ( 3i , ), ( 3i , ), ( 4k3i+1 , 4 +1 4k+1 4k+1 4k+1 4k+1 4k+1 1 3 j +2 3 j +3 3i +2 3 j +1 3i +3 3 j +1 3i 3i +1 3 j +2 3i +2 3 j +2 3i +3 3 j +2 3i 3i +1 ( 3ik++11 , 3 kj + +1 ), ( k+1 , k+1 ), ( k+1 , k+1 ), ( k+1 , k+1 ), ( k+1 , k+1 ), ( k+1 , k+1 ), ( k+1 , k+1 ), ( k+1 , k+1 ), ( k+1 , 3j

4

4

4

4

4

4

4

4

4

4

4

4

3j

4

4

3j

4

4

4

3 j +1 ), 4k+1 3 j +3 ), 4k+1

+2 3 j +3 +3 3 j +3 ( 3i , ) and ( 3i , ) respectively. Labelling of old and new points is shown in Fig. 2 which illustrates subdivi4k+1 4k+1 4k+1 4k+1

sion scheme (3.1) and notations (3.7). Let us suppose

  m  m      δ2 = max  aα ,s bβ,r , α , β = 0, 1, 2, 3 ,  α ,β  s =0

(3.2)

r =0

where bβ,r is defined by (2.4). Suppose further for α , β = 0, 1, 2, 3,



k +1 

1 k k k k

M α ,β = pni +α ,nj +β − 2 (n − α )(n − β) p i , j + α (n − β) p i +1, j + (n − α )β p i , j +1 + α β p i +1, j +1

, n     m m −1     α (4 − β)     + ηα1 ,β = aβ,0 aα ,t − a a˜ α ,s ,   β,0     16 t =1 s =1  m  m −1 m    β    ηα2 ,β =  aβ,t −  +  aα ,r a˜ β,s ,   4  t =1 r =0 s =1  m   m m m −1     α β     3 ηα ,β =  aα ,t aβ,t − aβ,t a˜ α ,s , +   16   k

t =1

t =1

where a˜ α ,s is defined by (2.6).

t =1

s =1

(3.3)

(3.4)

(3.5)

(3.6)

S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

163

Subdivision step of scheme (3.1) Fig. 2. Solid lines show one face of coarse polygon whereas doted lines are refined polygon. It can be obtain by subdividing one face into sixteen new faces by using sixteen rules of scheme (3.1).

Assume ki, j ,t , t = 1, 2, 3 are forward difference operators along the mesh directions defined by

⎧ k = pki+1, j − pki, j , ⎪ ⎪ ⎨ ki , j ,1



i , j ,2 = pki, j +1 − pki, j , χt = max 0i , j ,t , i, j ⎪ ⎪ ⎩ k i , j ,3 = pki+1, j +1 − pki, j +1 .

t = 1, 2, 3,

(3.7)

Furthermore suppose



 3   t  ϑ = max (χt ) ηα ,β , α , β = 0, 1, 2, 3 . α ,β

(3.8)

t =1

3.2. Main result In this paragraph we are going to estimate error bounds for quaternary subdivision surfaces. To show this, first we will prove the following two important lemmas. Lemma 3.1. Given initial control polygon p 0i , j = p i , j , i , j ∈ Z, let the values pki, j , k  1, be defined recursively by subdivision process (3.1) together with (2.2). Then









max ki, j ,t  (δ2 )k max 0i , j ,t , i, j

i, j

(3.9)

where δ2 , ki, j ,t , t = 1, 2, 3, are defined by (3.2) and (3.7) respectively. Proof. Proof is shown in Appendix A.

2

The proof of the following lemma is shown in Appendix B. Lemma 3.2. Given initial control polygon p 0i , j = p i , j , i , j ∈ Z, let the values pki, j , k  1, be defined recursively by subdivision process (3.1) together with (2.2). Then

M kα ,β  (δ2 )k

3    (χt ) ηαt ,β , t =1

t where δ2 , M kα ,β , ηα ,β , χt , α , β = 0, 1, 2, 3, are defined by (3.2)–(3.7).

