A family of bivariate rational Bernstein operators

Applied Mathematics and Computation 258 (2015) 162–171

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A family of bivariate rational Bernstein operators Chun-Gang Zhu ⇑, Bao-Yu Xia School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

a r t i c l e

i n f o

a b s t r a c t Rational Bernstein operators are widely used in approximation theory and geometric modeling but in general they do not reproduce linear polynomials. Based on the work of P. Pitßul and P. Sablonnière, we construct a new family of triangular and tensor product bivariate rational Bernstein operators, which are positive and reproduce the linear polynomials. The main result is a proof of convergence of the bivariate rational Bernstein operators defined on the square or triangle. Ó 2015 Elsevier Inc. All rights reserved.

Keywords: Rational approximants Bernstein operators Reproduction of polynomials

1. Introduction Let Bni ðxÞ ¼

  n i x ð1  xÞni be the Bernstein basis of degree n for i ¼ 0; 1; . . . ; n, and let i n   X i n Bi ðxÞ f n i¼0

Bn f ðxÞ ¼

ð1Þ

be the Bernstein operator defined for f 2 Cð½0; 1Þ, the space of all continuous functions on the unit interval ½0; 1. Bernstein operators were firstly presented by S.N. Bernstein in [13] for proving the Weierstrass Approximation Theorem. Bernstein operators have been investigated by many mathematicians, see [1–5,9,10,12], and they have several applications in Computer Aided Geometric Design, see [1,6,11,12]. For the history of theory and applications of Bernstein operators, we refer to Farouki’s paper [12]. It is standard fact that Bn mr ¼ mr for r ¼ 0; 1, where mr is defined as mr ðxÞ ¼ xr for x 2 ½0; 1 and r 2 N0 , that is, the operator Bn reproduces linear polynomials. The rational Bernstein operator is defined by the expression

Qn f ðxÞ ¼

n X

~ if x

i¼0

  n i Bi ðxÞ ; n Q n ðxÞ

x 2 ½0; 1;

ð2Þ

e i ; i ¼ 0; 1; . . . ; n, are are positive numbers and Q n ðxÞ 2 Pn is a given polynomial of degree less or equal to n which is where x strictly positive over ½0; 1. It is easy to achieve that Qn reproduces the constant function m0 by requiring that

Q n ðxÞ ¼

n X

~ i Bni ðxÞ: x

i¼0

⇑ Corresponding author. E-mail addresses: [email protected] (C.-G. Zhu), [email protected] (B.-Y. Xia). http://dx.doi.org/10.1016/j.amc.2015.02.010 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.

ð3Þ

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

163

In contrast to the classical case (1), the rational Bernstein operators are less studied, in particular no general convergence result is to be expected to hold. Simple examples show that Qn does not reproduce linear polynomials under the general assumptions (2) and (3). However, Pitßul and Sablonnière (see [1]) introduced a special class of univariate rational Bernstein operators of the form

Pn

i¼0

LRn f ðxÞ ¼

x i f ðxiðnÞ ÞBni ðxÞ Q n1 ðxÞ

;

x 2 ½0; 1

ð4Þ ðnÞ

 i and the abscissae xi for i ¼ 0; 1; . . . ; n are chosen in a way such that LRn reproduces linear where the positive numbers x polynomials. In order to achieve this, it is required that Q n1 ðxÞ 2 Pn1 , that is

Q n1 ðxÞ ¼

n1 X

xi Bn1 ðxÞ i

ð5Þ

i¼0

with given positive numbers xi ; i ¼ 0; 1; . . . ; n  1. Then the rational Bernstein operator LRn defined by (4) reproduces linear  i and xðnÞ polynomials if x are given by the formulae i



i n

 i xi ; n

 i ¼ xi1 þ 1  x

ðnÞ

xi

¼

i xi1 ; i n x

1 6 i 6 n  1: ðnÞ

 i are positive numbers but it should be noted that the abscissae xi are only increasing in the variable i under addiClearly x tional assumptions for the coefficients xi for i ¼ 0; 1; . . . ; n, see e.g. [1,9]. Pitßul and Sablonnière (see [1]) proved many interesting facts of this kind of rational Bernstein operator (4), among them a convergence result for LRn under the assumption that

Q n1 ðxÞ ¼ Bn1 uðxÞ

ð6Þ

for a given function u 2 Cð½0; 1Þ. Recently, H. Render (see [10]) removed the special requirement (6) and formulated convergence results under the assumption that

Dn ¼

   ðnÞ ðnÞ  sup xiþ1  xi 

i¼0;...;n1

converges to 0. The tensor product rational Bernstein operator over D ¼ ½0; 1  ½0; 1 is generally defined by

