Optimization of approximate integration of set-valued functions monotone with respect to inclusion
The best quadrature formula is found for the class of convex-valued functions defined on the interval [0, 1] and monotone with respect to an inclusion.
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508657599381504 |
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| author | Babenko, V. V. Babenko, V. F. Бабенко, В. В. Бабенко, В. Ф. Бабенко, В. В. Бабенко, В. Ф. |
| author_facet | Babenko, V. V. Babenko, V. F. Бабенко, В. В. Бабенко, В. Ф. Бабенко, В. В. Бабенко, В. Ф. |
| author_sort | Babenko, V. V. |
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| description | The best quadrature formula is found for the class of convex-valued functions defined on the interval [0, 1] and monotone with respect to an inclusion. |
| first_indexed | 2026-03-24T02:28:42Z |
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UDK 517.5
V. F. Babenko
(Dnepropetr. nac. un-t; Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
V. V. Babenko (Dnepropetr. nac. un-t)
OPTYMYZACYQ PRYBLYÛENNOHO YNTEHRYROVANYQ
MNOHOZNAÇNÁX FUNKCYJ,
MONOTONNÁX PO VKLGÇENYG
The best quadrature formula is found for the class of convex-valued functions defined on the interval
[0, 1] and monotone with respect to an inclusion.
Znajdeno najkrawu kvadraturnu formulu na klasi zadanyx na vidrizku [0, 1] opukloznaçnyx
funkcij, monotonnyx vidnosno vklgçennq.
Vvedenye. Teoryq mnohoznaçn¥x otobraΩenyj aktyvno razvyvaetsq v teçenye
poslednyx desqtyletyj v svqzy s potrebnostqmy teoryy optymyzacyy, teoryy
yhr, matematyçeskoj πkonomyky y druhyx oblastej matematyky. Obzor dosty-
Ωenyj v πtom napravlenyy y dal\nejßye ss¥lky moΩno najty v [1 – 3]. V po-
slednee vremq ynteres matematykov v¥z¥vagt zadaçy approksymacyy mnoho-
znaçn¥x otobraΩenyj (sm., naprymer, [4 – 8]). VaΩn¥m napravlenyem teoryy
approksymacyy y çyslennoho analyza qvlqetsq teoryq kvadraturn¥x formul
(sm., naprymer, [9]). Vmeste s tem avtoram neyzvestn¥ rabot¥, kasagwyesq op-
tymyzacyy pryblyΩennoho yntehryrovanyq mnohoznaçn¥x funkcyj. Dannaq
rabota posvqwena ymenno πtoj problematyke.
Suwestvuet mnoho razlyçn¥x podxodov k opredelenyg yntehralov ot
mnohoznaçn¥x funkcyj (sm. [10]). Odnym yz naybolee πlementarn¥x qvlqetsq
podxod Xukuxar¥ [11], kotor¥j predloΩyl rassmatryvat\ obobwenye yntehra-
la Rymana dlq funkcyj so znaçenyqmy v prostranstve K( )Rd
kompaktn¥x v¥-
pukl¥x podmnoΩestv prostranstva Rd
(nyΩe dlq polnot¥ yzloΩenyq budut
pryveden¥ opredelenye y πlementarn¥e svojstva πtoho yntehrala).
M¥ budem rassmatryvat\ zadaçu optymyzacyy pryblyΩennoho v¥çyslenyq
yntehralov v sm¥sle Xukuxar¥ na klasse monotonn¥x po vklgçenyg funkcyj
f d: [ , ] ( )0 1 → K R , kotor¥e na koncax otrezka [0, 1] prynymagt zadann¥e zna-
çenyq: f A( )0 = , f B( )1 = . Poluçenn¥e dlq πtoho klassa rezul\tat¥ obobwa-
gt yzvestn¥e dlq çyslov¥x funkcyj rezul\tat¥ Kyfera [12].
Kratko opyßem strukturu stat\y.
