Exact order of relative widths of classes $W^r_1$ in the space $L_1$
As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions.
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| Datum: | 2005 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509822029398016 |
|---|---|
| author | Parfinovych, N. V. Парфінович, Н. В. |
| author_facet | Parfinovych, N. V. Парфінович, Н. В. |
| author_sort | Parfinovych, N. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:18Z |
| description | As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions. |
| first_indexed | 2026-03-24T02:47:12Z |
| format | Article |
| fulltext |
UDK 517.5
N. V. Parfinovyç (Dnipropetr. un-t)
TOÇNYJ PORQDOK VIDNOSNYX POPEREÇNYKIV
KLASIV Wr
1 U PROSTORI L1
As n → ∞, the exact order of relative widths of classes W r
1 of periodic functions in the space L1 is
found under restrictions on higher derivatives of approximating functions.
Znajdeno toçnyj porqdok pry n → ∞ vidnosnyx popereçnykiv klasiv W r
1 periodyçnyx funkcij
u prostori L1 pry obmeΩennqx na starßi poxidni nablyΩagçyx funkcij.
Nexaj Lp , 1 ≤ p ≤ ∞, — prostory 2 π-periodyçnyx funkcij f : R → R z vidpo-
vidnymy normamy || f ||p . Qkwo r ∈ N, to çerez Wp
r
poznaçymo klas funkcij
f ∈ Lp , qki magt\ lokal\no absolgtno neperervnu poxidnu f
(
r
–
1
)
i taki, wo
|| f (
r
)
||p ≤ 1.
Najkrawym nablyΩennqm klasu M ⊂ L p mnoΩynog H ⊂ L p v metryci
prostoru Lp nazyva[t\sq velyçyna
E ( M, H )p : = sup sup
f M h H
pf h
∈ ∈
− .
Velyçyna
dn ( M, Lp ) : = inf ( , )
H
n p
n
E M H ,
de Hn — pidprostory prostoru Lp taki, wo dim Hn ≤ n, nazyva[t\sq n-
popereçnykom za Kolmohorovym [1] klasu M u prostori Lp .
Nexaj M
′ ⊂ Lp — deqkyj klas funkcij. Poklademo
dn ( M, Lp , M
′
) : =
inf ( , )
H
n p
n
E M H M∩ ′ ,
de Hn — pidprostory prostoru Lp , dim Hn ≤ n. Velyçyny typu dn ( M, Lp , M
′
)
vvedeni do rozhlqdu V. M. Konovalovym [2] i nazyvagt\sq vidnosnymy
popereçnykamy.
Vidomo (dyv., napryklad, [3, c. 249]), wo dlq vsix r ∈ N i p ∈ [ 1, ∞ ]
d W Ln p
r
p,( ) � n
–
r
, n → ∞. (1)
Zrozumilo takoΩ, wo dlq dovil\noho r ∈ N
d W L Wn
r r
2 2 2, ,( ) � n
–
r
, n → ∞. (2)
Prote povedinka vidnosnyx popereçnykiv d W L Wn
r r
∞ ∞ ∞( ), , i d W L Wn
r r
1 1 1, ,( ) pry
n → ∞ istotno vidriznq[t\sq vid (1) i (2). V. M. Konovalovym [2] bulo dovedeno,
wo dlq vsix r = 2, 3, …
d W L Wn
r r
∞ ∞ ∞( ), , � n
–
2
, n → ∞. (3)
Pizniße V. F. Babenko [4] doviv, wo pry r = 3, 4, …
d W L Wn
r r
1 1 1, ,( ) � n
–
2
, n → ∞. (4)
© N. V. PARFINOVYÇ, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10 1409
1410 N. V. PARFINOVYÇ
S. B. St[çkin vyslovyv prypuwennq, wo dlq dovil\noho fiksovanoho dodatnoho
çysla ε pry vsix r = 3, 4, … ma[ misce spivvidnoßennq
d W L Wn
r r
∞ ∞ ∞+( ), , ( )1 ε � n
–
r
, n → ∞.
V. F. Babenko [5] doviv istynnist\ ci[] hipotezy, a v [6] otrymav analohiçni re-
zul\taty dlq popereçnykiv d W L Wn
r r
1 1 11, , ( )+( )ε .