Pn Pm

n m i j ~ j¼0 xi;j f ð ; ÞBi ðxÞBj ðyÞ Pn Pm n mn ; m ~ i¼0 j¼0 xi;j Bi ðxÞBj ðyÞ

i¼0

Qn;m f ðx; yÞ ¼

ðx; yÞ 2 D:

ð7Þ

It is easy to see that the operator Qn;m f reproduces the constants. But in general it does not reproduce the linear polynomials x and y. The aim of the paper is to extend the results of Pitßul and Sablonnière to the rational Bernstein operators defined on the square D ¼ ½0; 1  ½0; 1 and on the triangle. In the following sections we will restrict the discussion to the case of a square which is technically easier and omit the proofs to the case of triangle and provided only the essential formulae. To achieve the reproduction of linear polynomials, we assumed that the polynomial

Q n1;m1 ðx; yÞ ¼

n1 X m1 X

xi;j Bn1 ðxÞBm1 ðyÞ i j

i¼0 j¼0

has positive coefficients xi;j and, in analogy to the construction of Pitßul and Sablonnière (see [1]), we set

 i;j ¼ x

       i j i j i j i j 1 xi1;j1 þ 1  xi1;j þ 1  xi;j1 þ 1  xi;j nm n m n m n m

ð8Þ

for 0 6 i 6 n; 0 6 j 6 m, where

x1;1 ¼ xi;1 ¼ x1;j ¼ 0; 0 6 i 6 n; 0 6 j 6 m: We construct a family of tensor product rational Bernstein operators Rn;m over D ¼ ½0; 1  ½0; 1 for f 2 CðDÞ by

Rn;m f ðx; yÞ ¼

n X m X

 i;j f ðxi;j ; yi;j Þ x

i¼0 j¼0

Bni ðxÞBm Pn;m ðx; yÞ j ðyÞ ; ¼ Q n1;m1 ðx; yÞ Q n1;m1 ðx; yÞ

ð9Þ

164

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

where xi;j and yi;j for 0 6 i 6 n; 0 6 j 6 m are given by

xi;j ¼

ijxi1;j1 þ iðm  jÞxi1;j ;  i;j x

yi;j ¼

ijxi1;j1 þ ðn  iÞjxi;j1 :  i;j x

ð10Þ

These assumptions ensure that the operator Rn;m reproduces constants and linear polynomials x and y. The abscissae i # xi;j and j # yi;j are increasing if and only if the coefficients xi;j satisfy the additional inequality (11). For all large n and m, this condition holds under the assumption that



xi;j ¼ u

 i j ; ; n1 m1

0 6 i 6 n  1;

06j6m1

for some u 2 C 2 ðDÞ. Under this assumption for xi;j , the operator Rn;m f converges uniformly to f 2 CðDÞ for n; m ! 1. 2. Tensor product rational Bernstein operators Theorem 2.1. The rational Bernstein operator Rn;m defined in (9) reproduces constants. Proof. For f ðx; yÞ ¼ 1, we only need to prove

Pn Pm m  n Pn;m ðx; yÞ i¼0 j¼0 xi;j Bi ðxÞBj ðyÞ ¼ Pn1 Pm1 ¼ 1: n1 Q n1;m1 ðx; yÞ ðxÞBm1 ðyÞ j i¼0 j¼0 xi;j Bi m1 Using the identities ð1  yÞBm1 ðyÞ ¼ ð1  mj ÞBm ðyÞ ¼ jþ1 Bm j j ðyÞ and yBj jþ1 ðyÞ one obtains m

  j jþ1 m m1 m1 Bm ðyÞ þ B ðyÞ: Bm1 ðyÞ ¼ B ðyÞ þ yB ðyÞ ¼ 1  j j j m j m jþ1 Using this identity, the denominator Q n1;m1 ðx; yÞ of Rn;m can be written in the form

Q n1;m1 ðx; yÞ ¼

n1 m 1 X X



xi;j 1 

i¼0 j¼0

 n1 X m1 X j j þ 1 n1 Bn1 ðxÞBm B ðxÞBm xi;j j ðyÞ þ jþ1 ðyÞ: m i m i i¼0 j¼0

Rearranging the summation in second sum for j ¼ 1; . . . ; m, it is easy to see that

Q n1;m1 ðx; yÞ ¼

n1 m 1 X X



xi;j 1 

i¼0 j¼0

 n1 X m X j j Bn1 ðxÞBm xi;j1 Bn1 ðxÞBm i j ðyÞ þ i j ðyÞ: m m i¼0 j¼0

By using similar arguments for Bn1 ðxÞ one arrives i

Q n1;m1 ðx; yÞ ¼ ¼

        n X m  X i j i j i j i j 1 xi1;j1 þ 1  xi1;j þ 1  xi;j1 þ 1  xi;j Bni ðxÞBmj ðyÞ n m n m n m n m i¼0 j¼0 n X m X

 i;j Bni ðxÞBm x j ðyÞ ¼ P n;m ðx; yÞ:

i¼0 j¼0

This completes the proof.

