On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$
We consider and study properties of the smoothness characteristics $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, of functions $f(x)$ that belong to the space $S^p,\quad 1 \leq p < \infty$, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and th...
Gespeichert in:
| Datum: | 2006 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2006
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3455 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509550639054848 |
|---|---|
| author | Vakarchuk, S. B. Shchitov, A. N. Вакарчук, С. Б. Щитов, А. Н. Вакарчук, С. Б. Щитов, А. Н. |
| author_facet | Vakarchuk, S. B. Shchitov, A. N. Вакарчук, С. Б. Щитов, А. Н. Вакарчук, С. Б. Щитов, А. Н. |
| author_sort | Vakarchuk, S. B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:07Z |
| description | We consider and study properties of the smoothness characteristics $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, of functions $f(x)$ that belong to the space $S^p,\quad 1 \leq p < \infty$, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$ are calculated. |
| first_indexed | 2026-03-24T02:42:53Z |
| format | Article |
| fulltext |
UDK 517.5
S. B. Vakarçuk, A. N. Wytov (Akad. tamoΩ. sluΩb¥ Ukrayn¥, Dnepropetrovsk)
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX
TEORYY APPROKSYMACYY FUNKCYJ
V PROSTRANSTVAX Sp
, 1 ≤≤≤≤ p < ∞∞∞∞
Properties of smoothness characteristics Ωm S
f t p( , ) , m ∈N , t > 0 , of functions f x( ) which belong
to the space S p , 1 ≤ < ∞p , introduced by A. I. Stepanets are considered and investigated. Exact
Jackson-type inequalities are obtained and exact values of widths of classes of functions defined by
means of Ωm S
f t p( , ) are computed.
Rozhlqnuto ta doslidΩeno vlastyvosti hladkisnyx xarakterystyk Ωm S
f t p( , ) , m ∈N , t > 0 ,
funkcij f x( ), wo naleΩat\ uvedenomu O. I. Stepancem prostoru S p , 1 ≤ < ∞p . OderΩano
toçni nerivnosti typu DΩeksona ta obçysleno toçni znaçennq popereçnykiv klasiv funkcij, vy-
znaçenyx za dopomohog Ωm S
f t p( , ) .
1. Pust\ Lp ≡ Lp
( [ – π, π ] ) , 1 ≤ p < ∞ , — prostranstvo 2π -peryodyçeskyx
yzmerym¥x na [ – π, π ] funkcyj f ( x ) , ymegwyx koneçnug normu
f p = f x dxp
p
( )
/
−
∫
π
π 1
.
Pry reßenyy v Lp mnohyx zadaç teoryy approksymacyy funkcyj v kaçestve xa-
rakterystyky skorosty stremlenyq k nulg velyçyn yx nayluçßyx polynomy-
al\n¥x pryblyΩenyj yspol\zugt, naprymer, modul\ neprer¥vnosty m-ho po-
rqdka
ωm pf t( , ) = sup ( ) :∆h
m
p
f h t⋅ ≤ ≤{ }0 , t ≥ 0, (1)
hde ∆h
m f x( ) = ( ) ( )− +−
=∑ 1
0
m j
m
j
j
m
C f x jh — koneçnaq raznost\ m-ho porqdka
funkcyy f v toçke x s ßahom h. Odnako v nekotor¥x zadaçax narqdu s (1) ys-
pol\zuetsq velyçyna [1, 2]
Ωm pf t( , ) =
1
0
1
t
f dh
t
h
m
p
p
p
∫ ⋅
∆ ( )
/
, t > 0. (2)
Naprymer, v rabote [1] nayluçßee pryblyΩenye funkcyj polynomamy po syste-
me Xaara ocenyvaetsq ne s pomow\g modulq neprer¥vnosty pervoho porqdka, a
posredstvom velyçyn¥ Ω1( , )f t p, kotoraq, kak pokazano v [3], πkvyvalentna
ω1( , )f t p .