Nexaj εn n{ } =
∞
1 — nezrostagça, αn n{ } =
∞
1 — nespadna poslidovnosti dodat-
nyx çysel. U robotax [6, 7] otrymano ocinky zverxu dlq poslidovnostej vely-
çyn d W L Wn
r
n
r
1 1 11, , ( )+( )ε i d W L Wn
r
n
r k
1 1 1, , α +( ) vidpovidno.
U danij roboti pokazano, wo ocinky znyzu dlq cyx velyçyn zbihagt\sq z ocin-
kamy zverxu, tobto vstanovleno porqdkovi rivnosti pry n → ∞ dlq poslidov-
nostej velyçyn d W L Wn
r
n
r
1 1 11, , ( )+( )ε i d W L Wn
r
n
r k
1 1 1, , α +( ).
Teorema 1. Nexaj r = 3, 4, … i εn n{ } =
∞
1 — nezrostagça poslidovnist\ do-
datnyx çysel. Todi pry n → ∞ magt\ misce spivvidnoßennq
d W L Wn
r
n
r
1 1 11, , ( )+( )ε �
n n
n n O
r
n
r
n n
n
− −
−
/ → ∞ →
=
ε ε ε
ε
1 2 2
2 2
0
1
, , ,
, ( ).
Teorema 2. Nexaj r = 2, 3, … , k = 1, 2, … i αn n{ } =
∞
1 — nespadna
poslidovnist\ dodatnyx çysel. Todi pry n → ∞ magt\ misce spivvidnoßennq
d W L Wn
r
n
r k
1 1 1, , α +( ) > C > 0,
qkwo αn [ obmeΩenog;
d W L Wn
r
n
r k
1 1 1, , α +( ) �
1
εn
r k rn/ ,
qkwo αn = εn n
k
, εn n
k → ∞, εn → 0, i
d W L Wn
r
n
r k
1 1 1, , α +( ) �
1
nr ,
qkwo αn ≥ C n
k
.
Dovedennq teoremy 1 . Ocinky zverxu dlq popereçnykiv
d W L Wn
r
n
r
1 1 11, , ( )+( )ε vyplyvagt\ iz rezul\tativ V. F. Babenka [6]. Znajdemo
ocinku znyzu. Zafiksu[mo poslidovnist\ n-vymirnyx pidprostoriv { Hn } pros-
toru L1 . Vykorystovugçy teoremu dvo]stosti dlq najkrawyx L1-nablyΩen\
opuklog mnoΩynog [8] (teorema 1.4.1), ma[mo
En : = E W H Wr
n n
r
1 1 1
1, ( )∩ +( )ε =
=
sup sup ( ) ( ) ( ) sup ( ) ( )
f W g
n
u H Wr
n
r
f t g t dt u t g t dt
∈ ≤ ∈∞
∫ ∫− +
1 11 0
2
0
2
1
π π
ε
∩
=
=
sup sup ( ) ( ) ( ) sup ( ) ( )( )
g W f
f
n
u H W
r
r
n
r
f t g t dt u t g t dt
∈ ≤
⊥
∈∞
∫ ∫− +
1 1
1
1
0
2
0
2
1
π π
ε
∩
=
=
sup ( ) ( ) sup ( ) ( )( )
g W
n
u H W
r
r
n
r
E g u t g t dt
∈
∞
∈∞
− +
∫1
1 0
2
ε
π
∩
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
TOÇNYJ PORQDOK VIDNOSNYX POPEREÇNYKIV KLASIV W r
1 … 1411
Nexaj dlq bud\-qkoho t ∈ R ϕλ, 0 ( t ) = sgn sin λ t i ϕλ, r ( ⋅ ) — r-j 2π / λ-perio-
dyçnyj intehral vid funkci] ϕλ, 0 ( ⋅ ), wo ma[ nul\ove seredn[ znaçennq na perio-
di. Zaznaçymo, wo dlq bud\-qkoho α ∈ R i l ∈ N ϕl, r ( ⋅ + α ) ∈ W r
∞ , tomu
En ≥
sup sup ( ) ( ) sup ( ) ( ),
( )
,
l
l r n
u H W
r
l r
n
r
u t t dt
∈ ∈
∞
∈
⋅ + − + +
∫
N Rα
π
ϕ α ε ϕ α1
1 0
2
∩
=
=
sup ( ) inf sup ( ) ( ),
( )
,
l
l r n
u H W
r
l r
n
r
u t t dt
∈
∞
∈
− + +
∫
N
ϕ ε ϕ α
α
π
1
1 0
2
∩
. (5)
Qkwo l > n, to
ϕl r, ∞ –
( ) inf sup ( ) ( )( )
,1
1 0
2
+ +
∈
∫ε ϕ α
α
π
n
u H W
r
l r
n
r
u t t dt
∩
< ϕn r, ∞,
ale, qk vidomo,
d W L Wn
r
n
r
1 1 11, , ( )+( )ε ≥ d W Ln
r
1 1,( ).