(3.10)

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S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

Here is our main result to estimate error bounds for quaternary subdivision surfaces. Theorem 3.3. Given initial control polygon p 0i , j = p i , j , i , j ∈ Z, let the values pki, j , k  1, be defined recursively by subdivision process

(3.1) together with (2.2). Suppose P k is the piecewise linear interpolant to the values pki, j and P ∞ is the limit surface of the subdivision process (3.1). If δ2 < 1, then the error bounds between the limit surface and its control polygon after k-fold subdivision are

k

P − P ∞



ϑ ∞

(δ2 )k , 1 − δ2

(3.11)

where δ2 and ϑ are defined by (3.2) and (3.8) respectively. Proof. Let .∞ denote the uniform norm. Since the maximum difference between P k+1 and P k is attained at a point on the (k + 1)th mesh, we have

k +1

P − P k



  max M kα ,β , α , β = 0, 1, 2, 3 ,

(3.12)

α ,β

where M kα ,β is defined by (3.3). Using (3.10) and (3.12) we get



 3   t   max (χt ) ηα ,β , α , β = 0, 1, 2, 3 (δ2 )k . ∞

k +1

P − P k

α ,β

t =1

This implies

k +1

P − P k



 ϑ(δ2 )k ,

where δ2 and ϑ are defined by (3.2) and (3.8) respectively. By triangle inequality we get (3.11). This completes the proof.

2

Corollary 3.4. Given p 0i , j = p i , j , i , j ∈ Z, let the values pki, j , k  0, be defined recursively by tensor product form of 4PQASS. Suppose P k is the piecewise linear interpolant to the values pki, j and P ∞ is the limit surface of tensor product 4PQASS. Then by Theorem 3.3

k

P − P ∞

ϑ ∞

(δ2 )k , 1 − δ2

where δ2 = 1/4, ϑ = 3.47763 at w = 3/4 and χt = 0.1, t = 1, 2, 3. Remark 3.1. Theorem 7 in [4] and Theorem 3.2 in [5] are particularly designed to evaluate the error bounds of tensor product binary (i.e. each face of control polygon divided into four sub-faces) and tensor product ternary (i.e. each face divided into nine sub-faces) surface schemes respectively. But in tensor product quaternary schemes each face of control polygon is divided into sixteen sub-faces therefore it is not possible to evaluate error bounds of tensor product quaternary schemes (particularly tensor product 4PQASS as well) by [4, Theorem 7] and [5, Theorem 3.2]. In view of the above, contribution of our Theorem 3.3 will be very helpful and justified. 4. Future work The authors are looking, as a future work, to extend the estimating error bound technique for n-ary subdivision schemes. This could be quite useful in advance in many engineering applications such as surface/surface intersection, surface rendering and so on. Appendix A. Proof of Lemma 3.1 Proof. From (2.2), (3.1) and using similar approach as we did for (2.12), we obtain

pk4i +α +1,4 j +β pk4i +α +1,4 j +4

− −

pk4i +α ,4 j +β pk4i +α ,4 j +4

=

m 

 aβ,s

s =0

=

m  s =0

 a 0, s

m  r =0

m  r =0



−1 bα ,r pki + r +1 , j + s





−1 bα ,r pki + r +1 , j + s +1

−1 pki + r , j +s





 (A.1)

,

−1 pki + r , j + s +1



 ,

(A.2)

S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

pk4i +α ,4 j +β+1 − pk4i +α ,4 j +β = pk4i +4,4 j +β+1



pk4i +4,4 j +β

=



m 

aα ,r

r =0

m 



a0,r

r =0

m  s =0

m  s =0



−1 k −1 bβ,s pki + r , j +s+1 − p i +r , j +s



−1 bβ,s pki + r +1 , j + s +1





165

 (A.3)

,

−1 pki + r +1 , j + s



 (A.4)

,

where bβ,r is defined by (2.4) and α , β = 0, 1, 2, 3. Now using (A.1) recursively together with notations defined by (3.7), we get

m k  m   

k



  max i , j ,1  max aα ,s bβ,r  max 0i , j ,1 .  α ,β  i, j i, j s =0

r =0

From (3.2) and using the above inequality we get









max ki, j ,1  (δ2 )k max 0i , j ,1 . i, j

i, j

Again using (A.2) recursively and by utilizing (3.2) and (3.7) we have









max ki, j ,3  (δ2 )k max 0i , j ,3 . i, j

i, j

Similarly, using (A.3) and (A.4) recursively together with (3.2) and (3.7)









max ki, j ,2  (δ2 )k max 0i , j ,2 . i, j

i, j

This completes the proof.

2

Appendix B. Proof of Lemma 3.2 Proof. From (2.2) and (3.1) we get

pk4i+,41 j



pki, j

=

m 

 a0,r

r =0

m  s =0



a0,s pki+r , j +s



pki, j



 (B.1)

.