h

Corollary 2.1. The rational Bernstein operators Rn;m can be written as

Pn Pm Pn Pm n m n m   i¼0 j¼0 xi;j f ðxi ; yj ÞBi ðxÞBj ðyÞ i¼0 j¼0 xi;j f ðxi ; yj ÞBi ðxÞBj ðyÞ P P Rn;m f ðx; yÞ ¼ Pn1 Pm1 ¼ n m m n1  n ðxÞBm1 ðyÞ i¼0 j¼0 xi;j Bi ðxÞBj ðyÞ i¼0 j¼0 xi;j Bi j by the degree elevation of denominator. Theorem 2.2. The rational Bernstein operator Rn;m reproduces linear polynomials. Proof. For f ðx; yÞ ¼ x, we only need to prove

Rn;m x ¼

Pn Pm n m  Pn;m ðx; yÞ i¼0 j¼0 xi;j xi Bi ðxÞBj ðyÞ ¼ Pn1 Pm1 ¼ x: n1 m1 Q n1;m1 ðx; yÞ ðxÞBj ðyÞ i¼0 j¼0 xi;j Bi

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

165

By the identities xBn1 ðxÞ ¼ iþ1 Bniþ1 ðxÞ, we have i n

xQ n1;m1 ðx; yÞ ¼

m1 X j¼0

! ! m1 n X X iþ1 n i n m1 n B ðxÞ Bj ðyÞ ¼ xi;j xi1;j Bi ðxÞ þ 0Bi ðxÞ Bm1 ðyÞ: j n iþ1 n i¼0 j¼0 i¼1

n1 X

n m1 X i n X ðyÞ ¼ Bi ðxÞ xi1;j Bm1 j n i¼0 j¼0

Rearranging the summation and by (8), it is easy to see that

!  m1 n n m  X X X X i n m1 i n i n X j j m1 m1 B Bm ðxÞ x B ðyÞ ¼ x ðxÞ ðyÞ ¼ ðxÞ x þ ð1  Þ x B B B i1;j i1;j i1;j1 i1;j i j i j i j ðyÞ i¼0 n n n m m j¼0 j¼0 i¼0 i¼0 j¼0

Xn

¼

n X m X

 i;j x

i¼0 j¼0

n X m X ijxi1;j1 þ iðm  jÞxi1;j n  i;j xi Bni ðxÞBm Bi ðxÞBm x j ðyÞ ¼ j ðyÞ ¼ P n;m ðx; yÞ  i;j x i¼0 j¼0

and this leads to Rn;m x ¼ x. For f ðx; yÞ ¼ y, we can prove the result in the same way. h

  For applications, the points xi;j ; yi;j should be monotone as in the case of the points ni ; mj , ie, xi;j < xiþ1;j ; yi;j < yi;jþ1 , for 1 6 i 6 n  2; 1 6 j 6 m  2. By (10), xi;j < xiþ1;j implies

ijxi1;j1 þ iðm  jÞxi1;j ði þ 1Þjxi;j1 þ ði þ 1Þðm  jÞxi;j < :  i;j  iþ1;j x x After simplification, this leads to the following inequality holds

"  #    2 2  j j j  j iðn  i  1Þ xi1;j1 xiþ1;j1 þ 1 xi1;j1 xiþ1;j þ xi1;j xiþ1;j1 þ 1  xi1;j xiþ1;j m m m m "  #  2 2 j j j j < ði þ 1Þðn  iÞ x2i;j1 þ 2 ð1  Þxi;j1 xi;j þ 1  x2i;j : m m m m

ð11Þ

Let the weights be defined by a given positive continuous function u 2 C 2 ðDÞ as

xi;j ¼ u



 i j ; ; n1 m1

0 6 i 6 n  1;

0 6 j 6 m  1;

ð12Þ

j 1 1 i m and let Dx ¼ n1 ; Dy ¼ m1 ; x ¼ n1 ; y ¼ m1 . We multiply both sides of inequality (11) by ½ðm1Þðn1Þ 2 , then we have

xð1  xÞ½y2 uðx  Dx; y  DyÞuðx þ Dx; y  DyÞ þ yð1  y þ DyÞðuðx  Dx; y  DyÞuðx þ Dx; yÞ þ uðx  Dx; yÞuðx þ Dx; y  DyÞÞ þ ð1  y þ DyÞ2 uðx  Dx; yÞuðx þ Dx; yÞ < ðx þ DxÞð1  x þ DxÞ½y2 u2 ðx; y  DyÞ þ 2yð1  y þ DyÞuðx; y  DyÞuðx; yÞ þ ð1  y þ DyÞ2 u2 ðx; yÞ:

ð13Þ

Suppose u is sufficiently smooth and by using Taylor’s expansions, we obtain

uðx  Dx; y  DyÞ ¼ u  Dx

! @u @u 1 @2u @2u @2u 3  Dy þ ðDxÞ2 2  2DxDy þ ðDyÞ2 2 þ Oðh Þ; @x @y 2 @x @x@y @y

" !  2 # @u @2u @u @2u @u @u 2 þ DxDy u uðx  Dx; y  DyÞuðx þ Dx; yÞ ¼ u  Dy u þ ðDxÞ u 2   @y @x @x @x@y @x @y 2

1 @2u 3 þ ðDyÞ2 u 2 þ Oðh Þ; 2 @y and

" !  2 # @u @2u @u @2u @u @u 2  DxDy u uðx  Dx; yÞuðx þ Dx; y  DyÞ ¼ u  Dy u þ ðDxÞ u 2   @y @x @x @x@y @x @y 2

1 @2u 3 þ ðDyÞ2 u 2 þ Oðh Þ: 2 @y

ð14Þ

166

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

We have the following inequality after simplifying (13)

" xð1  xÞDx2 u

 2 # @2u @u 3  ð1 þ DyÞ2 þ Oðh Þ < Dxð1 þ DxÞ½yuðx; y  DyÞ þ ð1  y þ DyÞuðx; yÞ2 : @x2 @x

By the continuity of u and uðx; y  DyÞ ! uðx; yÞ, when Dy ! 0, we have

"

u2 þ Dx u2  xð1  xÞ u

@2u @u 2 ð Þ 2 @x @x

!# 2

þ Oðh Þ > 0:

ð15Þ

Similarly, the result for yi;j < yi;jþ1 can be deduced

"

u2 þ Dy u2  yð1  yÞ u

  2 !# @2u @u 2 þ Oðh Þ > 0:  @y2 @y

ð16Þ

Corollary 2.2. The points ðxi;j ; yi;j Þ induced by (12) are monotone in vertical and horizontal directions if u 2 C 2 ðDÞ and n; m are large enough. Let the tensor product Bernstein operator for f 2 CðDÞ be

Bn;m f ðx; yÞ ¼

 n X m  X i j Bni ðxÞBm f ; j ðyÞ; ðx; yÞ 2 D n m i¼0 j¼0

and xi;j be defined by a positive continuous function u 2 CðDÞ by (12). From the convergence property of operator Bn;m in the multivariate approximation theory ([7]), the denominator of Rn;m defined by (9) can be written as Bn1;m1 u and satisfies Bn1;m1 u ! u when n; m ! 1. Therefore, we have the following result. Theorem 2.3. For the given positive continuous function u 2 CðDÞ and the weights defined by (12), the sequence of rational operators Rn;m f converges uniformly to f when n; m ! 1 for f 2 CðDÞ.  n  m Proof. Let un;m ðx; yÞ ¼ u n1 x; m1 y . By (8) and (10), we have

   i j i j ¼ un;m ; ; ; n1 m1 n m



j j i1 j1 i1 j i m un;m n ; m þ ð1  mÞun;m n ; m ; xi;j ¼  i;j n x



i i1 j1 i i j1 j n un;m n ; m þ ð1  nÞun;m n ; m yi;j ¼ ;  i;j m x

xi;j ¼ u



where

 i;j ¼ x

        i j i1 j1 i j i1 j i j i j1 þ þ ð1  Þ un;m un;m ; 1 un;m ; ; nm n m n m n m n m n m      i j i j 1 un;m ; : þ 1 n m n m

It is convenient to define a new function



1 n

 n;m ðx; yÞ ¼ xyun;m x  ; y  x

     1 1 1 þ xð1  yÞun;m x  ; y þ ð1  xÞyun;m x; y  þ ð1  xÞð1  yÞun;m ðx; yÞ; m n m

and

 n;m ðx; yÞf wn;m ðx; yÞ ¼ x

       ! xyun;m x  1n ; y  m1 þ xð1  yÞun;m x  1n ; y xyun;m x  1n ; y  m1 þ ð1  xÞyun;m x; y  m1 : ;  n;m ðx; yÞ  n;m ðx; yÞ x x

Then the numerator P n;m ðx; yÞ of Rn;m defined in (9) is equal to Bn;m wðx; yÞ. If we can prove that wn;m converges uniformly to uf when n; m ! 1, then Bn;m wn;m approaches to uf too. Then the denominator of Rn;m converging to u implies Rn;m f uniformly converges to f for any continuous function f.

