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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 )

m i

xα_, = + / +i

α

m

α

,

( ) ( 2 )

n j

yβ_, = + / +j

β

n

β

,

( ) ( 2 )

r k

zγ_, = + / +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 b_{1 1}] [a b_{2 2}] … [a b_{s s}] for a function f D: _s →R. It is known that by linear changes of variables, the domain Ds can be transformed into a hypercube [0 1]

s s

Ω = , . 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

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(Ωs)

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

s

m k m k s

k k s

k k

… p x …p x f …

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

m

p x x x i s

k

 

  −

  , _{ }  

= − , = , .

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 k

f x … x dx …dx … f … 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 s x …x

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), , .

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