Tomu zovnißng toçnu verxng meΩu v (5) moΩna braty lyße po 1 ≤ l ≤ n. Vyko-
rystavßy metody z [9], pokaΩemo, wo
inf sup ( ) ( )( )
,
α
π
ϕ α
u H W
r
l r
n
r
u t t dt
∈
∫ +
∩ 1 0
2
≤ ϕ π
l r t
n, max +
8
, (6)
de tmax take, wo ϕl r, ∞ = ϕl r t, max( ) .
Prypustymo, wo, navpaky, dlq koΩnoho α ∈ R
sup ( ) ( )( )
,
u H W
r
l r
n
r
u t t dt
∈
∫ +
∩ 1 0
2π
ϕ α > ϕ π
l r t
n, max +
8
.
Dovil\nomu α ∈ R postavymo u vidpovidnist\ element uα z Hn ∩ W r
1 takyj,
wo
0
2π
α∫ u x dxr( )( ) = 1
i
0
2π
α ϕ α∫ +u t t dtr
l r
( )
,( ) ( ) > ϕ π
l r t
n, max +
8
. (7)
Oçevydno, wo pry zroblenyx prypuwennqx element uα isnu[.
Funkciq ϕl, r nabuva[ ekstremal\nyx znaçen\ v 2 l toçkax vidrizka [ 0, 2 π ].
Poznaçymo, ci toçky ti
max , 1 ≤ i ≤ 2 l. Nexaj
Un, α =
i
n
i ix
n
x
n=
− − − +
1
2
4 4∪ max max;α π α π
.
PokaΩemo, wo dlq bud\-qkoho α ∈ R
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1412 N. V. PARFINOVYÇ
0
2π
α∫ u x dxr( )( ) –
U
r
n
u x dx
,
( )( )
α
α∫ < 2– 1
0
2π
α∫ u x dxr( )( ) =
1
2
, (8)
tobto
U
r
n
u x dx
,
( )( )
α
α∫ >
1
2
.
Prypustymo suprotyvne. Todi z (7) otryma[mo
ϕ π
l r t
n, max +
8
<
0
2π
α ϕ α∫ +u x x dxr
l r
( )
,( ) ( ) =
=
U
r
l r
n
u x x dx
,
( )
,( ) ( )
α
α ϕ α∫ + +
0 2,
( )
,
\ ,
( ) ( )
π
α
α
ϕ α
[ ]
∫ +
U
r
l r
n
u x x dx ≤
≤ ϕ
α
αl r
U
rt u x dx
n
, max
( )( ) ( )
,
∫ + ϕ π
π
α
α
l r
U
rt
n
u x dx
n
, max
,
( )
\ ,
( )+
[ ]
∫4
0 2
=
= λϕl r t, max( ) + ( ) , max1
4
− +
λ ϕ π
l r t
n
,
de zhidno zi zroblenymy prypuwennqmy
λ =
0
2π
α∫ u x dxr( )( ) ≤
1
2
.
Oskil\ky funkciq ϕl, r ( tmax + ⋅ ) opukla dohory v pravomu okoli nulq, to
ϕ π
l r t
n, max +
8
< λϕl r t, max( ) + ( ) , max1
4
− +
λ ϕ π
l r t
n
≤
≤ ϕ λ π
l r t
n, max ( )+ −
1
4
≤ ϕ π
l r t
n, max +
8
.
Takym çynom, otrymaly supereçnist\
ϕ π
l r t
n, max +
8
< ϕ π
l r t
n, max +
8
.
OtΩe, (8) spravdΩu[t\sq.
Teper rozhlqnemo mnoΩynu
Mn = α π= ∈ ≤ ≤ −{ }k
n
k k n
2
1 2 1, ,N .