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

Let

     1 1 1 þ ð1  yÞun;m x  ; y ; u1 ðx; yÞ ¼ x yun;m x  ; y  n m n     1 þ ð1  yÞun;m ðx; yÞ ; v 1 ðx; yÞ ¼ ð1  xÞ yun;m x; y  m      1 1 1 þ ð1  xÞun;m x; y  ; u2 ðx; yÞ ¼ y xun;m x  ; y  n m m     1 v 2 ðx; yÞ ¼ ð1  yÞ xun;m x  ; y þ ð1  xÞun;m ðx; yÞ : n

167

ð17Þ

It is clear that

 n;m ðx; yÞ; u1 ðx; yÞ þ v 1 ðx; yÞ ¼ u2 ðx; yÞ þ v 2 ðx; yÞ ¼ x   u1 ðx; yÞ u2 ðx; yÞ : ; wn;m ðx; yÞ ¼ ðu1 ðx; yÞ þ v 1 ðx; yÞÞf u1 ðx; yÞ þ v 1 ðx; yÞ u2 ðx; yÞ þ v 2 ðx; yÞ Then we have

 u1 ðx; yÞ u2 ðx; yÞ ; u1 ðx; yÞ þ v 1 ðx; yÞ u2 ðx; yÞ þ v 2 ðx; yÞ    f ðx; yÞ :

wn;m ðx; yÞ  uðx; yÞf ðx; yÞ ¼ ½u1 ðx; yÞ þ v 1 ðx; yÞ  uðx; yÞf   u1 ðx; yÞ u2 ðx; yÞ þuðx; yÞ f ; u1 ðx; yÞ þ v 1 ðx; yÞ u2 ðx; yÞ þ v 2 ðx; yÞ



For the first bracket in the right of (18), we have

ð18Þ

        nx  1 my  1 nx  1 my  uðx; yÞ þ xð1  yÞ u  uðx; yÞ u1 ðx; yÞ þ v 1 ðx; yÞ  uðx; yÞ ¼ xy u ; ; n1 m1 n1 m1     h nx i nx my  1 my

 uðx; yÞ þ ð1  xÞð1  yÞ u ; ;  uðx; yÞ : þ ð1  xÞy u n1 m1 n1 m1

Suppose m > n, then

  nx  1  1    n  1  x < n  1 ;

  my  1  1 1    m  1  y < m  1 < n  1 :

We have the upper bound of the function u on D

pffiffiffi !    nx  1 my  1  2 uð   n  1 ; m  1 Þ  uðx; yÞ < x u; n  1 ; pffiffiffi !       2  u nx  1 ; my ;  uðx; yÞ < x u;  n1 m1 n1 pffiffiffi !       u nx ; my  1  uðx; yÞ < x u; 2 ;   n1 m1 n1 pffiffiffi !  nx  my

2   ;  uðx; yÞ < x u; ; u n1 m1 n1 where x is the module of continuity. Then

pffiffiffi ! pffiffiffi ! 2 2 ¼ x u; : u1 ðx; yÞ þ v 1 ðx; yÞ  uðx; yÞ 6 ½xy þ xð1  yÞ þ ð1  xÞy þ ð1  xÞð1  yÞx u; n1 n1

For the second bracket in the right of (18), we consider the relation

u1 ðx; yÞ ð1  xÞu1 ðx; yÞ  xv 1 : x¼ u1 ðx; yÞ þ v 1 ðx; yÞ u1 ðx; yÞ þ v 1 ðx; yÞ The numerator in the right of the above equation is

      nx  1 my  1 nx my  1 u ð1  xÞu1 ðx; yÞ  xv 1 ðx; yÞ ¼ ð1  xÞx y u ; ; n1 m1 n1 m1     nx nx  1 my my

u ; ; : þð1  yÞ u n1 m1 n1 m1

168

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

From

pffiffiffi !        u nx  1 ; my  1  u nx ; my  1  < x u; 2 ;  n1 m1 n1 m1  n1 pffiffiffi !    