V pervom punkte pryveden¥ neobxodym¥e opredelenyq y fakt¥, kasagwyesq
prostranstva kompaktn¥x v¥pukl¥x mnoΩestv, vo vtorom — neobxodym¥e op-
redelenyq y fakt¥, kasagwyesq yntehryrovanyq mnohoznaçn¥x (v¥pukloznaç-
n¥x) funkcyj. V tret\em punkte reßaetsq zadaça o nayluçßej kvadraturnoj
© V. F. BABENKO, V. V. BABENKO, 2011
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2 147
148 V. F. BABENKO, V. V. BABENKO
formule na klasse monotonn¥x (otnosytel\no vklgçenyq) funkcyj
f d: [ , ] ( )0 1 → K R .
1. Prostranstvo v¥pukl¥x mnoΩestv v Rdd
. Çerez K K= ( )Rd
budem
oboznaçat\ sovokupnost\ nepust¥x kompaktn¥x v¥pukl¥x podmnoΩestv prost-
ranstva Rd
. V sovokupnosty K vvodqtsq sledugwye operacyy.
Pust\ A , B ∈K , α ≥ 0 . Tohda
A + B = : x y x A y B+ ∈ ∈{ }: , , α A = : αx x A: ∈{ } .
MnoΩestvo A + B naz¥vaetsq summoj Mynkovskoho mnoΩestv A y B .
V yssledovanyqx po approksymacyy v¥pukl¥x tel mnohohrannykamy ys-
pol\zugtsq razlyçn¥e metryky v K K= ( )Rd
(sm., naprymer, [13]). M¥ budem
yspol\zovat\ metryku Xausdorfa δH A B( , ) , kotoraq opredelqetsq sledug-
wym obrazom:
δH A B( , ) = max sup inf , sup inf
x A y B x B y A
x y x y
∈ ∈ ∈ ∈
− −
,
hde ⋅ — evklydova norma v Rd
.
Opornoj funkcyej v¥pukloho mnoΩestva A ∈K naz¥vaetsq opredelennaq
na edynyçnoj sfere Sd−1
prostranstva Rd
funkcyq
h uA( ) = sup ,
x A
x u
∈
〈 〉 , u Sd∈ −1 .
V termynax opornoj funkcyy metryku δH
moΩno predstavyt\ v vyde
δH C D( , ) = sup ( ) ( )
u S
C D
d
h u h u
∈ −
−
1
. (1)
Otmetym, çto po otnoßenyg k metryke δH
summa Mynkovskoho y operacyq
umnoΩenyq na neotrycatel\n¥e çysla neprer¥vn¥, a metryçeskoe prostranstvo
〈 〉K, δH
qvlqetsq poln¥m.
Dlq nas vaΩn¥my budut dva sledugwyx svojstva metryky δH
:
1) dlq lgb¥x A B C D, , , ∈K
δH A B C D( , )+ + ≤ δ δH HA C B D( , ) ( , )+ ; (2)
2) dlq lgb¥x A B, ∈K y lgboho α ≥ 0
δ α αH A B( , ) = αδH A B( , ) . (3)
Svojstvo (3) qvlqetsq oçevydn¥m. Proverym v¥polnenye svojstva (2). Dlq
proyzvol\n¥x v¥pukl¥x mnoΩestv A B C D, , , , yspol\zuq predstavlenye (1),
poluçaem
δH A B C D( , )+ + =
= max sup ( ) ( ) , sup
u S
A B C D
u S
C D
d d
h u h u h
∈
+ +
∈
+
− −
−( )
1 1
(( ) ( )u h uA B−( )
+ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
OPTYMYZACYQ PRYBLYÛENNOHO YNTEHRYROVANYQ MNOHOZNAÇNÁX … 149
= max sup ( ) ( ) ( ) ( )
u S
A C B D
d
h u h u h u h u
∈ −
− + −( )
1
,
sup ( ) ( ) ( ) ( )
u S
C A D B
d
h u h u h u h u
∈ −
− + −( )
1
.
