On the composite Bernstein type cubature formula

On the composite Bernstein type cubature formula

Dan B˘arbosu, Dan Micl˘au¸s

Abstract

Considering a given function f ∈ C([0,1]×[0,1]), the bivariate in- terval [0,1]×[0,1] is divided inmnequally spaced bivariate subintervals _k−1

m ,_m^k

×j−1 n ,_n^j

,k= 1, m, j= 1, n. On each such type of subinter- vals the Bernstein bivariate approximation formula is applied and a cor- responding Bernstein type cubature formula is obtained. Making the sum of mentioned cubature formulas, the composite Bernstein type cubature formula is obtained. The coefficients of above formula are determined and an upper-bound for its remainder term is given.

2010 Mathematics Subject Classification: 65D32, 41A10

Key words and phrases: Bernstein bivariate operator, Bernstein bivariate approximation formula, Bernstein quadrature formula, Bernstein cubature

formula, Bivariate divided difference, Remainder term

1 Preliminaries

Let us to denote N={1,2, ...} and N₀=N∪ {0}.

The Bernstein bivariate operatorBm,n :C([0,1]×[0,1])→C([0,1]×[0,1]) is defined for anyf ∈C([0,1]×[0,1]), any (x, y)∈[0,1]×[0,1] and anym, n∈N, by

(1) (Bm,nf)(x, y) =

m

X

k=0 n

X

j=0

p_m,k(x)pn,j(y)f k

m,j n

,

1Received 24 April, 2009

Accepted for publication (in revised form) 18 May, 2010

73

where

(2) p_m,k(x) =

m k

x^k(1−x)^m−k and

(3) pn,j(y) =

n j

y^j(1−y)^n−j

are the fundamental Bernstein’s polynomials. The polynomial (1) is called the Bernstein bivariate polynomial.

Letf ∈C([0,1]×[0,1]) be given. The following equality (4) f(x, y) = (Bm,nf) (x, y) + (Rm,nf) (x, y)

is known as the Bernstein bivariate approximation formula, where Rm,n is the remainder operator associated to the Bernstein bivariate operator B_m,n, i.e. Rm,nf is the remainder term of the bivariate approximation formula (4).

Regarding the remainder term of (4), D. B˘arbosu and O. T. Pop [7] established the following:

Theorem 1 For any f ∈C([0,1]×[0,1]) and any (x, y) ∈ [0,1]×[0,1], the remainder term of (4) can be expressed under the form

(Rm,nf)(x, y) =−x(1−x) m

m−1

X

k=0 n

X

j=0

p_m−1,k(x)pn,j(y)

"

x,_m^k,^k+1_m

j m

;f

# (5)

−y(1−y) n

m

X

k=0 n−1

X

j=0

p_m,k(x)pn−1,j(y)

" _k

m

y,_n^j,^j+1_n ;f

#

+xy(1−x)(1−y) mn

m−1

X

k=0 n−1

X

j=0

p_m−1,k(x)pn−1,j(y)

"

x,_m^k,^k+1_m y,_k^j,^j+1_n ;f

# .

In (5) the brackets denote the bivariate divided differences.

Theorem 2 ([7]) Let be p, q ∈ N₀, a ≤ x₀ < x₁ < ... < x_p ≤ b, c ≤ y₀ <

y₁ < ... < y_q ≤ d and the function f : [a, b]×[c, d] → R be given. If f ∈ C^{(p−1,q−1)}([a, b]×[c, d])and there exists _∂x^∂p+qp∂y^fq on]a, b[×]c, d[then, there exists (ξ, η)∈]a, b[×]c, d[ such that

(6)

"

x₀, x₁, . . . , xp

y₀, y₁, . . . , y_q ;f

#

= 1 p!q!

∂^p+qf

∂x^p∂y^q(ξ, η).

Theorem 3 ([7]) Let f : [0,1]×[0,1]→ R be a function with the properties that f ∈ C([0,1]×[0,1]), there exists _∂x^∂2⁴∂y^f2 on ]0,1[×]0,1[ and ^∂_∂x^2f2, ^∂_∂y^2f2,

∂⁴f

∂x²∂y² are bounded on]0,1[×]0,1[. Then, the inequalities

|(Rm,nf)(x, y)| ≤ x(1−x)

2m M₁[f] +y(1−y)

2n M₂[f] +xy(1−x)(1−y) 4mn M₃[f] (7)

≤ 1

8mM₁[f] + 1

8nM₂[f] + 1

64mnM₃[f] hold, for any (x, y)∈[0,1]×[0,1] and any m, n∈N, where

M₁[f] = sup

(x,y)∈]0,1[×]0,1[

∂²f

∂x²(x, y) ,

M₂[f] = sup

(x,y)∈]0,1[×]0,1[

∂²f

∂y²(x, y) (8) ,

M₃[f] = sup

(x,y)∈]0,1[×]0,1[

∂⁴f

∂x²∂y²(x, y) .