Oskil\ky dlq vsix n ≥ 2 ma[mo card Mn = 2 n – 1 > n, to funkci] uα , α ∈ Mn , a
z nymy i u r
α
( )
, budut\ linijno zaleΩnymy. OtΩe, znajdut\sq çysla Aα , α ∈ Mn ,
taki, wo α α α∈∑ M
r
n
A u( ) = 0 i α α∈∑ Mn
A = 1.
Na pidstavi oznaçen\ i dovedenyx vlastyvostej uα , qk v [9], ma[mo
0 =
0
2π
α
α α∫ ∑
∈M
r
n
A u x dx( )( ) ≥
β α
α α
β∈ ∈
∑ ∫ ∑
M U M
r
n n n
A u x dx
,
( )( ) ≥
≥
β
β β
α
α β
α α
β∈ ∈
≠
∑ ∫ ∑−
M U
r
M
r
n n n
A u x A u x dx
,
( ) ( )( ) ( ) >
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
TOÇNYJ PORQDOK VIDNOSNYX POPEREÇNYKIV KLASIV W r
1 … 1413
>
β
β
α
α β
α α
β∈ ∈
≠
∑ ∑ ∫−
M M U
r
n n n
A
A u x dx
2
,
( )( ) =
=
1
2 β
β
∈
∑
Mn
A –
β α
α β
α α
β∈ ∈
≠
∑ ∑ ∫
M M U
r
n n n
A u x dx
,
( )( ) =
=
1
2
–
α
α
β
β α
α
β∈ ∈
≠
∑ ∑ ∫
M M U
r
n n n
A u x dx
,
( )( ) ≥
≥
1
2
–
α
α
π
α
α∈ [ ]
∑ ∫
M U
r
n n
A u x dx
0 2,
( )
\ ,
( ) >
1
2
–
1
2
= 0.
Otrymana supereçnist\ dovodyt\ spivvidnoßennq (6). Dlq sprowennq zapysu, ne
zmenßugçy zahal\nosti, moΩemo vvaΩaty, wo v (6) tmax = 0. Pislq zroblenyx
zauvaΩen\ dlq En moΩemo zapysaty
En ≥ max ( )
1≤ ≤l n
nF l ,
de
Fn ( l ) = ϕl r, ∞ – ( ) ,1
8
+
ε ϕ π
n l r n
=
1
1
81 1
l
l
nr r n rϕ ε ϕ π
, ,( )∞ − +
{ }.
Rozhlqnemo porqd z Fn ( l ) funkcig dijsnoho arhumentu λ ∈ [ 1, n ]
Fn ( λ ) =
1
1
81 1λ
ϕ ε ϕ λπ
r r n r n, ,( )∞ − +
.
Doslidymo spoçatku vypadok, koly εn n
2 → ∞, εn → 0, n → ∞ . Dlq poxidno]
funkci] Fn ( λ ) ma[mo
′Fn( )λ = –
r
nr r n rλ
ϕ ε ϕ λπ
+ − +
1 1 10 1
8, ,( ) ( ) –
1
8 81 1 1
+
+ −
ε
λ
ϕ λπ λπn
r r n n, =
= –
( )
( ), ,
1
0
81 1 1
+ −
+
ε
λ
ϕ ϕ λπn
r r r
r
n
–
1
8 81 1 1
+
+ −
ε
λ
λπϕ λπn
r rn n, +
r n
r r
ε
λ
ϕ+ ∞1 =
=
1
8 81
0
8
1 1 1 1
+ −
+ − −
/
∫ε
λ
ϕ λπϕ λπλπ
n
r
n
r rr t dt
n n, ,( ) +
r n
r r
ε
λ
ϕ+ ∞1 1, .