Dlq yssledovanyq povedenyq nayluçßeho pryblyΩenyq funkcyj alhebray-
çeskymy polynomamy v prostranstve L a bp([ , ]), p ≥ 1, y C a b([ , ]) K. H. Yvanov
vvel v rassmotrenye nov¥j vyd m-modulej neprer¥vnosty τ λm p pf w( ; , ) ,′ , a
takΩe yzuçyl yx svojstva y vzaymosvqzy s yzvestn¥my dyfferencyal\no-raz-
nostn¥my xarakterystykamy funkcyj [4, 5].
Napomnym, çto τ-modulem neprer¥vnosty m-ho porqdka dlq f ( x )8∈
∈ L p pmax( , )′ , hde p, p ′ ≥ 1, naz¥vagt velyçynu [5, 6]
τ λm p pf w( , ; ) ,′ = w fm p p
( ) ( , ; ( ))⋅ ⋅ ⋅ ′ω λ , (3)
© S. B. VAKARÇUK, A. N. WYTOV, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3 303
304 S. B. VAKARÇUK, A. N. WYTOV
hde λ ( x ) — proyzvol\naq opredelennaq na [ 0, 2π ] poloΩytel\naq 2π-peryo-
dyçeskaq funkcyq, w ( x ) — neprer¥vnaq neotrycatel\naq peryoda 2π funk-
cyq,
ω λm pf x x( , ; ( )) ′ =
1
2
1
λ λ
λ
( )
( )
( )
( ) /
x
f x dh
x
x
h
m p
p
−
′
′
∫
∆ . (4)
Esly, naprymer, λ ( x ) ≡ t = const, f ( x ) ∈ Lp , w ( x ) ≡ 1 y p ′ ∈ [ 1, p ] , to
τm p pf t( , ; ) ,1 ′ � ωm pf t( , ) , hde symvol � oznaçaet otnoßenye slaboj πkvyva-
lentnosty.
Otmetym, çto velyçyn¥ τm f t( , ; ) ,1 2 2 b¥ly yspol\zovan¥ v [6] dlq reßenyq
rqda πkstremal\n¥x zadaç teoryy approksymacyy v L2 . Yz (3), (4), v çastnosty,
sleduet
τm p pf t( , ; ) ,1 =
1
2
1
t
f dh
t
t
h
m
p
p
p
−
∫ ⋅
∆ ( )
/
, t > 0. (5)
V sylu (2), (5) y toho fakta, çto
∆h
m f ( )⋅
2
2
= 2 12
1
ρk
m
k
f kh( )( cos )−
=
∞
∑ ,
hde ρk f2( ) = a f b fk k
2 2( ) ( )+ , a a fk ( ) y b fk ( ) — koπffycyent¥ Fur\e funk-
cyy f ( x ) , ymeem Ωm f t( , )2 = τm f t( , ; ) ,1 2 2 .
Na osnovanyy yzloΩennoho opredelenn¥j ynteres, s naßej toçky zrenyq,
predstavlqet yspol\zovanye xarakterystyk vyda (2) dlq reßenyq πkstremal\-
n¥x zadaç teoryy approksymacyy v normyrovann¥x prostranstvax S
p
, vveden-
n¥x A. Y. Stepancom v [7, 8].
2. Napomnym, çto pod prostranstvom S
p
, 1 ≤ p < ∞ , ponymagt prostran-
stvo 2π-peryodyçeskyx funkcyj f ( x ) , dlq kotor¥x
f S p = ˆ ( )
/
f k
p
k
p
∈
∑
Z
1
< ∞ ,
hde
ˆ ( )f k = ( ) ( )/2 1 2π
π
π
−
−
−∫ f x e dxikx
(6)
— koπffycyent¥ Fur\e funkcyy f ( x ) po tryhonometryçeskoj systeme
( ) exp( )/2 1 2π − ikx , k ∈ Z.