Uçyt¥vaq, çto
sup ( ) ( ) ( ) ( )
u S
A C B D
d
h u h u h u h u
∈ −
− + −( )
1
≤
≤ sup ( ) ( ) sup ( ) ( )
u S
A C
u S
B D
d d
h u h u h u h u
∈ ∈− −
−( ) + −
1 1
(( )
y
sup ( ) ( ) ( ) ( )
u S
C A D B
d
h u h u h u h u
∈ −
− + −( )
1
≤
≤ sup ( ) ( ) sup ( ) ( )
u S
C A
u S
D B
d d
h u h u h u h u
∈ ∈− −
−( ) + −
1 1
(( ) ,
naxodym
δH A B C D( , )+ + ≤
≤ max sup ( ) ( ) sup ( )
u S
A C
u S
B D
d d
h u h u h u h
∈ ∈− −
−( ) + −
1 1
(( )u( )
,
sup ( ) ( ) sup ( ) ( )
u S
C A
u S
D B
d d
h u h u h u h u
∈ ∈− −
−( ) + −
1 1
(( )
≤
≤ max sup ( ) ( ) , sup ( )
u S
A C
u S
C A
d d
h u h u h u h
∈ ∈− −
−( ) −
1 1
(( )u( )
+
+ max sup ( ) ( ) , sup ( )
u S
B D
u S
D B
d d
h u h u h u h
∈ ∈− −
−( ) −
1 1
(( )u( )
=
= δ δH HA C B D( , ) ( , )+ .
V zaklgçenye punkta otmetym sledugwee. Esly m¥ summu Mynkovskoho
rasprostranym na proyzvol\noe koneçnoe çyslo mnoΩestv Ak , k = 1, … , n ,
poloΩyv
Ak
k
n
=
∑
1
: = A Ak
k
n
n
=
−
∑ +
1
1
,
to po yndukcyy yz svojstva (2) metryky δH
poluçym
δH
k
k
n
k
k
n
A B
= =
∑ ∑
1 1
, ≤ δH
k k
k
n
A B,( )
=
∑
1
. (4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
150 V. F. BABENKO, V. V. BABENKO
2. Yntehryrovanye mnohoznaçn¥x funkcyj. Napomnym, çto razbyenyem
P otrezka [ , ]a b , a b< , naz¥vaetsq koneçnaq systema toçek x xn0, ,… πtoho
otrezka, takaq, çto a x x x bn= < <…< =0 1 . Parametrom razbyenyq P naz¥va-
etsq çyslo
λ( )P : = max
,i n
i ix x
=
−−
1
1 .
Esly v kaΩdom yz otrezkov [ , ]x xi i−1 razbyenyq P v¥brano po toçke
ξi i ix x∈ −[ , ]1 , i = 1, … , n , to poluçaem razbyenye ( , )P ξ otrezka [ , ]a b s
otmeçenn¥my toçkamy ξ ξ ξ= …( )( , , )1 n .
Pust\ zadana funkcyq f a b: [ , ] → K . KaΩdomu razbyenyg ( , )P ξ s otme-
çenn¥my toçkamy postavym v sootvetstvye yntehral\nug summu
σ ξf P; ( , )( ) = f xi i
i
n
( )ξ ∆
=
∑
1
,
hde ∆x x xi i i= − −1 .
Opredelenye. Esly suwestvuet πlement I ∈K takoj, çto dlq lgboho
ε > 0 najdetsq δ > 0 takoe, çto dlq lgboho razbyenyq ( , )P ξ s otmeçen-
n¥my toçkamy, parametr kotoroho λ δ( )P < , ymeet mesto sootnoßenye
δ ξH
i i
i
n
I f x, ( )∆
=
∑
1
< ε ,
to hovorqt, çto funkcyq f a b: [ , ] → K yntehryruema na otrezke [ , ]a b , a
πlement I naz¥vaetsq ee yntehralom. Pry πtom pyßut
I = f x dx
a
b
( )∫ .
Sovokupnost\ vsex yntehryruem¥x funkcyj f a b: [ , ] → K budem obozna-
çat\ çerez R K[ , ],a b( ) .
Analohyçno sluçag çyslov¥x funkcyj ustanavlyvagtsq sledugwye ut-
verΩdenyq.