 2 nx my  u nx  1 ; my u ; < x u; ;  n1 m1 n1 m1  n1 we have

jð1  xÞu1 ðx; yÞ  xv 1 ðx; yÞj 6 xð1  xÞx u;

pffiffiffi ! pffiffiffi ! 2 2 1 6 x u; : 4 n1 n1

This implies

       

1 1 1 1 þ ð1  yÞun;m x  ; y þ ð1  xÞ yun;m x; y  þð1  yÞun;m ðx; yÞ u1 ðx; yÞ þ v 1 ðx; yÞ ¼ x yun;m x  ; y  n m n m P t; where t :¼ inf ðx;yÞ2D uðx; yÞ. Therefore, we obtain

pffiffiffi !     2 u1 ðx; yÞ 1   u1 ðx; yÞ þ v 1 ðx; yÞ  x 6 4t x u; n  1 ; pffiffiffi !     u2 ðx; yÞ 1 2   u ðx; yÞ þ v ðx; yÞ  y 6 4t x u; n  1 : 2 2 For the second bracket in the right of (18) and f 2 CðDÞ, we have

pffiffiffi pffiffiffi !!       u1 ðx; yÞ u2 ðx; yÞ 2 2 f  :  u ðx; yÞ þ v ðx; yÞ ; u ðx; yÞ þ v ðx; yÞ  f ðx; yÞ 6 x f ; 4t x u; n  1 1 1 2 2 This leads to

pffiffiffi ! pffiffiffi pffiffiffi !! 2 2 2 jwn;m  uf j 6 kf k1 x u; x u; þ kuk1 x f ; : n1 4t n1 and which implies the uniform convergence of wn;m to uf when n; m ! 1. 3. Triangular rational Bernstein operators To construct triangular rational Bernstein operators, we recall the barycentric coordinates over the triangle region T (see [8]). Let the vertices of the triangle T be v 1 ¼ ðx1 ; y1 Þ; v 2 ¼ ðx2 ; y2 Þ, v 3 ¼ ðx3 ; y3 Þ and an arbitrary point v ¼ ðx; yÞ 2 R2 . We assume the triangle areas of Mv 1 v 2 v 3 , Mvv2 v 3 , Mv 1 vv3 and Mv 1 v 2 v are A; A1 ; A2 ; A3 , respectively. It is clear that

  1 x1 1  A ¼  1 x2 2  1 x3

  1 x1 1  A2 ¼  1 x 2  1 x3

 y1   y2 ;  y3 

 1 x 1  A 1 ¼  1 x2 2  1 x3

 y1   y ;  y  3

  1 x1 1  A 3 ¼  1 x2 2 1 x

 y   y2 ;  y3 

 y1   y2 :  y

The barycentric coordinates of v ¼ ðu; v ; wÞ is defined by u ¼ AA1 ; v ¼ AA2 ; w ¼ AA3 , and u þ v þ w ¼ 1; v ¼ uv 1 þ vv 2 þ wv 3 . The barycentric coordinates of three vertices are v 1 ¼ ð1; 0; 0Þ; v 2 ¼ ð0; 1; 0Þ; v 3 ¼ ð0; 0; 1Þ. Setting s ¼ ðu; v ; wÞ; k ¼ ði; j; kÞ; jkj ¼ i þ j þ k, k! :¼ i!j!k!, sk :¼ ui v j wk , the Bernstein basis functions of degree n over the triangle T are defined by [8],

Bnk ðsÞ ¼ Bni;j;k ðu; v ; wÞ ¼

n! k n! i j k s ¼ uv w ; k! i!j!k!

where i; j; k P 0; u; v ; w P 0; jsj ¼ u þ v þ w ¼ 1; jkj ¼ n. For given f 2 CðTÞ, the rational Bernstein operators of degree n over triangle are generally defined by (see [4,8])

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

P Qn f ðsÞ ¼



169



~ i;j;k f ni ; nj ; nk Bni;j;k ðu; v ; wÞ x P ; n ~ iþjþk¼n xi;j;k Bi;j;k ðu; v ; wÞ

iþjþk¼n

ð19Þ

where the weights xi;j;k P 0 for 0 6 i; j; k 6 n, and u þ v þ w ¼ 1; u; v ; w P 0. It is a fact that the operator Qn f reproduces constants, but does not reproduce the linear polynomials in general. To achieve the reproduction of linear polynomials, we define a new family of rational Bernstein operators of degree n by

P

 i;j;k f ðui ; v j ; wk ÞBni;j;k ðu; v ; wÞ x ; n1 iþjþk¼n1 xi;j;k Bi;j;k ðu; v ; wÞ

iþjþk¼n

Rn f ðsÞ ¼

P

ð20Þ

 i;j;k and ðui ; v j ; wk Þ are given by where xi;j;k are positive numbers, x

i n

j n

k n

 i;j;k ¼ xi1;j;k þ xi;j1;k þ xi;j;k1 ; x

ð21Þ

xi1;j;k i ixi1;j;k ¼ ;  i;j;k n ixi1;j;k þ jxi;j1;k þ kxi;j;k1 x x jxi;j1;k j v j ¼ i;j1;k ¼ ; xi;j;k n ixi1;j;k þ jxi;j1;k þ kxi;j;k1 xi;j;k1 k kxi;j;k1 ¼ ; wk ¼  i;j;k n ixi1;j;k þ jxi;j1;k þ kxi;j;k1 x ui ¼

