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Optimizationofintervalformulasfor
approximateintegrationofset-valued
functionsmonotonewithrespecttoinclusion
ARTICLEinUKRAINIANMATHEMATICALJOURNAL·APRIL2012
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1AUTHOR:
V.V.Babenko
UniversityofUtah
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Ukrainian Mathematical Journal, Vol. 63, No. 11, April, 2012 (Ukrainian Original Vol. 63, No. 11, November, 2011)
BRIEF COMMUNICATIONS
OPTIMIZATION OF INTERVAL FORMULAS FOR APPROXIMATE INTEGRATION
OF SET-VALUED FUNCTIONS MONOTONE WITH RESPECT TO INCLUSION
V. V. Babenko
UDC 517.5
The best interval quadrature formula is obtained for the class of convex set-valued functions defined on
the segment Œ0; 1 and monotone with respect to inclusion.
In [1], the problem of the best quadrature formula was solved on the class of monotonically nondecreasing
functions f W Œ0; 1 ! R such that f .0/ D 0 and f .1/ D 1: The problem of the best interval quadrature formula
on this class of functions was solved in [2]. In [3], the problem of optimization of approximate calculation of
integrals in the sense of Hukuhara [4] was solved with the use of “point” quadrature formulas on classes of setvalued functions defined on Œ0; 1 and monotone with respect to inclusion. The aim of the present paper is to solve
the problem of optimization of interval quadrature formulas on the classes of functions considered in [3].
We use the definitions and facts presented in [3]. Below, we briefly describe some of them.
Let K.Rd / denote the collection of nonempty, compact, convex subsets of the space Rd : Assume that
A1 ; : : : ; An 2 K.Rd /; ˛1 ; : : : ; ˛n 2 R; Ai ¤ ¿; and ˛i 0 for i D 1; n: As usual, we set
n
X
˛i Ai D
¼ n
X
i D1
½
˛i xi W xi 2 Ai ; i D 1; n :
i D1
The Hausdorff metric in K.Rd / is defined by the formula
º
»
ı.A; B/ D max sup inf jx
x2A y2B
yj; sup inf jx
x2B y2A
yj ;
where j j is the Euclidean norm in the space Rd :
The rigorous statement of the problem considered here is as follows:
Let sets A; B 2 K.Rd / .A B/ be given. Let MA;B denote the class of functions f W Œ0I 1 ! K.Rd /
monotone with respect to inclusion . i.e., the relation 0 x1 < x2 1 implies that f .x1 / f .x2 // and such
that f .0/ D A and f .1/ D B:
Dnepropetrovsk National University, Dnepropetrovsk, Ukraine.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 11, pp. 1565–1569, November, 2011. Original article submitted July 1,
2011.
0041-5995/12/6311–1781
c 2012
Springer Science+Business Media, Inc.
1781
V. V. BABENKO
1782
For given numbers n 2 N and H 2 .0; 1/; we denote by Qn;H the collection of quadrature formulas of the
form
q.f / D C C
n
X
kD1
1
ck
jIk j
Z
f 2 MA;B ;
f .x/dx;
(1)
Ik
where C 2 K.Rd /; ck 0; k D 1; n; fIk gnkD1 is the collection of segments contained in Œ0; 1 and such that
n
X
jIk j H;
kD1
and jIk j is the length of the segment Ik :
We set
0
R.MA;B ; q/ D
ı@
sup
f 2MA;B
Rn;H .MA;B / D
Z1
1
f .x/dx; q.f /A ;
0
inf
q2Qn;H
(2)
R.MA;B ; q/:
The problem of a quadrature formula from Qn;H that is the best on the class MA;B consists of determining
quantity (2) and finding a quadrature formula of the form (1) that realizes the infimum on the right-hand side of (2).
Theorem 1. Among all quadrature formulas q 2 Qn;H ; the following formula is the best on the class MA;B W
H
Z
n
1C
X
.1 H /
n
n
qn;H .f / D
.A C B/ C
f .x/dx;
2.n C 1/
nC1
H
kD1
Ik
where
H
6
n
I k D 4k
nC1
2
H3
H
n 7;
; k
5
n
nC1
1C
1C
k D 1; nI
moreover,
0 1
1
Z
1 H
Rn;H .MA;B / D R.MA;B ; qn;H / D sup ı @ f .x/dx; qn;H .f /A D
ı.A; B/:
2.n C 1/
f 2MA;B
0
Proof. We need the following result on integrals of monotone functions (see [3], Proposition 5):
Zb
f .a/.b
a/ f .x/dx f .b/.b
a
a/:
(3)
O PTIMIZATION OF I NTERVAL F ORMULAS FOR A PPROXIMATE I NTEGRATION OF S ET-VALUED F UNCTIONS
1783
In what follows, for simplicity, we set
H
lD
2n
xk D
and
H
1 C H=n
Ck
D
2n
nC1
1 C 2l
l Ck
;
nC1
k D 1; n:
Taking into account the additivity of the integral (see [3], Proposition 4), for f 2 MA;B we get
xZ1 Cl
Z1
f .x/dx D
0
For k D 0; n
xZ2 Cl
f .x/dx C
x0 Cl
xZn Cl
f .x/dx C : : : C
x1 Cl
xn
Z1
f .x/dx C
f .x/dx:
xn Cl
1 Cl
1; using (3), we obtain
xkC1
Z Cl
xkC1
Z l
f .x/dx D
xk Cl
xkC1
Z Cl
f .x/dx C
xk Cl
f .x/dx f .xkC1
xkC1 l
n 1 H
H nC1
Z
H
Z
n n
f .x/dx D
nC1 H
xkC1
Z Cl
f .x/dx
xkC1 l
1C
Z
f .x/dx C
I kC1
1 H
l/
C
nC1
I kC1
f .x/dx:
I kC1
Furthermore,
Z1
f .x/dx xn Cl
1 H
B:
nC1
Then
Z1
H
n Z
X
1 H
n
f .x/dx f .x/dx C B
:
nC1
nC1
1C
kD1
0
Ik
By analogy, we obtain
Z1
0
H
Z
n
X
n
1 H
n
f .x/dx f .x/dx C A
:
nC1
H
nC1
1C
kD1
Ik
Thus,
H
H
Z
Z1
Z
n
n
1C
1C
X
X
1 H
n
1
H
n
n
n
A
C
f .x/dx f .x/dx B
C
f .x/dx:
nC1
nC1
H
nC1
nC1
H
kD1
Ik
0
kD1
Ik
(4)
V. V. BABENKO
1784
It was proved in [3] that if X; Y; Z 2 K.Rd / and X Y Z; then
X CZ
1
ı Y;
ı.X; Z/:
2
2
Using this result and (4), we obtain
0
1
H
Z1
Z
n
1
C
X n
1 H
1 H
B
C
n
ı @ f .x/dx;
.A C B/ C
f .x/dx A ı.A; B/:
2.n C 1/
nC1
H
2.n C 1/
kD1
0
Ik
Thus, we have proved that
Rn;H .MA;B / R.MA;B I qn;H / 1 H
ı.A; B/:
2.n C 1/
(5)
We now show that, for any quadrature formula of the form (1), one has
0 1
1
Z
1 H
R.MA;B ; q/ D sup ı @ f .x/dx; q.f /A ı.A; B/:
2.n
C 1/
f 2MA;B
0
This and relation (5) yield the statement of the theorem.