UtverΩdenye81. Lgbaq neprer¥vnaq na [ , ]a b funkcyq f a b: [ , ] → K qv-
lqetsq yntehryruemoj na [ , ]a b .
UtverΩdenye82. Lgbaq funkcyq f a b: [ , ] → K , monotonnaq v tom sm¥s-
le, çto
a x x b≤ < ≤1 2 ⇒ f x f x( ) ( )1 2⊂ , (5)
qvlqetsq yntehryruemoj na [ , ]a b .
UtverΩdenye83. Esly f, g prynadleΩat R K[ , ],a b( ) , to yx lynejnaq
kombynacyq α βf g+ s neotrycatel\n¥my koπffycyentamy takΩe yntehry-
ruema na [ , ]a b , pryçem
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
OPTYMYZACYQ PRYBLYÛENNOHO YNTEHRYROVANYQ MNOHOZNAÇNÁX … 151
( )( )α βf g x dx
a
b
+∫ = α βf x dx g x dx
a
b
a
b
( ) ( )∫ ∫+ .
UtverΩdenye84. Esly a b c< < y f a c∈ ( )R K[ , ], , to f a b[ , ] ∈
∈ ( )R K[ , ],a b , f b cb c[ , ] [ , ],∈ ( )R K y ymeet mesto ravenstvo
f x dx
a
c
( )∫ = f x dx f x dx
a
b
b
c
( ) ( )∫ ∫+ .
UtverΩdenye85. Esly funkcyq f monotonna na [ , ]a b v sm¥sle (5), to
spravedlyv¥ vklgçenyq
f a b a f x dx f b b a
a
b
( )( ) ( ) ( )( )− ⊂ ⊂ −∫ .
Yspol\zuq opredelenye yntehrala y svojstvo (4) metryky Xausdorfa, ne-
trudno ustanovyt\ sledugwee utverΩdenye.
UtverΩdenye86. Esly f, g prynadleΩat R K[ , ],a b( ) , to
δH
a
b
a
b
f x dx g x dx( ) , ( )∫ ∫
≤ δH
a
b
f x g x dx( ), ( )( )∫ .
3. Optymyzacyq kvadraturn¥x formul na klassax monotonn¥x mnoho-
znaçn¥x funkcyj. Pust\ M A B, ( )A B⊂ — klass funkcyj
f d: [ ; ] ( )0 1 → K R , monotonn¥x v sm¥sle (5) y takyx, çto f A( )0 = , f B( )1 = ,
hde A , B — zadann¥e mnoΩestva.
Rassmotrym zadaçu o nayluçßej na klasse M A B, kvadraturnoj formule
vyda
q f( ) = C c f xk k
k
n
+
=
−
∑ ( )
1
1
, (6)
hde C d∈K( )R , c cn1 1 0, ,… ≥− , 0 11 1≤ <…< ≤−x xn . Sovokupnost\ vsex ta-
kyx formul oboznaçym çerez Q . Zadaça formulyruetsq sledugwym obrazom.
PoloΩym
R Mn A B−1( ), = inf sup ( ) , ( )
,
q Q f M
H
A B
f x dx q f
∈ ∈
∫
δ
0
1
. (7)
Trebuetsq najty velyçynu (7) y kvadraturnug formulu vyda (6), realyzugwug
toçnug nyΩngg hran\ v pravoj çasty (7). Ymenno takaq formula naz¥vaetsq
nayluçßej na klasse M A B, .
Spravedlyva sledugwaq teorema.
Teorema. Sredy vsex kvadraturn¥x formul vyda (6) nayluçßej na klasse
M A B, qvlqetsq formula
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
152 V. F. BABENKO, V. V. BABENKO
q fn−1( ) =
A B
n n
f
k
nk
n+
+
=
−
∑
2
1
1
1
,
pry πtom
R Mn A B−1( ), = sup ( ) , ( )
,f M
H
n
A B
f x dx q f
∈
−∫
δ 1
0
1
=
1
2n
A BHδ ( , ) .
