Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces
We consider the problem of characterization for subspaces of the uniqueness of element of the best nonsymmetric
Saved in:
| Date: | 2008 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3205 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509253431721984 |
|---|---|
| author | Babenko, V. F. Tkachenko, M. E. Бабенко, В. Ф. Ткаченко, М. Є. |
| author_facet | Babenko, V. F. Tkachenko, M. E. Бабенко, В. Ф. Ткаченко, М. Є. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:23Z |
| description | We consider the problem of characterization for subspaces of the uniqueness of element of the best
nonsymmetric |
| first_indexed | 2026-03-24T02:38:10Z |
| format | Article |
| fulltext |
UDK 517.5
V. F. Babenko (Dnepropetr. un-t,
Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
M. E. Tkaçenko (Dnepropetr. un-t)
VOPROSÁ EDYNSTVENNOSTY ∏LEMENTA
NAYLUÇÍEHO NESYMMETRYÇNOHO L1-PRYBLYÛENYQ
NEPRERÁVNÁX FUNKCYJ SO ZNAÇENYQMY
V KB-PROSTRANSTVAX
We consider the problem of characterization for subspaces of the uniqueness of element of the best
nonsymmetric L1-approximation of functions that are continuous on a metric compact and have values
in the KB-space. We establish classes of test functions which characterize the uniqueness of element of
the best nonsymmetric approximation.
Rozhlqnuto zadaçu xarakteryzaci] pidprostoriv [dynosti elementa najkrawoho nesymetryçnoho
L1-nablyΩennq neperervnyx na metryçnomu kompakti funkcij zi znaçennqmy v K B-prostori.
Znajdeno klasy testovyx funkcij, qki xarakteryzugt\ [dynist\ elementa najkrawoho nesymet-
ryçnoho nablyΩennq.
V dannoj stat\e rassmatryvagtsq vopros¥ xarakteryzacyy podprostranstv
edynstvennosty πlementa nayluçßeho nesymmetryçnoho L1-pryblyΩenyq ne-
prer¥vn¥x na metryçeskom kompakte funkcyj so znaçenyqmy v KB-prostran-
stve. Rezul\tat¥ dannoj stat\y obobwagt nekotor¥e rezul\tat¥ yz rabot
[15–54].
Pryvedem neobxodym¥e opredelenyq yz teoryy uporqdoçenn¥x vektorn¥x
prostranstv (podrobnee ob πtom sm. v [5]).
Pust\ X — çastyçno uporqdoçennoe vektornoe prostranstvo, v kotorom po-
rqdok sohlasovan s alhebrayçeskymy operacyqmy.
Dlq nepustoho mnoΩestva E ⊂ X πlement y X∈ , udovletvorqgwyj uslo-
vyqm:
1) x ≤ y (x ≥ y) ∀ ∈x E ;
2) esly πlement z X∈ takov, çto x ≤ z (x ≥ z) dlq lgboho x Z∈ , to y ≤ z
(y ≥ z),
naz¥vaetsq supremumom (ynfymumom) mnoΩestva E y oboznaçaetsq sup E
(inf E ) ; esly Ωe mnoΩestvo E sostoyt yz koneçnoho çysla πlementov x1,
x2 , … , xn , to yx supremum y ynfymum oboznaçagtsq sootvetstvenno x1 ∨
∨ x2 ∨ … ∨ xn y x1 ∧ x2 ∧ … ∧ xn .
Esly v X dlq lgb¥x dvux πlementov x, y X∈ suwestvuet yx supremum x ∨
∨ y, to πlement x+ = x ∨ 0 naz¥vagt poloΩytel\noj çast\g πlementa x X∈ ,
πlement x− = (– x ) ∨ 0 — eho otrycatel\noj çast\g y πlement x = x+ + x−
— modulem πlementa x. Dva πlementa x , y X∈ naz¥vagtsq dyzægnktn¥my
(oboznaçenye x ∆ y), esly x ∧ y = 0.
Çastyçno uporqdoçennoe vektornoe prostranstvo X, v kotorom porqdok so-
hlasovan s alhebrayçeskymy operacyqmy y dlq lgb¥x dvux πlementov x, y X∈
suwestvuet x ∨ y, naz¥vaetsq KN-lynealom yly normyrovannoj reßetkoj,
esly v X opredelena monotonnaq norma, t.5e. norma, udovletvorqgwaq uslovyg
x ≤ y , vleçet x X ≤ y X . KN-lynealamy, v çastnosty, qvlqgtsq pro-
© V. F. BABENKO, M. E. TKAÇENKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7 867
868 V. F. BABENKO, M. E. TKAÇENKO
stranstva C a b[ , ], L a bp[ , ], 1p y dr.
KN-lyneal, v kotorom lgboe (lgboe sçetnoe) nepustoe ohranyçennoe sverxu
yly snyzu mnoΩestvo ymeet sootvetstvenno verxngg yly nyΩngg hran\,
naz¥vaetsq KN-prostranstvom (K σ N -prostranstvom).
Pust\ X — KN-lyneal y X*
— prostranstvo lynejn¥x y neprer¥vn¥x v
ob¥çnom sm¥sle funkcyonalov na X. Tohda X*
qvlqetsq poln¥m v ob¥çnom
sm¥sle KN-prostranstvom.
K σ N -prostranstvo, v kotorom norma udovletvorqet dvum dopolnytel\n¥m
uslovyqm:
1) esly xn ↓ 0, to xn X → 0;
2) esly xn ↑ + ∞ ( xn ≥ 0), to xn X → + ∞,
naz¥vaetsq KB-prostranstvom.
Pust\ Q — metryçeskyj kompakt s metrykoj ρ, Σ — σ-pole borelevskyx
podmnoΩestv metryçeskoho kompakta Q, µ — neotrycatel\naq, koneçnaq,
bezatomnaq mera, poloΩytel\naq na lgbom nepustom otkr¥tom podmnoΩestve
Q. Pust\ takΩe X — KB-prostranstvo s normoj ⋅ X .
Oboznaçym çerez C Q X( , ) prostranstvo neprer¥vn¥x funkcyj f : Q → X.