Consider the set
Œ0; 1 n
n
[
Ik :
kD1
Since
n
X
jIk j H;
kD1
this set contains an interval .a; b/ whose length is at least
f1 .x/ D
8
<A;
x a;
:
B;
x > a;
1 H
: We set
nC1
f2 .x/ D
and
8
<A;
x < b;
B;
x b:
:
Then
Z1
Z1
f1 .x/dx D Aa C B.1
0
f2 .x/dx D Ab C B.1
a/;
0
and
q.f1 / D q.f2 /:
b/;
O PTIMIZATION OF I NTERVAL F ORMULAS FOR A PPROXIMATE I NTEGRATION OF S ET-VALUED F UNCTIONS
1785
Therefore,
0
Z1
1
ı@
sup
f 2MA;B
f .x/dx; q.f /A
0
8 0 1
1
0 1
19
Z
< Z
=
max ı @ f1 .x/dx; q.f1 /A ; ı @ f2 .x/dx; q.f2 /A
:
;
0
0
8 0 1
1
0 1
19
Z
Z
<
=
1
@
A
@
A
ı
f1 .x/dx; q.f1 / C ı
f2 .x/dx; q.f2 /
;
2:
0
0
0 1
1
Z
Z1
1
1
ı @ f1 .x/dx; f2 .x/dx A D ı.Aa C B.1
2
2
0
a/; Ab C B.1
b//:
0
For sets C; D K.Rd /; we put
e.C; D/ D sup inf jx
yj;
x2C y2D
so that
ı.C; D/ D max.e.C; D/; e.D; C //:
Consider e.Aa C B.1
a/; Ab C B.1
e.Aa C B.1
a/; Ab C B.1
b//: Using the duality theorem (see, e.g., [5], Theorem 2.3.1), we get
b//
¼
D
sup
sup
½
f .z/
f .w/
sup
z2AaCB.1 a/ jjf jj1
w2.AbCB.1 b//
¼
½
D sup
.af .x/ C .1
sup
jjf jj1
a/f .y//
x2A;y2B
sup
.bf .u/ C .1
b/f .v//
u2A;v2B
!
D sup
a sup f .x/ C .1
D sup ..a
jjf jj1
D .b
a/ sup f .y/
x2A
jjf jj1
b sup f .x/
y2B
b/ sup f .x/ C .b
x2A
a/ sup sup .f .y/
y2B jjf jjD1
.1
x2A
a/ sup f .y//
y2B
sup f .x// D .b
x2A
a/e.B; A/:
b/ sup f .y/
x2B
V. V. BABENKO
1786
Since A B; it is easy to verify that
Ab C B.1
b/ Aa C B.1
a//
and, hence, e.A; B/ D 0; so that
ı.Aa C B.1
a/; Ab C B.1
b// D e.B; A/ D ı.A; B/:
Thus, for any quadrature formula q 2 Qn;H ; we have
0
sup
f 2MA;B
Z1
ı@
1
1
f .x/dx; q.f /A .b
2
a/ı.A; B/ 0
1 H
ı.A; B/:
2.n C 1/
The theorem is proved.
REFERENCES
1. J. Kiefer, “Optimum sequential search and approximation methods under minimum regularity assumptions,” J. Soc. Indust. Appl. Math.,
5, No. 3, 105–136 (1957).
2. V. F. Babenko and S. V. Borodachev, “On optimization of cubature monotone functions of several variables,” Visn. Dnipropetr. Univ.,
Ser. Mat., Issue 7, 3–7 (2002).
3. V. B. Babenko and V. V. Babenko, “Optimization of approximate integration of set-valued functions monotone with respect to inclusion,” Ukr. Mat. Zh., 63, No. 2, 147–155 (2011); English translation: Ukr. Math. J., 63, No. 2, 177–186 (2011).
4. M. Hukuhara, “Intégration des applications mesurables dont la valeur est un compact convexe,” Funkc. Ekvac., 10, 205–223 (1967).
5. N. P. Korneichuk, Extremal Problems in Approximation Theory [in Russian], Nauka, Moscow (1976).
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