Dokazatel\stvo. Uçyt¥vaq addytyvnost\ yntehrala (utverΩdenyeL4), mo-
notonnost\ funkcyy f y monotonnost\ yntehrala (utverΩdenyeL5), ymeem
f x dx( )
0
1
∫ = f x dx
k n
k n
k
n
( )
/
( )/+
=
−
∫∑
1
0
1
⊂
1
1
1
n
f
k
n
B
nk
n
+
=
−
∑ .
Analohyçno
A
n n
f
k
nk
n
+
=
−
∑1
1
1
⊂ f x dx
k n
k n
k
n
( )
/
( )/+
=
−
∫∑
1
0
1
= f x dx( )
0
1
∫ .
Takym obrazom,
A
n n
f
k
nk
n
+
=
−
∑1
1
1
⊂ f x dx( )
0
1
∫ ⊂
B
n n
f
k
nk
n
+
=
−
∑1
1
1
. (8)
Ymegt mesto sledugwye vklgçenyq:
A
n n
f
k
nk
n
+
=
−
∑1
1
1
⊂
A B
n n
f
k
nk
n+
+
=
−
∑
2
1
1
1
⊂
B
n n
f
k
nk
n
+
=
−
∑1
1
1
. (9)
Ocenym
δH
k
n
f x dx
A B
n n
f
k
n
( ) ,
0
1
1
1
2
1
∫ ∑+
+
=
−
.
DokaΩem, çto esly
X ⊂ Y ⊂ Z, (10)
to
δH Y
X Z
,
+
2
≤
1
2
δH X Z( , ) . (11)
Otsgda s uçetom (8) y (9) poluçym
δH
k
n
f x dx
A B
n
f
k
n
( ) ,
0
1
1
1
2∫ ∑+
+
=
−
≤
1
2n
A BHδ ( , ) . (12)
Ytak, dokaΩem (11).
Dlq X Y, ∈K budem yspol\zovat\ oboznaçenye
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
OPTYMYZACYQ PRYBLYÛENNOHO YNTEHRYROVANYQ MNOHOZNAÇNÁX … 153
e X Y( , ) = sup inf
x X y Y
x y
∈ ∈
− .
Rassmotrym
δH Y
X Z
,
+
2
= max , , ,e Y
X Z
e
X Z
Y
+
+
2 2
.
V sylu (10) y svojstva (2) metryky Xausdorfa ymeem
e Y
X Z
,
+
2
≤ e Z
X Z
,
+
2
≤ δH Z
X Z
,
+
2
≤
1
2
δH X Z( , ) .
Dalee
e
X Z
Y
+
2
, ≤ e
X Z
X
+
2
, ≤ δH X Z
X
+
2
, ≤
1
2
δH X Z( , ) .
Takym obrazom,
δH Y
X Z
,
+
2
= max , , ,e Y
X Z
e
X Z
Y
+
+
2 2
≤
1
2
δH X Z( , ) .
Sootnoßenye (11), a s nym y sootnoßenye (12) dokazan¥.
Teper\ pokaΩem, çto dlq lgboj kvadraturnoj formul¥ vyda (6)
sup ( ) , ( )
,f M
H
A B
f x dx q f
∈
∫
δ
0
1
≥
1
2n
A BHδ ( , ) .
Otsgda y budet sledovat\ utverΩdenye teorem¥.
Dlq proyzvol\noho nabora toçek 0 10 1 1= ≤ < … < ≤ =−x x x xn n najdetsq
k n= … −0 1 1, , , takoe, çto x x nk k+ − ≥1 1 / . PoloΩym
f x1( ) =
A x x
B x x
k
k
, ,
, ,
≤
>
y
f x2( ) =
A x x
B x x
k
k
, ,
, .