Dlq kaΩdoho x Q∈ y poloΩytel\n¥x çysel α, β poloΩym
f x f x f x( ) ( ) ( ),α β α β= ⋅ + ⋅+ − ,
f x f x f xX X( ) ( ) ( ); ,α β α β= ⋅ + ⋅+ − ,
hde f x±( ) = ±( )f x( ) ∨ 0.
Opredelym v C Q X( , ) nesymmetryçnug L1-normu, poloΩyv
f f x d x
Q
X1; , ; ,( ) ( )α β α β µ= ∫ .
Pust\ f C Q X∈ ( , ), H C Q X⊂ ( , ). Velyçynu
E f H f g g H( , ) inf :; , ; ,1 1α β α β= − ∈{ } (1)
budem naz¥vat\ nayluçßym ( , )α β -pryblyΩenyem funkcyy f mnoΩestvom H v
metryke L1. ∏lement yz H , realyzugwyj toçnug nyΩngg hran\ v (1), naz¥-
vaetsq πlementom nayluçßeho ( , )α β -pryblyΩenyq. MnoΩestvo πlementov na-
yluçßeho ( , )α β -pryblyΩenyq funkcyy f v H oboznaçym çerez P fH
( , )( )α β
, a
mnoΩestvo nulej funkcyy f na Q — çerez Z f .
Dlq f, g C Q X∈ ( , ) poloΩym
τ α β
− ( )( , ) ( ), ( )f x g x = lim
( ) ( ) ( ); , ; ,
t
X Xf x tg x f x
t→ −
+ −
0
α β α β
.
Pry α = β = 1 takoj funkcyonal rassmatryvaetsq, naprymer, v [6, s.53]. Kak
y v [6], lehko pokazat\, çto, tak kak funkcyq
r f g t xα β, , ; ,( ) =
f x tg x f x
t
X X( ) ( ) ( ); , ; ,+ −α β α β
ne ub¥vaet po t y ohranyçena sverxu na ( , )−∞ 0 , τ α β
−
( , )( , )f g suwestvuet dlq
proyzvol\n¥x f, g C Q X∈ ( , ), a takΩe obladaet sledugwymy svojstvamy:
i) dlq lgboho x Q∈ y lgboj poloΩytel\noj dejstvytel\noj funkcyy
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
VOPROSÁ EDYNSTVENNOSTY ∏LEMENTA NAYLUÇÍEHO NESYMMETRYÇNOHO … 869
γ( )x
τ γ τα β α β
− −( ) = ( )( , ) ( , )( ) ( ), ( ) ( ), ( )x f x g x f x g x ;
ii) dlq lgboho x Q∈
τ α β
β α− ( ) ≥ −( , )
; ,( ), ( ) ( )f x g x g x X .
V dal\nejßem nam ponadobytsq sledugwee obobwenye yzvestnoho kryteryq
πlementa nayluçßeho L1-pryblyΩenyq (sm., naprymer, [6] (teorema 2.1), [7]
(teorema 4.1)) na sluçaj nayluçßeho ( , )α β -pryblyΩenyq funkcyj yz C Q X( , ).
Teorema 1. Pust\ H — podprostranstvo prostranstva C Q X( , ). ∏le-
ment p H∈ qvlqetsq πlementom nayluçßeho ( , )α β -pryblyΩenyq funkcyy
f C Q X∈ ( , ) v H tohda y tol\ko tohda, kohda
Q Z Z
X
f p f p
f p x g x d x g x d x
\
( , )
; ,( )( ), ( ) ( ) ( ) ( )
− −
∫ ∫− −( ) ≤τ µ µα β
β α ∀ ∈g H . (2)
Dokazatel\stvo. Pust\ p P fH∈ ( , )( )α β
. Poskol\ku H — podprostranstvo,
dlq lgboho g H∈
f p f p tg− ≤ − +1 1; , ; ,α β α β ∀ ∈t R .
Tohda, s odnoj storon¥, vsledstvye toho, çto r f p g t xα β, ( , ; , )− ne ub¥vaet po t
y ohranyçena sverxu na ( , )−∞ 0 , sohlasno teoreme B.5Levy
Q
f p x g x d x∫ − −( )τ µα β( , ) ( )( ), ( ) ( ) =
Q
t
r f p g t x d x∫ → −
−lim ( , ; , ) ( ),
0
α β µ =
= lim
( )( ) ( ) ( )( ) ( ); , ; ,
t
Q X Xf p x tg x f p x d x
t→ −
∫ − + − −( )
0
α β α β µ
=
= lim ; , ; ,
t
f p tg f p
t→ −
− + − −
0
1 1α β α β ≤ 0 ∀ ∈g H .
S druhoj storon¥,
Q
f p x g x d x∫ − −( )τ µα β( , ) ( )( ), ( ) ( ) =
=
Q Z f p
f p x g x d x
\
( , ) ( )( ), ( ) ( )
−
∫ − −( )τ µα β –
Z
X
f p
g x d x
−
∫ ( ) ( ); ,β α µ ∀ ∈g H . (3)
Sopostavlqq dva poslednyj sootnoßenyq, poluçaem (2).
Neobxodymost\ dokazana.
Pust\ teper\ v¥polnqetsq neravenstvo (2). V sylu (3) ymeem
Q
f p x g x d x∫ − −( )τ µα β( , ) ( )( ), ( ) ( ) ≤ 0 ∀ ∈g H .
Poskol\ku funkcyq
r f p g tα β, ( , ; )− =
Q
r f p t x d x∫ −α β µ, ( ; , ) ( )
takΩe ne ub¥vaet na ( , )−∞ 0 , polahaq t = – 1, poluçaem
f p f g f p x g p x d x
Q
− − − ≤ − −( ) ≤∫ −1 1 0; , ; ,
( , ) ( )( ), ( )( ) ( )α β α β
α βτ µ ∀ ∈g H ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
870 V. F. BABENKO, M. E. TKAÇENKO
y, sledovatel\no, p P fH∈ ( , )( )α β
.
Teorema dokazana.
Pust\ dalee X — stroho normyrovannoe KB-prostranstvo, t.5e. X — KB-
prostranstvo, v kotorom ravenstvo x y X+ = x X + y X dlq x ≠ 0, y ≠ 0
vozmoΩno tol\ko kohda x = cy, hde c ∈R , c > 0.