ð22Þ ð23Þ ð24Þ

for 0 6 i; j; k 6 n; ui þ v j þ wk ¼ 1; 0 6 i; j; k 6 n, and x1;j;k ¼ xi;1;k ¼ xi;j;1 ¼ 0. Note that the degree of denominator in (20) is n  1 whereas the degree of numerator is n, we can increase the degree of the denominator such that Rn has the same expression as the classical rational Bernstein approximant. Corollary 3.1. The rational Bernstein operator Rn can be written as

P

 i;j;k f ðui ; v j ; wk ÞBni;j;k ðu; v ; wÞ x ¼ n1 iþjþk¼n1 xi;j;k Bi;j;k ðu; v ; wÞ

iþjþk¼n

Rn f ðsÞ ¼

P

P

 i;j;k f ðui ; v j ; wk ÞBni;j;k ðu; v ; wÞ x : n  iþjþk¼n xi;j;k Bi;j;k ðu; v ; wÞ

iþjþk¼n

P

ð25Þ

The reproduction of linear polynomials, the convergence of operator over triangle Rn f can be deduced in the same way as the tensor product operator Rn;m f . The proofs are omitted and only the essential formulae are provided. Theorem 3.1. The rational Bernstein operator Rn reproduces constants and linear polynomials. For applications, the points ðui ; v j ; wk Þ in (20) should be monotone generally, that is ui < uiþ1 ; v j1 < v j ; wk < wkþ1 , for 1 6 i; j; k 6 n  2. In fact, by the symmetric property of u; v ; w, we need to prove ui < uiþ1 and v j1 < v j only, i.e.,

ixi1;j;k ði þ 1Þxi;j1;k < ; ixi1;j;k þ jxi;j1;k þ kxi;j;k1 ði þ 1Þxi;j1;k þ ðj  1Þxiþ1;j2;k þ kxiþ1;j1;k1 ðj  1Þxiþ1;j2;k jxi;j1;k < : ði þ 1Þxi;j1;k þ ðj  1Þxiþ1;j2;k þ kxiþ1;j1;;k1 ixi1;j;k þ jxi;j1;k þ kxi;j;k1 After simplification, it leads the inequalities,

iðj  1Þxi1;j;k xiþ1;j2;k þ ikxi1;j;k xiþ1;j1;k1 < ði þ 1Þjx2i;j1;k þ ði þ 1Þkxi;j1;k xi;j;k1 ; iðj  1Þxi1;j;k xiþ1;j2;k þ ðj  1Þkxi;j;k1 xiþ1;j2;k < ði þ 1Þjx2i;j1;k þ jkxi;j1;k xiþ1;j1;k1 : We assume that the weights xi;j;k in the denominator in (20) are chosen in the following way: there exists a positive continuous function u such that



xi;j;k ¼ u

 i j k ; ; ; n1 n1 n1

0 6 i; j; j 6 n  1;

j 1 i k where i þ j þ k ¼ n  1. Setting h ¼ n1 ; x ¼ n1 ; y ¼ n1 ; z ¼ n1 ¼ 1  x  y, we obtain



xi;j;k ¼ u

 i j k ¼ uðx; y; 1  x  yÞ: ; ; n1 n1 n1

Let /ðx; yÞ ¼ uðx; y; 1  x  yÞ, we have

xðy  hÞ/ðx  h; yÞ/ðx þ h; y  2hÞ þ xð1  x  y þ hÞ/ðx  h; yÞ/ðx þ h; y  hÞ

ð26Þ

170

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

< ðx þ hÞy/2 ðx; y  hÞ þ ðx þ hÞð1  x  y þ hÞ/ðx; y  hÞ/ðx; yÞ;

ð27Þ

xðy  hÞ/ðx  h; yÞ/ðx þ h; y  2hÞ þ ðy  hÞð1  x  y þ hÞ/ðx; yÞ/ðx þ h; y  2hÞ < ðx þ hÞy/2 ðx; y  hÞ þ yð1  x  y þ hÞ/ðx; y  hÞ/ðx þ h; y  hÞ:

ð28Þ

Suppose / 2 C 2 ðTÞ, by Taylor’s expansions and (27), (28), we have

(

" !# " !# )  2 @/ @/ @/ @2/ @2/ @/ @/ @/ @/ 2 @2/ @2/ 2 h xy  /  þ xð1  xÞ ð Þ þ/  þ ðx þ y þ 1Þ/ / @x @y @y @x@y @y2 @y @x @y @x @x2 @x@y 2

þ Oðh Þ < /2 ;