Integrating the Bernstein bivariate approximation formula (4) one arrives to the following Bernstein’s cubature formula [6]:

(9)

Z1

0

Z1

0

f(x, y)dxdy =

m

X

i=0 n

X

j=0

A_i,jf i

m,j n

+R_m,n[f], where

(10) A_i,j = 1

(m+ 1)(n+ 1), i= 0, m, j= 0, n and

(11) |R_m,n[f]| ≤ 1

12mM₁[f] + 1

12nM₂[f] + 1

144mnM₃[f], whereM₁[f], M₂[f] and M₃[f] were defined at (8).

The focus of the present paper is to construct the composite Bernstein type cubature formula. For this aim, the bivariate interval [0,1]×[0,1] will be divided in mn equally spaced bivariate subintervals_k−1

m ,_m^k

×h

j−1 n ,_n^ji

,k= 1, m, j = 1, n. On each such type of subintervals _k−1

m ,_m^k

×h_j−

1 n ,_n^ji

, the Bernstein’s cubature formula (9) will be applied. Next, adding the mentioned cubature formulas, the composite Bernstein cubature formula on [0,1]×[0,1]

will be obtained.

2 Main results

We start by the simple results contained in the following two lemmas.

Lemma 1 Suppose that a, b, c, d∈R, a < b and c < d, f ∈C([a, b]×[c, d]).

Then, the bivariate Bernstein polynomial associated to the function f is defined by

(B_m,nf)(x, y) = 1

(b−a)^m(d−c)ⁿ

m

X

k=0 n

X

j=0

m k

n j

(x−a)^k(b−x)^m−k (12)

·(y−c)^j(d−y)^n−jf

a+kb−a

m , c+jd−c n

.

Proof. The parametric extensions [5] of the Bernstein’s univariate operator [8] are defined by

(B_m^xf)(x, y) = 1

(b−a)^m(d−c)ⁿ

m

X

k=0 n

X

j=0

m k

n j

(x−a)^k(b−x)^m−k (13)

·(y−c)^j(d−y)^n−jf

a+kb−a m , y

,

respectively

(B_n^yf)(x, y) = 1

(b−a)^m(d−c)ⁿ

m

X

k=0 n

X

j=0

m k

n j

(x−a)^k(b−x)^m−k (14)

·(y−c)^j(d−y)^n−jf

x, c+jd−c n

.

Their tensorial product [9] is the bivariate Bernstein operator (12).

Lemma 2 Suppose thata, b, c, d∈R,a < b,c < dand letf : [a, b]×[c, d]→R be a function with the properties thatf ∈C^2,2([a, b]×[c, d]), there exists _∂x^∂2⁴∂y^f2

on ]a, b[×]c, d[ and ^∂_∂x^2f2, ^∂_∂y^2f2, _∂x^∂2⁴∂y^f2 are bounded on ]a, b[×]c, d[. Then, the remainder term of the Bernstein bivariate approximation formula on [a, b]× [c, d]verifies the following inequality

|(R_m,nf)(x, y)| ≤ (x−a)(b−x)

2m(b−a)² M₁⁰[f] + (y−c)(d−y) 2n(d−c)² M₂⁰[f] (15)

+(x−a)(b−x)(y−c)(d−y) 4mn(b−a)²(d−c)² M₃⁰[f],

for any(x, y)∈]a, b[×]c, d[and any m, n∈N, where M₁⁰[f] = sup

(x,y)∈]a,b[×]c,d[

∂²f

∂x²(x, y) ,

M₂⁰[f] = sup

(x,y)∈]a,b[×]c,d[

∂²f

∂y²(x, y) (16) ,

M₃⁰[f] = sup

(x,y)∈]a,b[×]c,d[

∂⁴f

∂x²∂y²(x, y) .

Proof. One applies (7) and the method of parametric extensions [9].

In what follows, let us to consider the bivariate interval [0,1]×[0,1] divided in themnequally spaced bivariate subintervals_k−1

m ,_m^k

×h

j−1 n ,_n^ji

,k= 1, m, j= 1, n. In each interval_k−1

m ,_m^k

×h

j−1 n ,_n^ji

one considers the distinct knots x_h = ^kp−p+h_mp ,h= 0, p, respectivelyy_l= ^jq−q+l_nq ,l= 0, q.

Applying Lemma 1 yields the following Bernstein type bivariate polynomial

(Bp,k,q,jf)(x, y) =m^pn^q

p

X

h=0 q

X

l=0

p h

q

l x−k−1 m

hk m −x

p−h

(17)

·

y−j−1 n

l j n−y

q−l

f

kp−p+h

mp ,jq−q+l nq

.

The following Bernstein type bivariate approximation formula (18) f(x, y) = (B_p,k,q,jf)(x, y) + (R_p,k,q,jf)(x, y) holds, on any interval_k−1

m ,_m^k

×h_j−

1 n ,_n^ji

,k= 1, m,j= 1, n.