Opredelym v X stroho monotonnug normu sledugwym obrazom: x < y
vleçet x X < y X .
Pust\ H — podprostranstvo prostranstva C Q X( , ). PoloΩym
H ′ = h C Q X g H x Q h x g xh h∈ ∃ ∈ ∀ ∈ = ±{ }( , ): ( ) ( ) .
Sledugwaq teorema obobwaet teoremu 1 yz [1], a takΩe teorem¥ 1 yz [3] y 2
yz [4] na sluçaj nesymmetryçnoho pryblyΩenyq funkcyj yz C Q X( , ).
Teorema 2. Pust\ X — stroho normyrovannoe KB-prostranstvo so stro-
ho monotonnoj normoj. KaΩdaq funkcyq f C Q X∈ ( , ) ymeet ne bolee odnoho
πlementa nayluçßeho ( , )α β -pryblyΩenyq πlementamy yz H tohda y tol\ko
tohda, kohda kaΩdaq funkcyq h H∈ ′ ymeet ne bolee odnoho πlementa nayluç-
ßeho ( , )α β -pryblyΩenyq v H.
Dlq dokazatel\stva teorem¥ nam ponadobytsq sledugwaq lemma.
Lemma 1. Pust\ h H∈ ′ { }\ 0 , 0 ∈P hH
( , )( )α β
y gh — funkcyq yz H takaq,
çto h x( ) = ±g xh( ) ∀ ∈x Q. Tohda g P hh H∈ ( , )( )α β
.
Dokazatel\stvo. Qsno, çto Zh ⊂ Zh gh− y h x( ) – g xh( ) = 2 h x( ) ∀ x ∈
∈ Q Zh gh
\ − .
V sylu svojstva ii) funkcyonala τ α β
−
( , )( , )f g y teorem¥ 1, tak kak 0 ∈
∈ P hH
( , )( )α β
, poluçaem
Q Z
h
h gh
h g x g x d x
\
( , ) ( )( ), ( ) ( )
−
∫ − −( )τ µα β =
=
Q Zh gh
h x g x d x
\
( , ) ( ), ( ) ( )
−
∫ − ( )τ µα β 2 =
=
Q Zh
h x g x d x
\
( , ) ( ), ( ) ( )∫ − ( )τ µα β –
Z Zh gh h
h x g x d x
−
∫ − ( )
\
( , ) ( ), ( ) ( )τ µα β ≤
≤
Z
X
h
g x d x∫ ( ) ( ); ,β α µ +
Z Z
X
h gh h
g x d x
−
∫
\
; ,( ) ( )β α µ =
=
Z
X
h gh
g x d x
−
∫ ( ) ( ); ,β α µ ∀ ∈g H .
Sledovatel\no, g P hh H∈ ( , )( )α β
, y lemma dokazana.
Dokazatel\stvo teorem¥ 2. Neobxodymost\ oçevydna. DokaΩem dosta-
toçnost\. Pust\ kaΩdaq funkcyq h H∈ ′ ymeet ne bolee odnoho πlementa
nayluçßeho ( , )α β -pryblyΩenyq v H v metryke L1, no suwestvuet funkcyq
f C Q X∈ ( , ) takaq, çto p1, p2 ∈ P fH
( , )( )α β , p1 ≠ p2 . Tohda
p p1 2
2
+
∈ P fH
( , )( )α β
y
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
VOPROSÁ EDYNSTVENNOSTY ∏LEMENTA NAYLUÇÍEHO NESYMMETRYÇNOHO … 871
f
p p
f p f p− + = − + −1 2
1
1 1 2 12
1
2
1
2; ,
; , ; ,
α β
α β α β
yly
Q X
f
p p
x d x∫ − +
1 2
2
( ) ( )
; ,α β
µ =
=
Q
X Xf p x f p x d x∫ − + −( )1
2 1 2( )( ) ( )( ) ( ); , ; ,α β α β µ .
Poskol\ku pod¥ntehral\n¥e funkcyy neprer¥vn¥ y neotrycatel\n¥, a mera
µ poloΩytel\na na lgbom nepustom otkr¥tom podmnoΩestve Q, to
( )( ) ( )( ) ; ,f p x f p x X− + −1 2 α β =
= ( )( ) ( )( ); , ; ,f p x f p xX X− + −1 2α β α β ∀ ∈x Q . (4)
Sohlasno teoreme III.4.3 yz [5] ( )f g+ ± ≤ f± + g± . Takym obrazom, uçyt¥vaq
monotonnost\ norm¥ y neravenstvo treuhol\nyka dlq norm, poluçaem
( )( ) ( )( ) ; ,f p x f p x X− + −1 2 α β =
= α β( )( ) ( )( ) ( )( ) ( )( )f p x f p x f p x f p x
X
− + −[ ] + − + −[ ]+ −1 2 1 2 ≤
≤ α α β β( )( ) ( )( ) ( )( ) ( )( )f p x f p x f p x f p x X− + − + − + −+ + − −1 2 1 2 ≤
≤ ( )( ) ( )( ); , ; ,f p x f p xX X− + −1 2α β α β ∀ ∈x Q ,
y tak kak v¥polnqetsq (4), to vezde ymeet mesto znak ravenstva. Uçyt¥vaq
strohug monotonnost\ norm¥ v X, ymeem
( )( ) ( )( ) ( )( ) ( )( )f p x f p x f p x f p x− + −[ ] = − + −± ± ±1 2 1 2 ∀ ∈x Q (5)
yly
( )( ) ( )( ) ( )( ) ( )( )f p x f p x f p x f p x− + − = − + −1 2 1 2 ∀ ∈x Q (6)
y
( )( ) ( )( ), ,f p x f p x
X
− + −1 2α β α β =
= α β( )( ) ( )( )f p x f p x X− + −+ −1 1 +
+ α β( )( ) ( )( )f p x f p x X− + −+ −2 2 ∀ ∈x Q . (7)
V sylu strohoj normyrovannosty X ravenstvo (7) vozmoΩno tohda y tol\ko
tohda, kohda dlq kaΩdoho x Q∈ lybo odna yz velyçyn ( )( ) ,f p x− 1 α β y
( )( ) ,f p x− 2 α β ravna nulg, lybo ( )( ) ,f p x− 1 α β = c x f p x( ) ( )( ) ,− 2 α β , hde
c x( ) — poloΩytel\naq dejstvytel\naq funkcyq.