Vraxovugçy, wo
r t dt
n
r
0
8
1 1
π
ϕ
/
∫ −, ( ) –
π ϕ π
8 81 1n nr, −
� n
–
2
pry n → ∞ , v rozhlqnutomu vypadku ( εn n
2 → ∞ , εn → 0, n → ∞ ) pry vsix
dostatn\o velykyx n ′Fn( )1 > 0. Rozhlqnemo ′F nn( ) :
′F nn( ) =
1
8 81
0
8
1 1 1 1
+ −
+ − −
/
∫ε ϕ πϕ ππ
n
r r r
n
r t dt, ,( ) +
r
n
n
r r
ε ϕ+ ∞1 1, .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1414 N. V. PARFINOVYÇ
Oskil\ky εn → 0, n → ∞, to pry dosyt\ velykyx n znak ′F nn( ) bude vyznaça-
tys\ znakom perßoho dodanka. Vraxovugçy te, wo funkciq ϕ1, r – 1 ( ⋅ ) opukla
donyzu na [ 0, π / 2 ] i nabuva[ na ( 0, π / 2 ] vid’[mnyx znaçen\, pry dosyt\ vely-
kyx n otrymu[mo ′F nn( ) < 0.
Rozhlqnemo teper funkcig
Gn ( λ ) = λ λr
nF+ ′1 ( ).
Dlq ]] poxidno] ma[mo
′Gn( )λ = r
n nn r( ) ,1
8 81 1+
−ε ϕ λπ π
–
π ε ϕ λπ( )
,
1
8 81 1
+
−
n
rn n
–
–
( )
,
1
8 8 81 2
+
−
ε πλπϕ λπn
rn n n
=
=
π ε ϕ λπ λπϕ λπ
8
1 1
8 8 81 1 1 2n
r
n n nn r r( ) ( ) , ,+ −
−
− − =
=
π ε ϕ λπϕ λπλπ
8
1 1
8 8
0
8
1 2 1 2n
r t dt
n nn
n
r r( ) ( ) ( ), ,+ − −
/
∫ − − .
Oskil\ky funkciq ϕ1, r – 2 ( ⋅ ) opukla donyzu na [ 0, π / 2 ] (linijna pry r = 3), to
′Gn( )λ < 0 dlq vsix λ ∈ [ 1, n ]. OtΩe, funkciq Gn ( λ ), a z neg i ′F nn( ) , [ spad-
nog na c\omu promiΩku. Vraxovugçy toj fakt, wo na kincqx vidrizka [ 1, n ]
′Fn( )λ nabuva[ znaçen\ riznyx znakiv, moΩemo stverdΩuvaty, wo isnu[ [dyna
toçka λn ∈ ( 1, n ) taka, wo
sup ( )
1≤ ≤λ
λ
n
nF = Fn ( λn ).
Neobxidnu umovu ekstremumu funkci] Fn ( λ ) ( ′Fn n( )λ = 0) zapyßemo u vyhlqdi
( ) ( ), ,1
8 8
0
8
1 1 1 1+ −
/
∫ − −ε ϕ λ πϕ λ π
λ π
n
n
r
n
r
nr t dt
n n
n
= – r n rε ϕ1 0, ( ) . (9)
Oskil\ky funkciq ϕ1, r – 1 ( ⋅ ) opukla donyzu na promiΩku [ 0, π / 2 ] i ϕ1, r – 1 ( t ) <
< 0 ∀ t ∈ ( 0, π / 2 ], to dlq λn ∈ [ 1, n ] vykonu[t\sq nerivnist\
λ πϕ λ πn
r
n
n n8 81 1, −
≤
0
8
1 1
λ π
ϕ
n n
r t dt
/
∫ −, ( ) .
Zrozumilo takoΩ, wo dlq λn ∈ [ 1, n ]
ϕ λ π
1 1 8,r
n
n−
≥ ′ −ϕ λ π
1 1 0
8, ( )r
n
n
.