<
≥
+
+
1
1
Tohda
f x dx1
0
1
( )∫ = Ax B xk k+ −( )1 ,
f x dx2
0
1
( )∫ = Ax B xk k+ ++ −1 11( )
y
q f( )1 = q f( )2 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
154 V. F. BABENKO, V. V. BABENKO
Sledovatel\no,
sup ( ) , ( )
,f M
H
A B
f x dx q f
∈
∫
δ
0
1
≥
≥ max ( ) , ( ) , ( ) , ( )δ δH Hf x dx q f f x dx q f1 1
0
1
2 2∫
00
1
∫
≥
≥
1
2
1 1
0
1
2 2
0
δ δH Hf x dx q f f x dx q f( ) , ( ) ( ) , ( )∫
+
11
∫
≥
≥
1
2 1
0
1
2
0
1
δH f x dx f x dx( ) , ( )∫ ∫
=
=
1
2
1 11 1δH
k k k kAx B x Ax B x+ − + −( )+ +( ), ( ) .
Rassmotrym e Ax B x Ax B xk k k k( ( ), ( ))+ − + −+ +1 11 1 . Yspol\zuq teoremu dvojst-
vennosty (sm., naprymer, [14], § 2.3), ymeem
e Ax B x Ax B xk k k k( ( ), ( ))+ − + −+ +1 11 1 =
= sup sup ( ) sup
( ) (z Ax B x f Ax B xk k k
f z
∈ + − ≤ ∈ + −
−
+1 1 11ω kk
f
+
1 )
( )ω =
= sup sup ( ) ( ) ( ) sup
, ,
( )
f x A y B
k k
u A
x f x x f y
≤ ∈ ∈ ∈
+ − −
1
1
vv
v
∈
+ ++ −
B
k kx f u x f( )( ) ( ) ( )1 11 =
= sup ( ) ( ) ( ) ( ) (
f
k A k B k A kx h f x h f x h f x
≤
+ ++ − − − −
1
1 11 1 )) ( )h fB( ) =
= sup ( ) ( ) ( ) ( )
f
k k A k k Bx x h f x x h f
≤
+ +− + −( )
1
1 1 =
= ( ) sup ( ) ( )x x h f h fk k
f
B A+
≤
− −( )1
1
= ( ) ( , )x x e B Ak k+ −1 .
Analohyçno
e Ax B x Ax B xk k k k+ ++ − + −( )1 11 1( ), ( ) =
= sup sup ( ) sup
( ) (z Ax B x f Ax Bk k k
f z
∈ + − ≤ ∈ ++ +
−
1 11 1 1ω −−
xk
f
)
( )ω =
= sup sup ( ) ( ) ( ) sup
,
( )
f x A y B
k kx f x x f y
≤ ∈ ∈
+ ++ − −
1
1 11
uu A B
k kx f u x f
∈ ∈
+ −
,
( )( ) ( ) ( )
v
v1 =
= sup ( ) ( ) ( ) ( ) (
f
k A k B k A kx h f x h f x h f x
≤
+ ++ − − − −
1
1 11 1 )) ( )h fB( ) =
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
OPTYMYZACYQ PRYBLYÛENNOHO YNTEHRYROVANYQ MNOHOZNAÇNÁX … 155
= sup ( ) ( ) ( ) ( )
f
k k A k k Bx x h f x x h f
≤
+ +− − −( )
1
1 1 =
= ( ) sup ( ) ( )x x h f h fk k
f
A B+
≤
− −( )1
1
= ( ) ( , )x x e A Bk k+ −1 .
Poskol\ku B ßyre A , to e A B( , ) = 0.
Takym obrazom, dlq lgboj kvadraturnoj formul¥ q Q∈
sup ( ) , ( )
,f M
H
A B
f x dx q f
∈
∫
δ
0
1
≥
1
2
1( ) ( , )x x A Bk k
H
+ − δ ≥
1
2n
A BHδ ( , ) .
Teorema dokazana.
1. Borysovyç G. H., Hel\man B. D., M¥ßkys A. D., Obuxovskyj V. V. Mnohoznaçn¥e
otobraΩenyq // Ytohy nauky y texnyky. Mat. analyz. – 1982. – 19. – S.L127 – 230.
2. Borysovyç G. H., Hel\man B. D., M¥ßkys A. D., Obuxovskyj V. V. O nov¥x rezul\tatax v
teoryy mnohoznaçn¥x otobraΩenyj. I. Topolohyçeskye xarakterystyky y razreßymost\
operatorn¥x sootnoßenyj // Tam Ωe. – 1987. – 25. – S.L123 – 197.