Poslednee ravenstvo perepyßem v sledugwem vyde:
α ( )( ) ( )( )( )f p x c x f p x− − −[ ]+ +1 2 +
+ β ( )( ) ( )( )( )f p x c x f p x− − −[ ]− −1 2 = 0 ∀ ∈x Q .
Yz (5) sleduet, çto
( ) ( ) ( ) ( )f p x f p x− −±1 2∆ ∓ ∀ ∈x Q, (8)
y v sylu teorem¥ III.6.3 yz [5]
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
872 V. F. BABENKO, M. E. TKAÇENKO
α β( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )f p x c x f p x f p x c x f p x− − −[ ]( ) − − −[ ]( )+ + − −1 2 1 2∆ ∀ ∈x Q .
Sohlasno sledstvyg 1 yz teorem¥ III.6.6 yz [5] naxodym, çto dlq lgboho x Q∈
α ( ) ( ) ( )( ) ( )f p x c x f p x− − −[ ]+ +1 2 = 0
y
β ( ) ( ) ( )( ) ( )f p x c x f p x− − −[ ]− −1 2 = 0.
Sledovatel\no, ravenstvo (7) vozmoΩno tohda y tol\ko tohda, kohda dlq
kaΩdoho x Q∈ lybo odna yz velyçyn ( )( )f p x− 1 y ( )( )f p x− 2 ravna nulg,
lybo ( )( )f p x− 1 = c x f p x( )( )( )− 2 , hde c x( ) — poloΩytel\naq dejstvytel\-
naq funkcyq, pryçem c x( ) ≠ 1 ∀ ∈ −x Q Zp p\
1 2
. Teper\, polahaq
f x f x
p p x
0
1 2
2
( ) ( )
( )( )= − +
,
poluçaem, çto suwestvuet dejstvytel\naq funkcyq γ( )x takaq, çto dlq x ∈
∈ Q Zp p\
1 2−
f x x p p x0 1 2( ) ( )( )( )= −γ ,
pryçem 0 ∈ P fH
( , )( )α β
0 .
Pust\
h x( ) =
sgn ( )( )( ), \ ,
, .
γ x p p x x Q Z
x Z
p p
p p
1 2 1 2
1 2
0
− ∈
∈
−
−
Yspol\zuq ravenstvo (6), lehko pokazat\, çto Z f0
⊂ Zp p1 2− . Poπtomu h ∈
∈ C Q X( , ) y h H∈ ′ , hde gh = p1 – p2, h ≠ 0.
Poskol\ku 0 ∈ P fH
( , )( )α β
0 , sohlasno teoreme 1, uçyt¥vaq svojstva funkcyo-
nala τ α β
−
( , )( , )f g , poluçaem
Q Zh
h x g x d x
\
( , ) ( ), ( ) ( )∫ − ( )τ µα β =
Q Zh
f x g x d x
\
( , ) ( ), ( ) ( )∫ − ( )τ µα β
0 =
=
Q Zf
f x g x d x
\
( , ) ( ), ( ) ( )
0
0∫ − ( )τ µα β –
Z Zh f
f x g x d x
\
( , ) ( ), ( ) ( )
0
0∫ − ( )τ µα β ≤
≤
Z
X
f
g x d x
0
∫ ( ) ( ); ,β α µ +
Z Z
X
h f
g x d x
\
; ,( ) ( )
0
∫ β α µ =
=
Z
X
h
g x d x∫ ( ) ( ); ,β α µ ∀ ∈g H .
Takym obrazom, 0 ∈ P hH
( , )( )α β
, y po lemme 1 p1 – p2 ∈ P hH
( , )( )α β
, çto proty-
voreçyt predpoloΩenyg.
Teorema dokazana.
Kak sledstvye teorem¥ 2, poluçaem sledugwug teoremu, qvlqgwugsq
obobwenyem teorem¥ 8 yz rabot¥ [1]. Dlq koneçnomern¥x podprostranstv (a
takye podprostranstva, kak yzvestno, qvlqgtsq mnoΩestvamy suwestvovanyq
πlementa nayluçßeho pryblyΩenyq) prostranstva C Q X( , ), hde X — banaxovo
prostranstvo, y α = β = 1 analohyçn¥j rezul\tat poluçen v [2] (teorema 1).
Teorema 3. Pust\ X — stroho normyrovannoe KB-prostranstvo so
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
VOPROSÁ EDYNSTVENNOSTY ∏LEMENTA NAYLUÇÍEHO NESYMMETRYÇNOHO … 873
stroho monotonnoj normoj, H — podprostranstvo prostranstva C Q X( , ).
KaΩdaq funkcyq f ∈ C Q X( , ) budet ymet\ ne bolee odnoho πlementa nayluç-
ßeho ( , )α β -pryblyΩenyq v H tohda y tol\ko tohda, kohda dlq kaΩdoj funk-
cyy h H∈ ′ \ { }0 nulevoj πlement ne budet πlementom nayluçßeho ( , )α β -pry-
blyΩenyq v H.
Rassmotrym dalee prostranstva sledugweho vyda. Pust\ u xi i
n( ){ } =1 — sys-
tema lynejno nezavysym¥x funkcyj yz C Q( , )R . PoloΩym
Hn = p x a u x a X i n
i
n
i i i( ) ( ): , , ,= ∈ = …
=
∑
1
1 .
Zametym, çto Hn qvlqetsq podprostranstvom slaboj razmernosty n. Vper-
v¥e podprostranstva koneçnoj slaboj razmernosty b¥ly opredelen¥ v [9].
Pryvedem πto opredelenye.