Skorystavßys\ cymy mirkuvannqmy i tym, wo ′ −ϕ1 1 0, ( )r = – ϕ1 2,r− ∞ , moΩemo
oderΩaty ocinky
r t dt
n n
r
0
8
1 1
λ π
ϕ
/
∫ −, ( ) –
λ πϕ λ πn
r
n
n n8 81 1, −
≥
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
TOÇNYJ PORQDOK VIDNOSNYX POPEREÇNYKIV KLASIV W r
1 … 1415
≥ ( ) ( ),r t dt
n n
r−
/
∫ −1
0
8
1 1
λ π
ϕ ≥ – ( ) ,r t dtr
nn
− − ∞
/
∫1 1 2
0
8
ϕ
λ π
=
= – ( )
,
r
n
r n−
− ∞1
2 8
1 2
2ϕ λ π
,
zvidky odrazu vyplyva[
λ πn
n8
2
≥
2 0
1 1
1
1 2
r
r
n r
r n
ε ϕ
ϕ ε
,
,
( )
( ) ( )− +− ∞
. (10)
Z inßoho boku,
r t dt
n n
r
0
8
1 1
λ π
ϕ
/
∫ −, ( ) –
λ πϕ λ πn
r
n
n n8 81 1, −
≤
≤ ( ) ( ),r t dt
n n
r−
/
∫ −2
0
8
1 1
λ π
ϕ ≤ –
( ) ,r
t dt
r
nn− − ∞
/
/
∫
2
2
1 1
0
8ϕ
π
λ π
=
= –
( ) ,r
n
r n
−
− ∞2
8
1 1
2ϕ
π
λ π
,
zvidky
λ πn
n8
2
≤
π ε ϕ
ϕ ε
r
r
n r
r n
1
1 1
0
2 1
,
,
( )
( ) ( )− +− ∞
. (11)
Zistavlqgçy (10) i (11), otrymu[mo
C n n1 ε ≤ λn ≤ C n n2 ε , (12)
de C1 > 0 i C2 > 0 — konstanty, qki ne zaleΩat\ vid n. Oberemo l ∈ { 1, … , n }
tak, wob
l – 1 ≤ λn ≤ l. (13)
Teper, vraxovugçy (9), (12) i (13), dlq En ma[mo
En = max ( )
1≤ ≤l n
nF l ≥ Fn ( l ) =
1
0 1
81 1
l nr r n rϕ ε ϕ λπ
, ,( ) ( )− +
=
= –
1
0
8
1 1
+ /
∫ −
ε ϕ
π
n
r
l n
r
l
t dt, ( ) –
ε ϕn
r r
l
1 0, ( ) ≥
≥ –
1
8
1
81 1
l n r nr
n n
r
nλ π ε ϕ λ π+
−, ≥
( )
( ) ,
l r
l l r n
r
r r
n
r
n−
−
+
− ∞1
1
8
2
1 1
2ε ϕ λ π
π
≥
≥
C
nn
r
n3
2
2λ
λ
=
C
nn
r
3
2 2λ − ≥
C
nn
r r
4
2 1ε / − ,
de C3
, C 4 — dodatni konstanty. OtΩe, v rozhlqnutomu vypadku neobxidni
ocinky znyzu otrymano.
Zaznaçymo, wo u vypadku, koly εn n
2 → ∞ i εn � 0 pry n → ∞ , porqdkova
rivnist\ d W L Wn
r
n
r
1 1 11, , ( )+( )ε � n
–
r
vyplyva[ odrazu z rezul\tatu V.LF.LBa-
benka [6].
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1416 N. V. PARFINOVYÇ
Nexaj teper εn n
2 = O ( 1 ), tobto εn ≤ C5 / n
2
( C5 — dodatna konstanta).
Dlq bud\-qkoho fiksovanoho l ∈ { 1, … , n } otryma[mo
En ≥ –
1
0
8
1 1
+ /
∫ −
ε ϕ
π
n
r
l n
r
l
t dt, ( ) –
ε ϕn
r r
l
1 0, ( ) ≥
≥
1 2
1 1
0
8+
− ∞
/
∫ε
π
ϕ
π
n
r r
l n
l
t dt, –
ϕ1 5
2
0, ( )r
rl
C
n
=
=
1
8
1 1
2+
− ∞ε ϕ
π
πn
r
r
l
l
n
,
–
ϕ1 5
2
0, ( )r
rl
C
n
≥
≥
1
64
1 0
2 2 1 1
1
5n l l
Cn
r r
r
r
π ε ϕ
ϕ+ −
− − ∞,
, ( )
.
Lehko pobaçyty, wo pry vsix fiksovanyx
l > 8
1 5
1 1
ϕ
ϕ π
,
,
r
r
C∞
− ∞
bude
C6 =
π ε ϕ
64
1
2 1 1
+
− − ∞
n
r r
l
, –
ϕ1
5
0, ( )r
rl
C > 0,
a En ≥ C6 / n
2
. OtΩe, neobxidnu ocinku znyzu otrymano i v c\omu vypadku.
Teoremu dovedeno.
Dovedennq teoremy 2. PokaΩemo, wo vykonu[t\sq perße spivvidnoßennq.
