CONSTRUCTION OF CUBATURE FORMULAS BY USING LINEAR POSITIVE OPERATORS

Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI 2004, Thessaloniki, Greece

438

CONSTRUCTION OF CUBATURE FORMULAS BY USING LINEAR POSITIVE OPERATORS

by Ioana Taşcu, Andrei Horvat-Marc

Abstract. We construct some cubature formulas for a hypercube using the polynomials approximation of Bernstein-Stancu type with the nodes M_{i j k, ,}^α having the coordinates

( ) ( 2 )

xm i^α_, = +i α / m+ α ,

( ) ( 2 )

yn j^β_, = j+β / +n β ,

( ) ( 2 )

zr k^γ_, = k+ / +γ r γ , when α β γ, , are non negative real parameters.

Keywords: Cubature formulas

Mathematic Subject Classification [2000]: 41A05, 65D30

1. Introduction

In order to construct cubature formulas for a parallelepipedic domain, we shall use some classes of linear positive interpolating operators for functions of several variables.

We want to approximate a multiple integral of order s extended to a hyperparallelepiped D_s =[a b₁, ×₁] [a b₂, ₂]× ×… a b[ _s, _s] for a function f D: _s →R. It is known that by linear changes of variables, the domain D_s can be transformed into a hypercubeΩ = ,_s [0 1]^s. Hence, we construct some cubature formulas for a function f : Ω →_s R. Such formulas have the following form:

( )

^{1 2} _{1 2 1 2 1 2}

1 2

(1) (2) ( )

1 2 1 2

0 0 0

, , , ^s _{s s s}( )

s s

m m m

s

s s i i …i i i i m m …m

i i i

w x x x dx dx dx … A, , , f x x …x^ ^ R , , , f

= = =

Ω

=

∑∑ ∑

, , , + ,

∫∫

^{K K}

(1.1)

Ioana Taşcu, Andrei Horvat-Marc - Construction of cubature formulas by using linear positive operators

439

wherew x x … x( ₁, , ,_{2 s}) is a weight function, w: Ω →_s R₊. The last term of this formula is the remainder, or the complementary term, of the cubature formula (1.1), the points

1 2, , ,_s i i …i

M 1 2

(1) (2) ( )

 

 

 , , , 

s

s

i i i

x x … x are the nodes and

1 2 s

i i …i

A_{, , ,} are the coefficients of this formula.

We remark that the nodes and the corresponding coefficients of this formula do not depend to the function f for which we approximate the weighted integral by using the multiple sum of order s from the second member of (1.1).

Such cubature formulas can be constructs in the following ways:

1) by using the undetermined coefficients method such that the formula has a given exactness degree and taking into account the number of parameters;

2) integrating the interpolation formulas of Lagrange, Newton or Biermann type. Also, we can use the extensions of the interpolation formulas of Bernstein or Hermite-Fejér type, which assure the uniform convergence of these interpolation procedures, when f ∈ ΩC( _s). It is known that for the s-dimensional Lagrange interpolation procedure the uniform convergence can’t be assured whatever is the select grid of nodes.

2. Cubature formula of Bernstein type in the space of functions C(Ω_s) Let us consider the extension of the Bernstein interpolation formula to s variables

(

1,..., _s

) (

_m1,...,_ms

) ⁽

1,..., _s

⁾ (

_m1,...,_ms

) ⁽

1,..., _s

⁾

,

f x x = B f x x + R f x x

where the Bernstein interpolation polynomial has the following expression

(

Bm … m1_{, ,} s f

) ⁽

x … x1, , s

⁾

¹

1 1 1

1 1

0 0 1

( ) ( )

s

s s

s

m m

m k m k s s

k k s

k

… p x …p x f k …

m m

, ,

= =

 

=  , , ,

 

∑ ∑

(2.2) and the basic interpolation polynomials are

( ) ⁱ(1 ) ^{i i} 1

i i

i k m k

m k i i i

i

p x m x x i s

k

 

  −

 

,  

= − , = , .

The remainder of the formula (2.1) can be expressed by the means of the divided differences of second order, see D. D. Stancu [3]. If we consider

Ioana Taşcu, Andrei Horvat-Marc - Construction of cubature formulas by using linear positive operators

440

the weight function identically equal with 1 on Ω_s and we integrate the interpolation formula (2.1), we obtain a cubature formula of the following form

( )

¹ ₁

1

1

1 1

0 0

1 1

1 ( )

( 1) ( 1)

s

s s s

m m

s

s s m …m

k k

s s

k

f x … x dx …dx … f k … R f

m … m _{= =} m m ^{, ,}

Ω

 

, , = + +

∑ ∑

 , , + ,

∫ ∫

^K

(2.4) because

1 0

( 1) ( 1) 1

( ) 1

( 2) 1

i i

i i i i

m k i i

i i i

m k m k

p x dx i s

k m m

 

 

 

,  

Γ + Γ − +

= = , = , .