Opredelenye 1. Pust\ O Q X( , ) — mnoΩestvo ohranyçenn¥x otobraΩenyj
f : Q → X , Fk{ } ⊂ O Q X( , ) — nekotoraq sovokupnost\ πlementov. ∏lement¥
Fk naz¥vagtsq slabo lynejno zavysym¥my, esly dlq lgboho nenulevoho funk-
cyonala G X∈ *
çyslov¥e funkcyy 〈 〉G Fk, arhumenta x Q∈ qvlqgtsq ly-
nejno zavysym¥my. V protyvopoloΩnom sluçae πlement¥ Fk naz¥vagtsq
slabo lynejno nezavysym¥my.
Opredelenye 2. Hovorqt, çto lynejnoe podprostranstvo H prostran-
stva O Q X( , ) ymeet slabug razmernost\ n, esly:
1) najdutsq n πlementov v H, kotor¥e slabo lynejno nezavysym¥;
2) lgb¥e n + 1 πlement yz H slabo lynejno zavysym¥.
V [4] pokazano, çto opredelennoe v¥ße podprostranstvo Hn qvlqetsq mno-
Ωestvom suwestvovanyq πlementa nayluçßeho L1-pryblyΩenyq. Analohyç-
n¥j fakt ymeet mesto dlq nayluçßyx ( , )α β -pryblyΩenyj. Dlq polnot¥ yz-
loΩenyq pryvedem eho s dokazatel\stvom.
Teorema 4. Pust\ X — KB -prostranstvo. Podprostranstvo Hn pro-
stranstva C Q X( , ) qvlqetsq mnoΩestvom suwestvovanyq πlementa nayluç-
ßeho ( , )α β -pryblyΩenyq dlq lgboj funkcyy g C Q X∈ ( , ).
Dokazatel\stvo. Pust\ g C Q X Hn∈ ( , ) \ . Dlq kaΩdoho j ∈N suwestvuet
πlement p xj ( ) =
i
n
i
j
ia u x=∑ 1
( ) ( ) yz Hn takoj, çto
g p E g H
jj n− < +
1 1
1
; , ; ,( , )
α β α β . (9)
Qsno, çto
p g p g E g H gj j n1 1 1 1 11
; , ; , ; , ; , ; ,( , )
β α α β β α α β β α≤ − + < + + ,
t.5e. suwestvuet postoqnnaq C ∈R takaq, çto
Q j X
p x d x∫ ( ) ( )
; ,β α
µ ≤ C, a sle-
dovatel\no, suwestvuet y postoqnnaq C1 ∈R takaq, çto
Q j X
p x d x∫ ( ) ( )µ ≤
≤ C1. Poπtomu dlq lgboho funkcyonala G X∈ *
s normoj G X* = 1
Q i
n
i
j
i
Q
j
Q
j X
G a u x d x G p x d x p x d x C∫ ∑ ∫ ∫
=
= ≤ ≤
1
1, ( ) ( ) , ( ) ( ) ( ) ( )( ) µ µ µ ,
y, sledovatel\no,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
874 V. F. BABENKO, M. E. TKAÇENKO
∃ ∈ +C2 R : ∀ ∈j N max ,
, ,
( )
i n
i
jG a C
= …
≤
1
2 .
Poskol\ku X*
qvlqetsq poln¥m KN-prostranstvom, to po teoreme Banaxa
– Ítejnxausa najdetsq postoqnnaq C3 ∈ +R takaq, çto ai
j
X
( ) ≤ C3 dlq vsex
i = 1, … , n; j ∈N . Tohda po teoreme Burbaky (sm., naprymer, [5, s.5279]) posle-
dovatel\nosty ai
j
j
( ){ } =
∞
1
, i = 1, … , n, kompaktn¥ v slaboj topolohyy. Znaçyt,
najdetsq posledovatel\nost\ jk ∈N , jk → ∞ pry k → ∞ , takaq, çto pry lg-
bom i = 1, … , n posledovatel\nost\ ai
j
k
k( ){ } =
∞
1
slabo sxodytsq k πlementu ai
0
yz X, y, sledovatel\no, suwestvuet posledovatel\nost\ πlementov p xjk
( ) =
=
i
n
i
j
ia u xk
=∑ 1
( ) ( ) , slabo sxodqwaqsq pry lgbom x Q∈ k πlementu p x0( ) =
=
i
n
i ia u x=∑ 1
0( ) ( ) yz Hn pry k → ∞ . A tak kak pry πtom dlq lgboho x Q∈
g x p xjk
( ) ( )−( )±
slabo sxodytsq k g x p x( ) ( )−( )±0 , to y g x p xjk
( ) ( )
,
−
α β
slabo
sxodytsq k g x p x( ) ( ) ,− 0 α β dlq kaΩdoho x Q∈ . Tohda (sm., naprymer, [8,
s.5217]) lim ( ) ( )
; ,
k
j X
g x p x
k
→∞
−
α β
≥ g x p x X( ) ( ) ; ,− 0 α β ∀ ∈x Q, pryçem nyΩnyj
predel koneçen.
Yspol\zuq teoremu B.5Levy o predel\nom perexode pod znakom yntehrala,
poluçaem
lim
; ,
k
jg p
k
→∞
−
1 α β
= lim ( ) ( ) ( )
; ,
k Q
j X
g x p x d x
k
→∞
∫ −
α β
µ ≥
≥
Q k
j X
g x p x d x
k∫
→∞
−lim ( ) ( ) ( )
; ,α β
µ ≥
Q
Xg x p x d x∫ −( ) ( ) ( ); ,0 α β µ =
= g p− 0 1; ,α β .
Tohda v sylu (9)
g p− 0 1; ,α β ≤ lim
; ,
k
jg p
k
→∞
−
1 α β
≤ E g Hn( , ) ; ,1 α β .
Sledovatel\no, p P gHn0 ∈ ( , )( )α β
.
Teorema dokazana.
Teper\ teoremu 2 dlq podprostranstva Hn moΩno sformulyrovat\ sledug-
wym obrazom.
Teorema 2*. KaΩdaq funkcyq f C Q X∈ ( , ), hde X — stroho normyrovan-
noe KB-prostranstvo so stroho monotonnoj normoj, ymeet edynstvenn¥j
πlement nayluçßeho ( , )α β -pryblyΩenyq v Hn tohda y tol\ko tohda, kohda
kaΩdaq funkcyq h H∈ ′ ymeet edynstvenn¥j πlement nayluçßeho ( , )α β -
pryblyΩenyq v Hn .