For any f ∈C^2,2([0,1]×[0,1]) the following upper-bound estimation for the bivariate remainder term

|(Rp,k,q,jf)(x, y)| ≤ m x−^k−1_{m k}

m −x

2 M₁⁰⁰[f] + n

y−^j−_n¹ _n^j −y 2 M₂⁰⁰[f] (19)

+

mn x−^k−_m¹ _k

m−x

y− ^j−_n¹ _n^j −y

4 M₃⁰⁰[f]

holds, where

M₁⁰⁰[f] = sup

(x,y)∈]^k_m^−1,_m^k[^×]^j−_n^1,_n^j[

∂²f

∂x²(x, y) ,

M₂⁰⁰[f] = sup

(x,y)∈]^k_m^−1,_m^k[^×]^j−_n^1,_n^j[

∂²f

∂y²(x, y) (20) ,

M₃⁰⁰[f] = sup

(x,y)∈]^k_m^−1,_m^k[^×]^j−_n^1,_n^j[

∂⁴f

∂x²∂y²(x, y) .

Theorem 4 If f ∈ C([0,1]×[0,1]), the coefficients of the Bernstein type cubature formula

(21)

k m

Z

k−1 m

j n

Z

j−1 n

f(x, y)dxdy =

p

X

h=0 q

X

l=0

A_h,k,l,jf

kp−p+h

mp ,jq−q+l nq

+R_k,j[f]

can be expressed under the form

(22) Ah,k,l,j = 1

mn(p+ 1)(q+ 1), h= 0, p, l= 0, q.

Proof. Integrating (18) on_k−1

m ,_m^k

×h_j−

1 n ,_n^ji

,k= 1, m,j= 1, nand taking (17) into account, yields

A_h,k,l,j =m^pn^q p

h q

l

k m

Z

k−1 m

j n

Z

j−1 n

x−k−1 m

h k m −x

p−h

·

y−j−1 n

l j n −y

q−l

dxdy

=m^pn^q p

h q

l

k m

Z

k−1 m

x−k−1 m

h k m −x

p−h

dx

·

j

Zn

j−1 n

y−j−1 n

l j n−y

q−l

dy

=m^pn^q p

h q

l Z¹

0

1

m^p+1t^h(1−t)^p−hdt

1

Z

0

1

n^q+1u^l(1−u)^q−ldu

= 1 mn

p h

q l

Z¹

0

t^h(1−t)^p−hdt

1

Z

0

u^l(1−u)^q−ldu.

The integral R1 0

xⁱ(1−x)ⁿ⁻ⁱdxis the Euler function of first kindB(i+1, n−i+1).

Taking the well known properties of Euler function of first kind into account, it follows that

A_h,k,l,j = 1 mn

p h

q l

B(h+ 1, p−h+ 1)B(l+ 1, q−l+ 1)

= 1 mn

p!

h!(p−h)!

h!(p−h)!

(p+ 1)!

q!

l!(q−l)!

l!(q−l)!

(q+ 1)!

= 1

mn(p+ 1)(q+ 1).

Theorem 5 Let f ∈C^2,2([0,1]×[0,1]) be given so that there exists _∂x^∂2²∂y^f2 on ]0,1[×]0,1[and ^∂_∂x^2f2, ^∂_∂y^2f2, _∂x^∂2⁴∂y^f2 are bounded on ]0,1[×]0,1[. Then the follow- ing upper-bound estimation for the bivariate remainder term of the Bernstein type cubature formula (21) is

(23) |R_k,j[f]| ≤ M₁⁰⁰[f]

12m²n+ M₂⁰⁰[f]

12mn² + M₃⁰⁰[f] 144m²n², where M₁⁰⁰[f], M₂⁰⁰[f], M₃⁰⁰[f]are given at (20).

Proof. The inequality (23) follows by integrating the Bernstein type bivariate approximation formula (18) and taking the inequality (19) into account.

Theorem 6 For anyf ∈C^2,2([0,1]×[0,1]), the following composite Bernstein type cubature formula

Z1

0

Z1

0

f(x, y)dxdy = 1

mn(p+ 1)(q+ 1)

m

X

k=1 n

X

j=1 p

X

h=0 q

X

l=0

f

kp−p+h

mp ,jq−q+l nq

(24)

+R_m,n[f].

holds, where |Rm,n[f]| were defined at (11).

Proof. Adding the relation (21) for any k = 1, m, j = 1, n, we get the following composite Bernstein type cubature formula (24).

Remark 1 It is easily to see that we get the same result for the bivariate remainder term of the composite Bernstein type cubature formula, as the result obtained by D. B˘arbosu and O. T. Pop in [6].

Corollary 1 The following equality

m,n→∞lim

1

mn(p+ 1)(q+ 1)

m

X

k=1 n

X

j=1 p

X

h=0 q

X

l=0

f

kp−p+h

mp ,jq−q+l nq

(25)

=

1

Z

0 1

Z

0

f(x, y)dxdy holds.

Proof. Yields immediately from Theorem 6, because lim

m,n→∞|Rm,n[f]|= 0.

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Dan B˘arbosu, Dan Micl˘au¸s North University of Baia Mare

Department of Mathematics and Computer Science Victoriei 76, 430122 Baia Mare, Romania

e-mail: [email protected], [email protected]