ON SOME ASPECTS REGARDING THE
GENERALIZATION OF THE OPTIMAL FORMULAS OF
SARD TYPE USING THE BOOLEAN-SUM TYPE
OPERATORS
by
C
ă
t
ă
lin Mitran
Abstract. In this paper will be presented some results about the optimal quadrature’s formulas of Sard type and there, starting from considerations, a generalization for the cubature’s formulas of boolean-sum type builts with the help of some operators who satisfy the optimal formulas mentioned above. A better generalization can be obtained using a boolean-sum type formula with a number of n operators and n corresponding remainders.
We shall consider the next quadrature’ formula:
( ) ( )
( )
( )
∫
=∑
= +b
a
m
k
m k
kf a R f
A x
d x f
0
λ (1),
where dλ is a positive measure of the integration integral ,
a
k are the nodes of the quadrature’s formula and satisfy the relations:b a a
a
a≤ 0 < 1<...< m≤ ,
k
A are named the coefficients of the formula and Rm
( )
f are named the remainderTheorem 1
If
f
∈
H
n[ ]
a
,
b
if we use the Peano’s theorem we shall have the next representation of the remainder:( )
=∫
( ) ( )
b
a
n n m
m f K t f t dt
R , ( ) (2),
where
( )
( )
( )
( ) ( )
( )
(
)
− −
− ⋅ − =
− −
= −
+ =
− + −
+
∫
∑
10 1
1
,
! 1 1 !
1
n k m
k k b
a n n
m n
m x t d x A a t
n n
t x R t
K λ
Remark 1
If the Peano’s kernel has a constant sign on the entire interval of the integration then using a midle formula we shall can write:
[ ]
( )( )
( )
. b a
, dt t K f
f R
b
a n , m m
m
< <
=
∫
ξ
ξ
(3)
Definition 1
The quadrature’s formula given by (1) will be named optimal of Sard type if:
( )
∫
b →a n m t dt
K , 2 min (4) Theorem 2
If
n
≤
m
the quadrature’s formula given by (1) who is considerated optimal of Sard type has an one solution and this solution will be obtained by the integration of the spline interpolation formula( ) ( )( )
x
Sf
x
R
( )
f
x
x
[ ]
a
b
f
=
+
m,
,
∈
,
that means
( ) ( ) ( )( ) ( )
∫
b =∫
a
b
a
x d x Sf x
d x
f
λ
λ
+∫
( ) ( )
b
a
m f x d x
R ,
λ
(5).We shall consider now some questions about the generalization of the optimal quadrature’s formulas of Sard type, more precisely about the cubature formulas of boolean-sum type who results from optimals quadrature’s formulas of Sard type.
Let’s consider the next decomposition of the unit operator:
(
1 2 1 2)
1 22 1 2 1
R
R
P
P
P
P
R
R
P
P
I
+
−
+
=
=
+
⊕
=
(6)
where P1,P2 are interpolation polynoms and R1,R2 are the corresponding remainders:
,I P R
R P
Remark 2
If we shall integrate the formula given by (6) on a plane domain
[ ] [ ]
a bxc dD= , , , where P1,P2 are supposed to be defined on
[ ]
a,b , respectively[ ]
c,d , we will obtain a cubature formula with the remainder given by:(
)( )
[ ] [ ]
∫∫
=abxcd D
dxdy
y
x
f
R
R
, ,
2
1
,
On R1,R2 we shall suppose that they have the corresponding Peano’s kernels
2 2 1 1,n
,
m,nm
K
K
given by:( )
( )
(
)
(
) ( )
( )
(
)
− −
− ⋅ − =
− −
= −
+ =
− + −
+
∫
∑
10 1 1
` 1 1
1
,
1 1 1
1
1
1 1
1 1
1 1!
1 !
1
n k m
k k b
a
n n
m n
m x t d x A a t
n n
t x R t
K λ ,
( )
( )
(
)
(
) ( )
( )
(
)
− −
− ⋅ − =
− −
= −
+ =
− + −
+
∫
∑
10 1
2 2
1
,
2 2 2
2 2 2
2
2 2
2 2
2 1!
1 !