Sledugwaq teorema qvlqetsq obobwenyem teorem¥ 2 yz [3] y teorem¥ 5 yz
[4] na sluçaj nesymmetryçnoho pryblyΩenyq funkcyj yz C Q X( , ) πlementamy
podprostranstva Hn .
Kak ob¥çno, çerez ω( , )u x oboznaçym modul\ neprer¥vnosty funkcyy u ∈
∈ C Q( , )R . V [9] dokazano, çto dlq toho çtob¥ ω( , )u x b¥l neub¥vagwej,
neprer¥vnoj y poluaddytyvnoj funkcyej, ravnoj nulg v nule, neobxodymo y
dostatoçno, çtob¥ Q b¥l metryçesky v¥pukl¥m, t.5e. dlq lgb¥x toçek x0 ,
x Q1 ∈ y lgboho λ ∈( , )0 1 suwestvovala toçka x Qλ ∈ takaq, çto ρ λ( , )x x0 =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
VOPROSÁ EDYNSTVENNOSTY ∏LEMENTA NAYLUÇÍEHO NESYMMETRYÇNOHO … 875
= λ ρ⋅ ( , )x x0 1 y ρ λ( , )x x1 = ( ) ( , )1 0 1− ⋅λ ρ x x . Poπtomu dalee budem sçytat\ Q
metryçesky v¥pukl¥m.
Dlq g C Q X∈ ( , ) poloΩym
g x
g x
g x
x Q Z
x Z
X
g
g
( )
( )
( )
, \ ,
, .
=
∈
∈
0
Pust\ takΩe ω( )x = max ( , )
, ,i n
iu x
= …1
ω , dlq x Q∈ y nepustoho mnoΩestva M ⊂ Q
E x M x y
y M
( , ) inf ( , )=
∈
ρ .
PoloΩym
′′ = ∈ ∃ ∈ ∀ ∈ = ± ⋅ ( )( ){ }H h C Q X p H x Q h x p x E x Zh n h ph
( , ): ( ) ( ) ,ω .
Teorema 5. Pust\ X — stroho normyrovannoe K B -prostranstvo so
stroho monotonnoj normoj, Q — metryçesky v¥pukl¥j kompakt. KaΩdaq
funkcyq f C Q X∈ ( , ) ymeet edynstvenn¥j πlement nayluçßeho ( , )α β -pry-
blyΩenyq v Hn tohda y tol\ko tohda, kohda kaΩdaq funkcyq h H∈ ′′ ymeet
edynstvenn¥j πlement nayluçßeho ( , )α β -pryblyΩenyq v Hn .
Dokazatel\stvo. Neobxodymost\ oçevydna. DokaΩem dostatoçnost\.
Pust\ kaΩdaq funkcyq h H∈ ′′ ymeet edynstvenn¥j πlement nayluçßeho
( , )α β -pryblyΩenyq v Hn , no suwestvuet f C Q X∈ ( , ) takaq, çto p1 =
=
i
n
i ia u=∑ 1
1 ∈ P fHn
( , )( )α β
y p2 =
i
n
i ia u=∑ 1
2 ∈ P fHn
( , )( )α β
, pryçem p1 ≠ p2 .
Kak y pry dokazatel\stve teorem¥ 2, polahaq
f x f x
p p x
0
1 2
2
( ) ( )
( )( )= − +
,
poluçaem, çto suwestvuet dejstvytel\naq funkcyq γ ( )x takaq, çto
f x x p p x0 1 2( ) ( )( )( )= −γ ∀ ∈ −x Q Zp p\
1 2
.
Pust\
h x
x p p x E x Z x Q Z
x Z
p p p p
p p
( )
sgn ( ) ( ) , , \ ,
, .
=
⋅ −( ) ( )( ) ∈
∈
− −
−
γ ω1 2 1 2 1 2
1 2
0
V sylu (6) Z f0
⊂ Zp p1 2− , poπtomu h H⊂ ′′ , pryçem ph = p1 – p2 .
Poskol\ku modul\ neprer¥vnosty ne ub¥vaet y neprer¥ven, a
Z
E x Zp pω ,
1 2−( )( ) ⊂ Zp p1 2− , kak lehko vydet\,
( )( ) ,p p x E x Z a a
X p p
i
n
i i X1 2
1
1 2
1 2
− ≤ ( )( ) −−
=
∑ω .
Sledovatel\no, esly δ > 0 dostatoçno malo, to dlq kaΩdoho x Q Zp p∈ −\
1 2
h x p p x h x
p p x
x E x Z
h x xX
p p
( ) ( )( ) ( )
( )( )
sgn ( ) ,
( ) ( )− − = −
−
( )( )
=
−
δ
δ
γ ω
µ1 2
1 21
1 2
,
pryçem funkcyq µ( )x > 0 ∀ ∈ −x Q Zp p\
1 2
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
876 V. F. BABENKO, M. E. TKAÇENKO
Prynymaq vo vnymanye znaçenye funkcyy f0 , poluçaem
h x
E x Z
x p p x
f xp p
X
( )
,
( ) ( )( )
( )=
( )( )
⋅ −
−ω
γ
1 2
1 2
0 ∀ ∈ −x Q Zp p\
1 2
.
Uçyt¥vaq poslednee ravenstvo y svojstva funkcyonala τ α β
−
( , )( , )f g , a takΩe
to, çto 0 0∈P fH
( , )( )α β
, dlq kaΩdoj funkcyy p Hn∈ ymeem
Q Zp p
h x p p x p x d x
\
( , ) ( ) ( )( ), ( ) ( )
1 2
1 2
−
∫ − − −( )τ δ µα β =
=
Q Zp p
f x p x d x
\
( , ) ( ), ( ) ( )
1 2
0
−
∫ − ( )τ µα β =
=
Q Z f
f x p x d x
\
( , ) ( ), ( ) ( )
0
0∫ − ( )τ µα β –
Z Zp p f
f x p x d x
1 2 0
0
−
∫ − ( )
\
( , ) ( ), ( ) ( )τ µα β ≤
≤
Z
X
f
p x d x
0
∫ ( ) ( ); ,β α µ +
Z Z
X
p p f
p x d x
1 2 0−
∫
\
; ,( ) ( )β α µ =
=
Z
X
p p
p x d x
1 2−
∫ ( ) ( ); ,β α µ .