ð29Þ

and

" !#   @/ @/ @/ 2 @2/ @2/ @/ @/ ð Þ þ/  þ 2 þ / ðx þ y  1Þ @x @y @x @x2 @x@y @x @y " !# ) 2 2 2 @/ @/ @/ @ / @ / 2  /2 þ Oðh Þ < /2 : ð Þ þ/  þ yð1  yÞ @x @y @y @y2 @x@y

(

h xy

ð30Þ

By above analysis, we have Corollary 3.2. The points ðui ; v j ; wk Þ induced by (22), (23), (24) and (26) are monotone if / 2 C 2 ðTÞ satisfying (29) and (30), and n is large enough. The operator Rn f can be written as

X

Rn f ðsÞ ¼

 i;j;k f ðui ; v j ; wk ÞBni;j;k ðu; v ; wÞ x

iþjþk¼n

X

xi;j;k Bn1 i;j;k ðu; v ; wÞ

¼

Pn f ðsÞ : Q n1 ðsÞ

iþjþk¼n1

We assume xi;j;k are given by (26) and the points ðui ; v j ; wk Þ are given by (22)–(24) for positive continuous function u, then

X

Q n1 ðsÞ ¼

iþjþk¼n1

u



 i j k Bn1 ðu; v ; wÞ ¼ Bn1 u; ; ; n  1 n  1 n  1 i;j;k

where Bn1 is the triangular Bernstein operator. It is a fact that then lim!1 Bn1 u ¼ u uniformly over the triangle T by the convergence of the Bn1 in approximation theory ([7]). For the convergence of the operator Rn , we have the following result. Theorem 3.2. For the given positive continuous function u 2 CðDÞ and xi;j;k and ðui ; v j ; wk Þ defined by (22), (23), (24) and (26), the sequence of rational operators Rn f converges uniformly to f when n ! 1 for f 2 CðDÞ. 4. Conclusions and future work In this paper, we construct a new family of tensor product and triangular bivariate rational Bernstein operators which reproduce linear polynomials. The convergence of the operators are presented. The convergence of Rn;m is established under

Q n1;m1 ðx; yÞ ¼ Bn1;m1 uðx; yÞ

ð31Þ

for a given function u 2 CðDÞ. Recently, H. Render (see [10]) removed the special requirement (6) in the univariate case and formulated convergence results under the assumption that

Dn ¼

   ðnÞ ðnÞ  sup xiþ1  xi 

i¼0;...;n1

converges to 0. How to remove the special requirement (31) and prove the convergence of Rn;m under the assumption generalized from Render’s work is our future work. In the next step, we will study on the variation diminishing property, convergence order, and explicit inequality of the error jRn;m  f j and jRn  f j of the operators. Moreover, our method can be generalized to construct the rational Bernstein operators over simplex and tensor product regions in higher dimensions. Notice that the weights induced in this paper is similar with the degree elevation of the Bézier curves. Another work for us is to study the geometric meaning of the weights and application of the presented operators in geometric modeling.

C.-G. Zhu, B.-Y. Xia / Applied Mathematics and Computation 258 (2015) 162–171

171

Acknowledgements We would like to thank an anonymous referee for a very careful reading, and his or her comments improved the paper considerably. This work is partly supported by the National Natural Science Foundation of China – China (No. 11271060), the Fundamental Research of Civil Aircraft – China (No. MJ-F-2012–04), the Program for Liaoning Excellent Talents in University – China (No. LJQ2014010), and the Fundamental Research Funds for the Central Universities – China (No. DUT14YQ111). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

P. Pitßul, P. Sablonnière, A family of univariate rational Bernstein operators, J. Approx. Theory 160 (2009) 39–55. P.J. Davis, Interpolation and Approximation, Dover Publications, New York, 1975. R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. G.G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953. M.J.D. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge, 1980. G. Farin, Mathematical Methods for CAGD, fifth ed., Morgan Kaufmann, Los Altos, 2002. R.H. Wang, X.Z. Liang, Approximation by Multivariate Functions, Science Press, Beijing, 1988. R.H. Wang, C.J. Li, C.G. Zhu, A Course for Computational Geometry, Science Press, Beijing, 2008. J.M. Aldaz, O. Kounchev, H. Render, Bernstein operators for extended Chebyshev systems, Appl. Math. Comput. 217 (2010) 790–800. H. Render, Convergence of rational Bernstein operators, Appl. Math. Comput. 232 (2014) 1076–1089. I. Selimovic, New bounds on the magnitude of the derivative of rational Bézier curves and surfaces, Comput. Aided Geom. Des. 22 (2005) 321–326. R. Farouki, The Bernstein polynomial basis: a centennial retrospective, Comput. Aided Geom. Des. 29 (2012) 379–419. S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Communications de la Socité Mathématique de Kharkov 2 XIII (1) (1912) 1–2.