Zafiksu[mo poslidovnist\ Hn n-vymirnyx pidprostoriv prostoru L1 . Oskil\ky
poslidovnist\ αn [ obmeΩenog, to znajdet\sq çyslo K > 0 take, wo αn ≤ K.
Zrozumilo, wo
E W H Wr
n n
r k
1 1 1
, ∩ α +( ) ≥ E W KWr r k
1 1 1
, +( ) = C.
Prypustymo, wo C = 0. Todi dlq koΩno] funkci] f ∈ W r
1 inf
u KW r k
f u
∈ +
−
1
1 = 0.
Oskil\ky mnoΩyna KW r k
1
+
[ lokal\no kompaktnog v L1, to dlq koΩno] fun-
kci] f ∈ W r
1 isnu[ uf ∈KW r k
1
+
taka, wo f u f−
1
= 0, tobto na [ 0, 2 π ] f ( t ) =
= uf ( t ) i f r k( )+
1
[ obmeΩenog. Rozhlqnemo funkcig fn ( t ) = sin n t / ( 4 n
r
).
Zrozumilo, wo fn ∈ W r
1 , prote f r k( )+
1
= n
k
i pry k ∈ N ne [ obmeΩenog. Ot-
rymana supereçnist\ dovodyt\, wo C > 0.
PokaΩemo teper, wo vykonu[t\sq i druhe spivvidnoßennq. Ocinky zverxu v
c\omu vypadku odrazu vyplyvagt\ z rezul\tatu [ 7 ]. Otryma[mo dlq
d W H Wn
r
n n
r k
1 1 1
, ∩ α +( ) ocinky znyzu. Dlq bud\-qkoho Hn ⊂ L1 na pidstavi teo-
remy dvo]stosti dlq najkrawoho L1-nablyΩennq v metryci prostoru L1 moΩe-
mo zapysaty
E W H Wr
n n
r k
1 1 1
, ∩ α +( ) =
=
sup sup ( ) ( ) sup ( ) ( )
f W g
g
n
u H Wr
n
r k
f t g t dt u t g t dt
∈ ≤
⊥
∈∞
+∫ ∫−
1 1
1
1
0
2
0
2π π
α
∩
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
TOÇNYJ PORQDOK VIDNOSNYX POPEREÇNYKIV KLASIV W r
1 … 1417
=
sup sup ( ) ( ) sup ( ) ( )( ) ( )
f
f
g W
k
n
u H W
r k
r k
n
r k
f t g t dt u t g t dt
1 1
1
1
0
2
0
2
≤
⊥
∈ ∈
+
∞
+ +∫ ∫−
π π
α
∩
=
= sup ( ) sup ( ) ( )( ) ( )
g W
k
n
u H W
r k
r k
n
r k
E g t u t g t dt
∈
∞
∈
+
∞
+ +
( ) −
∫α
π
∩ 1 0
2
≥
≥
sup max ( ) sup ( ) ( ),
( )
,
α
π
ϕ α α ϕ α
∈ ≤ ≤ ∈
+
++( ) − +
+ ∫
R 1 0
2
1l n
l r n
u H W
r k
l r kE t u t t dt
n
r k∩
=
=
max inf sup ( ) ( ),
( )
,
1 0
2
1≤ ≤
∞ ∈ ∈
+
+− +
+ ∫l n
l r n
u H W
r k
l r k
n
r k
u t t dtϕ α ϕ α
α
π
R ∩
≥
≥
max sup, ,
( )
1
1
1≤ ≤
∞
∈
+ ∞
+−
+l n
l r n
u H W
l r k
r k
n
r k
uϕ α ϕ
∩
≥
≥ max ,
,
,1
1
1
1
≤ ≤
∞
+ ∞
∞
−
l n
l r
n
k
k
r k
r
n
l
ϕ ε ϕ
ϕ
.
Nexaj l = 2 1εn
kn/[ ], de [ x ] — cila çastyna çysla x. Pry dostatn\o velykyx n
1 ≤ l ≤ n, i my moΩemo zapysaty
En ≥
ϕ
ε
ε
ε
1
12
1
2
,r
r
n
r k r
n
k
k
n
k kn
n
n
∞
/ /
−
[ ]
≥
C
nn
r k rε / ,
de C — dodatna konstanta. Zalyßylos\ zaznaçyty, wo ostann[ tverdΩennq te-
oremy vyplyva[ z [7] i rezul\tativ dlq vidpovidnyx popereçnykiv bez obmeΩen\.