Otsgda sleduet, çto dlq vsex dostatoçno mal¥x δ > 0 δ( )p p1 2− ∈ P hHn
( , )( )α β
,
çto protyvoreçyt predpoloΩenyg.
Teorema dokazana.
Teorema 6. Pust\ X — stroho normyrovannoe KB-prostranstvo so
stroho monotonnoj normoj, Q — metryçesky v¥pukl¥j kompakt. KaΩdaq
funkcyq h H∈ ′′ ymeet edynstvenn¥j πlement nayluçßeho ( , )α β -pryblyΩe-
nyq podprostranstvom Hn tohda y tol\ko tohda, kohda dlq kaΩdoj funkcyy
h H∈ ′′ nulevoj πlement ne budet πlementom nayluçßeho ( , )α β -pryblyΩenyq
podprostranstvom Hn .
Dokazatel\stvo. Pust\ kaΩdaq funkcyq h H∈ ′′ ymeet tol\ko odyn
πlement nayluçßeho ( , )α β -pryblyΩenyq v Hn v metryke L1, no suwestvuet
funkcyq h H1 ∈ ′′ , h1 ≠ 0, takaq, çto 0 1∈P hHn
( , )( )α β
. Poskol\ku h H1 ∈ ′′ , su-
westvuet πlement p1 =
i
n
i ia u=∑ 1
1 ∈ Hn takoj, çto
h x p x E x Zp1 1 1
( ) ( ) ,= ± ( )( )ω , x Q∈ .
Qsno, çto Zh1
= Zp1
⊂ Zp h1 1− .
Uçyt¥vaq, çto
p x E x Z aX p
i
n
i X1
1
1
1
( ) ,≤ ( )( )
=
∑ω ∀ ∈x Q,
poluçaem, çto suwestvuet konstanta δ0 > 0 takaq, çto dlq vsex δ δ∈( , )0 0
δ ω⋅ < ( )( )p x E x ZX p1 1
( ) , , x Q∈ .
Kak y pry dokazatel\stve teorem¥ 5, poluçaem, çto dlq kaΩdoho x Q Zp∈ \
1
h x p x x h x1 1 1( ) ( ) ( ) ( )− =δ µ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
VOPROSÁ EDYNSTVENNOSTY ∏LEMENTA NAYLUÇÍEHO NESYMMETRYÇNOHO … 877
hde µ( )x — dejstvytel\naq poloΩytel\naq funkcyq.
Uçyt¥vaq poslednee ravenstvo y svojstva funkcyonala τ α β
−
( , )( , )f g , a tak-
Ωe to, çto 0 0∈P fH
( , )( )α β
, po teoreme 1 dlq kaΩdoj funkcyy p Hn∈ ymeem
Q Zh p
h p x p x d x
\
( , ) )( ), ( ) ( )
1 1
1 1
−
∫ − −( )
δ
τ δ µα β =
=
Q Zh p
h x p x d x
\
( , ) ( ), ( ) ( )
1 1
1
−
∫ − ( )
δ
τ µα β =
=
Q Zh
h x p x d x
\
( , ) ( ), ( ) ( )
1
1∫ − ( )τ µα β –
Z Zh p h
h x p x d x
1 1 1
1
−
∫ − ( )
δ
τ µα β
\
( , ) ( ), ( ) ( ) ≤
≤
Z
X
h
p x d x
1
∫ ( ) ( ); ,β α µ +
Z Z
X
h p h
p x d x
1 1 1−
∫
δ
β α µ
\
; ,( ) ( ) =
=
Z
X
h p
p x d x
1 1−
∫
δ
β α µ( ) ( ); , .
Otsgda sleduet, çto δ p1 ∈ P hHn
( )1 dlq lgboho δ δ∈( , )0 0 , çto protyvoreçyt
predpoloΩenyg.
Pust\ teper\ dlq lgboj funkcyy h H∈ ′′ nulevoj πlement ne qvlqetsq
πlementom nayluçßeho ( , )α β -pryblyΩenyq podprostranstvom Hn v metryke
L1, no suwestvuet funkcyq h H0 ∈ ′′ takaq, çto p1, p2 ∈ P hHn
( , )( )α β
0 , p1 ≠ p2 .
Tohda
p p
P hHn
1 2
02
+ ∈ ( , )( )α β
y dlq vsex x Q∈ v¥polnqetsq ravenstvo
( )( ) ( )( ) ( )( ) ( )( ); , ; , ; ,h p x h p x h p x h p xX X X0 1 0 2 0 1 0 2− + − = − + −α β α β α β .
Polahaq H0 = h0 –
p p1 2
2
+
y rassuΩdaq, kak pry dokazatel\stve teorem¥ 2,
poluçaem, çto suwestvuet dejstvytel\naq funkcyq γ ( )x takaq, çto dlq x ∈
∈ Q Zp p\
1 2−
H x x p p x0 1 2( ) ( ) ( )( )= ⋅ −γ
y ymeet mesto vklgçenye Zh p p0 1 2 2− +( ) / ⊂ Zp p1 2− .
PoloΩym
h x1( ) =
sgn ( ) ( ) , , \ ,
, .
γ ωx p p x E x Z x Q Z
x Z
p p p p
p p
⋅ −( ) ( )( ) ∈
∈
− −
−
1 2 1 2 1 2
1 2
0
Qsno, çto h H1 ∈ ′′ , Zh1
= Zp p1 2− .
Uçyt¥vaq svojstva funkcyonala τ α β
−
( , )( , )f g , poluçaem, çto dlq lgboj
funkcyy p Hn∈
τ τ− −( ) = ( )h x p x H x p x1 0( ), ( ) ( ), ( ) ∀ ∈ −x Q Zp p\
1 2
.