Since m  s =0











a0,s pki+r , j +s − pki, j = a0,0 pki+r , j − pki, j + a0,1 pki+r , j +1 − pki, j



  + a0,2 pki+r , j +2 − pki+r , j +1 + pki+r , j +1 − pki, j   + a0,3 pki+r , j +3 − pki+r , j +2 + pki+r , j +2 − pki+r , j +1 + pki+r , j +1 − pki, j + · · ·   + a0,m pki+r , j +m − pki+r , j +m−1 + pki+r , j +m−1 − · · · − pki+r , j +1 + pki+r , j +1 − pki, j ,

therefore m  s =0









a0,s pki+r , j +s − pki, j = a0,0 pki+r , j − pki, j +

m  t =1





a0,t pki+r , j +1 − pki, j +

m −1  s =1





a˜ 0,s pki+r , j +s+1 − pki+r , j +s ,

where a˜ 0,s is defined by (2.6). Taking summation on both sides of above equation we get m 

 a0,r

r =0

= a0,0

m  s =0

m  r =0

Since



a0,s pki+r , j +s



a0,r pki+r , j





pki, j

pki, j





+



m  t =1

 a0,t

m  r =0



a0,r pki+r , j +1



pki, j



 +

m  r =0

a0,r

 m −1  s =1



a˜ 0,s pki+r , j +s+1



pki+r , j +s



 .

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S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

m  r =0











a0,r pki+r , j − pki, j = a0,1 pki+1, j − pki, j + a0,2 pki+2, j − pki+1, j + pki+1, j − pki, j



  + a0,3 pki+3, j − pki+2, j + pki+2, j − pki+1, j + pki+1, j − pki, j + · · ·   + a0,m pki+m, j − pki+m−1, j + pki+m−1, j − · · · + pki+2, j − pki+1, j + pki+1, j − pki, j ,

therefore m  r =0





a0,r pki+r , j − pki, j =

m  t =1



m −1 



a0,t pki+1, j − pki, j +

s =1





a˜ 0,s pki+s+1, j − pki+s, j .

Similarly m  r =0









a0,r pki+r , j +1 − pki, j = a0,0 pki, j +1 − pki, j +

m  t =1





a0,t pki+1, j +1 − pki, j +

m −1  s =1





a˜ 0,s pki+s+1, j +1 − pki+s, j +1 .

Substituting these summations into (B.1) then by (3.3) we obtain

 M k0,0

= a0,0

m 





a0,t

t =1

+

m  t =1

+

m 

pki+1, j





pki, j



 +

a0,r

r =0



s =1

2 a0,t



t =1

a0,t pki, j +1 − pki, j + a0,0

 m −1 

m 

m −1  s =1

pki+1, j +1 − pki, j +1





a˜ 0,s pki+s+1, j − pki+s, j +



a˜ 0,s pki+r , j +s+1 − pki+r , j +s







m 

a0,t

t =1

−1 m  s =1



a˜ 0,s pki+s+1, j +1 − pki+s, j +1



(B.2)

.

α , β = 0, 1, 2, 3 we obtain  m   m m      α (4 − β)  k αβ  k k k M α ,β = aβ,0 aα ,t − p i +1 , j − p i , j + aα ,t aβ,t − p i +1, j +1 − pki, j +1 16 16

Similarly from (2.2), (3.1) and (3.3) for



 +

t =1

m 

aβ,t −

t =1

+

m 

aβ,t

t =1

t =1



m −1  s =1

β  4



pki, j +1 − pki, j + aβ,0



a˜ α ,s pki+s+1, j +1



m −1  s =1

pki+s, j +1



+

where a˜ α ,s is defined by (2.6). Using (3.2), (3.9), (B.2) and (B.3) for

M kα ,β

t =1



a˜ α ,s pki+s+1, j − pki+s, j m  r =0

aα ,r

 m −1  s =1



a˜ β,s pki+r , j +s+1

α , β = 0, 1, 2, 3 we get     m m −1    



α (4 − β)     k  (δ2 ) aα ,t − a˜ α ,s  max 0i , j ,1

aβ,0  + aβ,0     i, j 16 t =1 s =1  m   m  −1 m   



β     +  aβ,t −  +  aα ,r a˜ β,s  max 0i , j ,2

  i, j 4  t =1 r =0 s =1     m  m −1 m m    

0

α β    



+  aα ,t aβ,t − aβ,t a˜ α ,s  max i , j ,3 . +   i, j 16   t =1

t =1

Utilizing notations (3.4)–(3.7)

M kα ,β  (δ2 )k

3    (χt ) ηαt ,β . t =1

This completes the proof.

2

t =1

s =1





pki+r , j +s



 ,

(B.3)

S. Hashmi, G. Mustafa / J. Math. Anal. Appl. 358 (2009) 159–167

167

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