Teoremu dovedeno.
1. Kolmohorov A. N. O nayluçßem pryblyΩenyy funkcyj zadannoho funkcyonal\noho klas-
sa // Matematyka y mexanyka. Yzbr. tr. – M.: Nauka, 1985. – S. 186 – 189.
2. Konovalov V. N. Ocenka popereçnykov typa Kolmohorova dlq klassov dyfferencyruem¥x
peryodyçeskyx funkcyj // Mat. zametky. – 1984. – 35, v¥p. 3. – S. 369 – 380.
3. Tyxomyrov V. M. Nekotor¥e vopros¥ teoryy pryblyΩenyj. – M.: Yzd-vo Mosk. un-ta, 1976.
– 304 s.
4. Babenko V. F. PryblyΩenye v srednem pry nalyçyy ohranyçenyj na proyzvodn¥e prybly-
Ωagwyx funkcyj // Vopros¥ analyza y pryblyΩenyj. – Kyev: Yn-t matematyky AN USSR,
1989. – S. 9 – 18.
5. Babenko V. F. O nayluçßyx ravnomern¥x pryblyΩenyqx splajnamy pry nalyçyy ohrany-
çenyj na yx proyzvodn¥e // Mat. zametky. – 1991. – 50, v¥p. 6. – S. 24 – 30.
6. Babenko V. F. O nayluçßyx L1-pryblyΩenyqx splajnamy pry nalyçyy ohranyçenyj na yx
proyzvodn¥e // Tam Ωe. – 1992. – 51, v¥p. 5. – S. 12 – 19.
7. Babenko V. F., Azar L., Parfynovyç N. V. O pryblyΩenyy klassov peryodyçeskyx funkcyj
splajnamy pry nalyçyy ohranyçenyj na yx proyzvodn¥e // Rqdy Fur’[: teoriq i zastosuvan-
nq: Pr. In-tu matematyky NAN Ukra]ny. – 1998. – 20. – S. 18 – 29.
8. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyq. – M.: Nauka, 1987. – 424 s.
9. Babenko V. F. On the relative widths of classes of functions with bounded mixed derivative // East
J. Approxim. – 1996. – 2, # 3. – P. 319 – 330.
OderΩano 25.03.2004,
pislq doopracgvannq — 17. 01. 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
|
| id | umjimathkievua-article-3694 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:47:12Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4d/1af79da92efec57608f2ef3a4127c94d.pdf |
| spelling | umjimathkievua-article-36942020-03-18T20:02:18Z Exact order of relative widths of classes $W^r_1$ in the space $L_1$ Точний порядок відносних поперечників класів $W^r_1$ у просторі $L_1$ Parfinovych, N. V. Парфінович, Н. В. As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions. Знайдено точний порядок при $n \rightarrow \infty$ відносних поперечників класів $W^r_1$ періодичних функцій у просторі $L_1$ при обмеженнях на старші похідні наближаючих функцій. Institute of Mathematics, NAS of Ukraine 2005-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3694 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 10 (2005); 1409–1417 Український математичний журнал; Том 57 № 10 (2005); 1409–1417 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3694/4114 https://umj.imath.kiev.ua/index.php/umj/article/view/3694/4115 Copyright (c) 2005 Parfinovych N. V. |
| spellingShingle | Parfinovych, N. V. Парфінович, Н. В. Exact order of relative widths of classes $W^r_1$ in the space $L_1$ |
| title | Exact order of relative widths of classes $W^r_1$ in the space $L_1$ |
| title_alt | Точний порядок відносних поперечників класів $W^r_1$ у просторі $L_1$ |
| title_full | Exact order of relative widths of classes $W^r_1$ in the space $L_1$ |
| title_fullStr | Exact order of relative widths of classes $W^r_1$ in the space $L_1$ |
| title_full_unstemmed | Exact order of relative widths of classes $W^r_1$ in the space $L_1$ |
| title_short | Exact order of relative widths of classes $W^r_1$ in the space $L_1$ |
| title_sort | exact order of relative widths of classes $w^r_1$ in the space $l_1$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3694 |
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