Poskol\ku 0 0∈P HHn
( , )( )α β
, sohlasno teoreme 1 dlq kaΩdoj funkcyy p Hn∈
ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
878 V. F. BABENKO, M. E. TKAÇENKO
Q Zh
h x p x d x
\
( , ) ( ), ( ) ( )
1
1∫ − ( )τ µα β =
Q Zh
H x p x d x
\
( , ) ( ), ( ) ( )
1
0∫ − ( )τ µα β =
=
Q ZH
H x p x d x
\
( , ) ( ), ( ) ( )
0
0∫ − ( )τ µα β –
Z Zh H
H x p x d x
1 0
0
\
( , ) ( ), ( ) ( )∫ − ( )τ µα β ≤
≤
Z
X
H
p x d x
0
∫ ( ) ( ); ,β α µ +
Z Z
X
h H
p x d x
1 0
\
; ,( ) ( )∫ β α µ =
=
Z
X
h
p x d x
1
∫ ( ) ( ); ,β α µ .
Takym obrazom, 0 1∈P hHn
( , )( )α β
, çto protyvoreçyt predpoloΩenyg.
Teorema dokazana.
Yz teorem 1, 5 y 6 v¥tekaet takoe sledstvye.
Sledstvye. Pust\ X — stroho normyrovannoe K B -prostranstvo so
stroho monotonnoj normoj, Q — metryçesky v¥pukl¥j kompakt. KaΩdaq
funkcyq f C Q X∈ ( , ) ymeet edynstvenn¥j πlement nayluçßeho ( , )α β -pry-
blyΩenyq podprostranstvom Hn tohda y tol\ko tohda, kohda dlq kaΩdoj
funkcyy h H∈ ′′ (h ≠ 0) suwestvuet funkcyq p Hn∈ takaq, çto
Q Z Z
X
h h
h x p x d x p x d x
\
( , )
; ,( ), ( ) ( ) ( ) ( )∫ ∫− ( ) >τ µ µα β
β α .
1. Strauß H. Eindeutigkeit in der L1-approximation // Math. Z. – 1981. – S. 63 – 74.
2. Kroo A. A general approach to the study of Chebyshev subspaces in L1-approximation of
continuous functions // J. Approxim. Theory. – 1987. – 51. – P. 98 – 111.
3. Babenko V. F., Hlußko V. N. O edynstvennosty πlementa nayluçßeho pryblyΩenyq v
metryke prostranstva L1 // Ukr. mat. Ωurn. – 1994. – 46, # 5. – S. 475 – 483.
4. Babenko V. F., Horbenko M. E. O edynstvennosty πlementa nayluçßeho L1-pryblyΩenyq
dlq funkcyj so znaçenyqmy v banaxovom prostranstve // Tam Ωe. – 2000. – 52 , # 1. –
S.5305–534.
5. Vulyx B. Z. Vvedenye v teoryg poluuporqdoçenn¥x prostranstv. – M.: Fyzmathyz, 1961. –
407 s.
6. Pinkus A. L1-approximation // Cambridge Tracts Math. – Cambridge: Cambridge Univ. Press,
1989. – 239 p.
7. Rosema E. Almost Chebyshev subspaces of L E1( , )µ // Pacif. J. Math. – 1974. – 53. – P. 585 –
604.
8. Lgsternyk L. A., Sobolev V. Y. ∏lement¥ funkcyonal\noho analyza. – M.: Nauka, 1965. –
519 s.
9. Babenko V. F., Pyçuhov S. A. Approksymacyq neprer¥vn¥x vektor-funkcyj // Ukr. mat.
Ωurn. – 1994. – 46, # 11. – S. 1435 – 1448.
Poluçeno 22.06.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
|
| id | umjimathkievua-article-3205 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:10Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/29/92a21f246bb16795fa51e20df3563229.pdf |
| spelling | umjimathkievua-article-32052020-03-18T19:48:23Z Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces Вопросы единственности элемента наилучшего несимметричного L 1-приближения непрерывных функций со значениями в KB-пространствах Babenko, V. F. Tkachenko, M. E. Бабенко, В. Ф. Ткаченко, М. Є. We consider the problem of characterization for subspaces of the uniqueness of element of the best nonsymmetric Institute of Mathematics, NAS of Ukraine 2008-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3205 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 7 (2008); 867 – 878 Український математичний журнал; Том 60 № 7 (2008); 867 – 878 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3205/3156 https://umj.imath.kiev.ua/index.php/umj/article/view/3205/3157 Copyright (c) 2008 Babenko V. F.; Tkachenko M. E. |
| spellingShingle | Babenko, V. F. Tkachenko, M. E. Бабенко, В. Ф. Ткаченко, М. Є. Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces |
| title | Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces |
| title_alt | Вопросы единственности элемента наилучшего несимметричного L 1-приближения непрерывных функций со значениями в KB-пространствах |
| title_full | Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces |
| title_fullStr | Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces |
| title_full_unstemmed | Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces |
| title_short | Problem of uniqueness of an element of the best nonsymmetric L 1-approximation of continuous functions with values in KB -spaces |
| title_sort | problem of uniqueness of an element of the best nonsymmetric l 1-approximation of continuous functions with values in kb -spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3205 |
| work_keys_str_mv | AT babenkovf problemofuniquenessofanelementofthebestnonsymmetricl1approximationofcontinuousfunctionswithvaluesinkbspaces AT tkachenkome problemofuniquenessofanelementofthebestnonsymmetricl1approximationofcontinuousfunctionswithvaluesinkbspaces AT babenkovf problemofuniquenessofanelementofthebestnonsymmetricl1approximationofcontinuousfunctionswithvaluesinkbspaces AT tkačenkomê problemofuniquenessofanelementofthebestnonsymmetricl1approximationofcontinuousfunctionswithvaluesinkbspaces AT babenkovf voprosyedinstvennostiélementanailučšegonesimmetričnogol1približeniânepreryvnyhfunkcijsoznačeniâmivkbprostranstvah AT tkachenkome voprosyedinstvennostiélementanailučšegonesimmetričnogol1približeniânepreryvnyhfunkcijsoznačeniâmivkbprostranstvah AT babenkovf voprosyedinstvennostiélementanailučšegonesimmetričnogol1približeniânepreryvnyhfunkcijsoznačeniâmivkbprostranstvah AT tkačenkomê voprosyedinstvennostiélementanailučšegonesimmetričnogol1približeniânepreryvnyhfunkcijsoznačeniâmivkbprostranstvah |