Γ + +

∫

(2.5)

According to the results contained in the papers [3], the remainder of the cubature formula (2.4) has the following expression:

2 2 2

1 1 2

1 1 1 2 1 2

2 2

1

1 2 1

1 1

1 1

1 1 1 1

( ) ( ) ( )

12 12

1 1

( )

12

s j j j

s

s s

m … m x s x x s

j j j j j j

s x …x s

s

R f f f …

m m m

f …

m …m

ξ ξ ξ ξ

ξ ξ

, , = , =

= − ,..., − ,..., −

− , , ,

∑ ∑

where the derivative points belong to the hypercube Ω_s.

3. Cubature formulas of Bernstein-Stancu type

By using the approximation polynomials of Bernstein-Stancu type

( )

0 0 0

( ) ( ) ( ) ( )

2 2 2

m n r

m n r m i n j r k

i j k

i j k

B f x y z p x p y p z f

m n r

α β γ α β γ

α β γ

, ,

, , , , ,

= = =

 + + + 

, , =

∑∑ ∑

 + , + , + 

(3.1) where α β γ, , are non negative real numbers and p_{m i,} ,p_{n j,} ,p_{r k,} are the basic interpolation polynomials of Bernstein type, we can construct a cubature formula for the 3-dimensional cube D₃= ,[0 1]³ which is given by

3

( )

D

f x y z dx dy dz, , =

∫∫∫

( )

, , , ,

0 0 0

1

( 1)( 1)( 1) 2 2 2

m n r

m n r

i j k

i j k

f R f

m n r m n r

α β γ

α β γ

α β γ

= = =

 + + + 

= + + +

∑ ∑ ∑

 + , + , + + ^. This formula has the exactness degree(1 1 1), , .

We remark that all coefficients are positive and equal. If we assume that f ∈C^{2 2 2, ,} ( )D₃ then the remainder can be expressed by the partial

Ioana Taşcu, Andrei Horvat-Marc - Construction of cubature formulas by using linear positive operators

441

derivatives of order (2 2 2), , as it can be shown by means of the Peano theorem. Because the Peano kernels have constant sign onD₃, we can use the mean value theorem of integral calculus and the remainder has a representation given by the partial derivatives at a point(ξ η ζ, , ∈) D₃.

When we construct an approximation polynomial of global degree m for a tetrahedron ∆ =₃ {(x y z x, , , ≥ , ≥ , ≥ , + + ≤) 0 y 0 z 0 x y z 1}, we obtain the following representation

0 0 0

( )( ) ( )

2 2 2

m i j m m i

m mi j k

i j k

i j k

B f x y z p x y z f

m m m

α β γ δ

α β γ

α β γ

− − , , , −

, , ,

= = =

 + + + 

, , =

∑∑∑

, ,  + , + , + ,^(3.3)

where

( ) (1 ) ^{− − −}

, , ,

− − −

   

, , =    − − − ,

   

i j k m i j k

m i j k

m m i m i j

p x y z x y z x y z

i j k

and α β γ, , are non negative real numbers.

By the Dirichlet trivariate integral

3

1 1 1 1

( , , , =) _∆ ^{p− q− r−}(1− − − )^s− , B p q r s u v w u v w du dv dw we can obtain the following cubature formula for ∆₃:

3 0 0 0

( ) 1 ( ) ( )

( 1)( 2)( 3) 2 2 2

m i j m m i

m n s

i j k

i j k

f x y z dxdydz f R f

m m m m m m

α β γ

α β γ

α β γ

− − −

, , , ,

= = =

∆

+ + +

, , = , , + ,

+ + +

∑∑ ∑

+ + +

∫∫∫

which has the exactness degree(1 1 1), , . The remainder can be represent by means of the partial derivatives of orders (2 0 0), , , (0 2 0), , , (0 0 2), , , (2 2 0), , ,

(2 0 2), , , (0 2 2), , , (2 2 2), , in a certain point (ξ η ζ, , ) of the tetrahedron ∆₃. References:

[1] STANCU, D. D., COMAN, GH., BLAGA, P., Analiză Numerică şi Teoria Aproximării, vol. I(2001), vol. II(2002), Presa Univ. Clujeană.

[2] STANCU,D.D., (1969) A new class of uniform approximating polynomial operators in two and several variables, Proc. Conf. Constr. Theory of Func., Budapest, 443-455.

[3] STANCU, D. D., (1964), The remainder of certain linear approximation formulas of two variables, J. SIAM Num. Anal. Ser. B, 1, 137-163.

Author:

Ioana Taşcu - North University of Baia Mare, Romania

Andrei Horvat-Marc - North University of Baia Mare, Romania