3. Hel\man B. D., Obuxovskyj V. V. O nov¥x rezul\tatax v teoryy mnohoznaçn¥x otobraΩenyj.
II. Analyz y pryloΩenyq // Tam Ωe. – 1987. – 25. – S.L107 – 159.
4. Vitale R. A. Approximations of convex set-valued functions // J. Approxim. Theory. – 1979. – 26.
– P. 301 – 316.
5. Zvi Artstein. Piecewise linear approximations of set-valued maps // Ibid. – 1989. – 56. – P. 41 – 47.
6. Nira Dyn, Alona Mokhov. Approximations of set-valued functions based on the metric average //
Rend. mat. Ser. VII. – 2006. – 26. – P. 249 – 266.
7. Nira Dyn, Elza Farkhi. Approximations of set-valued functions with compact images – an
overview, approximation and probability. – Warszawa: Banach Center Publ., 2006. – Vol. 72. –
P. 1 – 14.
8. Nira Dyn, Elza Farkhi, Alona Mokhov. Approximations of set-valued functions by metric linear
operators // Constr. Approxim. – 2007. – 25. – P. 193 – 209.
9. Nykol\skyj S. M. Kvadraturn¥e formul¥. – 4-e yzd., dop. s dobavlenyem N. P. Kornejçuka.
– M.: Nauka, 1988. – 256 s.
10. Zvi Artstein, John A. Burns. Integration of compact set-valued functions // Pacif. J. Math. – 1975. –
58, # 2. – P. 297 – 306.
11. Hukuhara M. Integration des applications mesurables dont la Valeur est un compact convexe //
Funkc. ekvacioj. – 1967. – 10. – P. 205 – 223.
12. Kiefer J. Optimum sequential search and approximation methods under minimum regularity
assumptions // J. Soc. Indust. Appl. Math. – 1957. – 5, # 3. – P. 105 – 136.
13. Gruber P. M. Aspects of approximation of convex bodies // Handb. Convex Geometry / Eds P. M.
Gruber, J. M. Wills. – 1993. – P. 319 – 345.
14. Kornejçuk N. P. ∏kstremal\n¥e zadaçy teoryy pryblyΩenyq. – M.: Nauka, 1976. – 320 s.
Poluçeno 06.08.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 2
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| id | umjimathkievua-article-2706 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:28:42Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/58/0b63d96b7c777fe7f8a2a738d8536658.pdf |
| spelling | umjimathkievua-article-27062020-03-18T19:34:23Z Optimization of approximate integration of set-valued functions monotone with respect to inclusion Оптимизация приближенного интегрирования многозначных функций, монотонных по включению Babenko, V. V. Babenko, V. F. Бабенко, В. В. Бабенко, В. Ф. Бабенко, В. В. Бабенко, В. Ф. The best quadrature formula is found for the class of convex-valued functions defined on the interval [0, 1] and monotone with respect to an inclusion. Знайдено найкращу квадратурну формулу на класі заданих на відрізку [0, 1] опуклозначних функцій, монотонних відносно включення. Institute of Mathematics, NAS of Ukraine 2011-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2706 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 2 (2011); 147-155 Український математичний журнал; Том 63 № 2 (2011); 147-155 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2706/2168 https://umj.imath.kiev.ua/index.php/umj/article/view/2706/2169 Copyright (c) 2011 Babenko V. V.; Babenko V. F. |
| spellingShingle | Babenko, V. V. Babenko, V. F. Бабенко, В. В. Бабенко, В. Ф. Бабенко, В. В. Бабенко, В. Ф. Optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| title | Optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| title_alt | Оптимизация приближенного интегрирования многозначных функций, монотонных по включению |
| title_full | Optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| title_fullStr | Optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| title_full_unstemmed | Optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| title_short | Optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| title_sort | optimization of approximate integration of set-valued functions monotone with respect to inclusion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2706 |
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