1
n k m
k k b
a
n n
m n
m x t d x A a t
n n
t x R t
K λ
Definition 2
About a cubature’s formula generated by the integration of a boolean-sum type formula we shall say that is optimal of Sard type if it verifies the relation:
( )
( )
∫∫
→D
n m n
m s K t dsdt
K , 2 , 2 min
2 2 1
1
Theorem 3
If P1,P2are the operators corresponding to an optimal quadrature’s formula and R1,R2 the corresponding remainders then the cubature’s formula obtained by the integration of a (6) type formula is optimal of Sard type in the sens of the definition 2. Remark 3
In order to demonstrate theorem 3 first is easily to see that
( )
( )
∫∫
=D
n m n
m s K t dsdt
K , 2 , 2
2 2 1
1
∫
( )
⋅∫
( )
b
a
d
c n m n
m s ds K t dt
K , 2 , 2
2 2 1
If we take care that the quadrature’s formulas generated from the 2
1,P
P operators and the corresponding remainders R1,R2 are optimals of Sard type, that means
( )
( )
∫
∫
→ →
d
c n m b
a n m
dt t K
ds s K
min min
2 ,
2 ,
2 2
1 1
will be evident taking care of (8) that
( )
( )
∫∫
→D
n m n
m s K t dsdt
K , 2 , 2 min
2 2 1
1
Remark 4
Such types of optimals quadrature’s formulas can be generalizated starting from a number of n operators
P
1,
P
2,...,
P
s and the corresponding remainderss
R
R
R
1,
2,...,
using an n-dimensional boolean-sum type formula:s s
R
R
R
P
P
P
I
=
1⊕
2⊕
...
⊕
+
1 2...
. (9) Remark 5A such type of boolean-sum type formula can be obtained supposing that we have verifieds the conditions
j i n j
i P P P
Pi j = j i,∀, =1,..., , ≠ and using a recursive formula
(
n)
nn P P P P
P P
P1⊕ 2⊕...⊕ = 1⊕ 2⊕...⊕ −1 ⊕ . (10) For example, using a number of 3 operators
P
1,
P
2,
P
3 we shall have:(
1 2)
3 1 2 3 1 2 1 3 2 3 1 2 33 2
1
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
Remark 6
It is easily to verify that the formula given by (10) has the property of associativity.
Definition 3
About a cubature’s formula generated by the integration of a boolean-sum type formula given by (9) we shall say that is optimal of Sard type if it verifies the relation:
( )
( )
( )
∫∫ ∫
→ D s s n m n m nm t K t K t dt dt dt
K
s
s ... min
...
... , 1 2 , 2 2 , 2 1 2
2 2 1
1
with
D
=
[ ] [
a
1,
b
1x
a
2,
b
2] [
x
...
x
a
s,
b
s]
.Supposing that the s quadrature’s formulas builts with the operators and the remainders mentionated above are optimal of Sard type, that means:
( )
( )
... min min 2 2 2 2 1 1 1 1 2 , 2 ,∫
∫
→ → b a n m b a n m dt t K dt t K (11)( )
∫
s →s s s b a n m t dt
K , 2 min
where
( )
( )
(
)
(
) ( )
( )
(
)
− − − ⋅ − = − − = − + = − + − +∑
∫
1 0 1 1 ` 1 1 1 , 1 1 1 1 1 1 1 1 11 1!
1 ! 1 n k m k k b a n n m n
m x t d x A a t
n n t x R t K λ
( )
( )
(
)
(
) ( )
( )
(
)
− − − ⋅ − = − − = − + = − + − +∑
∫
1 0 1 2 2 1 , 2 2 2 2 2 2 2 2 2 2 22 1!
1 ! 1 n k m k k b a n n m n
m x t d x A a t
n n t x R t K λ …
( )
( )
(
)
(
) ( )
( )
(
)
− − − ⋅ − = − − = − + = − + − +∑
∫
1 0 1 1 , ! 1 1 ! 1 s s s s s s s s s s s s n k m k k b a n s s n m nm x t d x A a t
n n t x R t K λ
( )
( )
∫
( )
∫
( )
∫
= 1 →1 1 1 1
1 ... ... ... min
2 , 1
2 1 , 1
2 , 2 1 ,
b
a
b
a
s s n m n
m D
s s
n m n
m
s
s s s s
s t dt dt K t dt K t dt
K t K
where
D
=
[ ] [
a
1,
b
1x
a
2,
b
2] [
x
...
x
a
s,
b
s]
REFERENCES
1. Coman Gh.,Analiză numerică, Ed. Libris, Cluj-Napoca, 1995
2. Coman Gh.,D.D.Stancu, Blaga Petru- Analiză numericăşi teoria aproximării, vol. I+II, Presa Universitară Clujeană, Cluj-Napoca, 2002
3. ***-Approximation and optimization: Proceedings of ICAOR-International Conference on Approximation and optimization, Transilvania Press, Cluj-Napoca, 1996