Pust\ Ψ ( k ) y β ( k ) = βk , k ∈ N, — suΩenye na N proyzvol\n¥x vewest-
venn¥x funkcyj Ψ ( x ) y β ( x ) , opredelenn¥x na polusehmente [ 1, ∞ ) . Pred-
poloΩym, çto rqd
k
k k k kk
a f kx b f kx
=
∞
∑ + + +{ }
1
1
2 2
Ψ ( )
( )cos( ) ( )sin( )/ /β π β π
qvlqetsq rqdom Fur\e nekotoroj summyruemoj funkcyy, kotorug, sohlasno
[9], oboznaçym symvolom f xβ
Ψ( ) y nazovem ( ),Ψ β -proyzvodnoj funkcyy f ( x ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX TEORYY APPROKSYMACYY … 305
Pod L S p
β
Ψ( ), 1 ≤ p < ∞ , ponymaem mnoΩestvo funkcyj f ( x ) ∈ Lβ
Ψ , ( ),Ψ β -
proyzvodn¥e f xβ
Ψ( ) kotor¥x prynadleΩat prostranstvu S p. V çastnosty,
esly Ψ( )x x r= − , 0 < r < ∞ , y β ( x ) ≡ r, to yspol\zuem oboznaçenye L Sr p( ) .
Dlq proyzvol\noho πlementa f ( x ) ∈ S
p
, 1 ≤ p < ∞ , po analohyy s (2) ras-
smotrym velyçynu
Ωm S
f t p( , ) =
1
0
1
t
f dh
t
h
m
S
p
p
p∫ ⋅
∆ ( )
/
, t > 0. (7)
S uçetom sootnoßenyq
∆h
m
S
p
f p( )⋅ = ˆ ( ) ( )f k
m
j
e
p
j
m
m j ijkh
k
p
=
−
∈
∑∑ −
0
1
Z
=
= 2 12 2mp p
k
mpf k kh/ /ˆ ( ) ( cos )
∈
∑ −
Z
(8)
dlq analoha xarakterystyky (5) v prostranstve S
p
, a ymenno,
τm S
f t p( , ; )1 =
1
2
1
t
f dh
t
t
h
m
S
p
p
p
−
∫ ⋅
∆ ( )
/
, t > 0,
ymeem
Ωm S
f t p( , ) = τm S
f t p( , ; )1 .
3. Otmetym nekotor¥e svojstva velyçyn¥ (7).
1. Esly funkcyy f ( x ) y g ( x ) prynadleΩat prostranstvu S
p
, 1 ≤ p <
< ∞ , to
Ωm S
f g t p( , )+ ≤ Ω Ωm S m S
f t g tp p( , ) ( , )+ , t > 0.
2. Esly ωm S
f t p( , ) = sup ( ) :∆h
m
S
f h tp⋅ ≤ ≤{ }0 , to Ωm S
f t p( , ) ≤
≤ ωm S
f t p( , ) dlq proyzvol\noho t > 0, hde f ( x ) ∈ S
p
, 1 ≤ p < ∞ .
Dokazatel\stva dann¥x svojstv ne pryvodqtsq, poskol\ku ony oçevydn¥.
3. Dlq lgboho n = 2, 3, … y proyzvol\noho t > 0 v¥polnqetsq neraven-
stvo
Ωm S
f nt p( , ) ≤ n f tm
m S pΩ ( , ) ,
hde f ( x ) ∈ S
p
, 1 ≤ p < ∞ .
Dokazatel\stvo. Nam potrebuetsq sledugwee sootnoßenye, spravedlyvoe
dlq lgb¥x natural\n¥x çysel m y n [10, c. 158]:
∆nH
m f x( ) =
j
n
j
n
j
n
H
m
m
m
f x H j
1 20
1
0
1
0
1
1=
−
=
−
=
−
=
∑ ∑ ∑ ∑… +
∆
ν
ν .
Otsgda ymeem
∆nH
m
S
f p( )⋅ ≤
j
n
j
n
j
n
H
m
m
Sm
p
f H j
1 20
1
0
1
0
1
1=
−
=
−
=
−
=
∑ ∑ ∑ ∑… ⋅ +
∆
ν
ν . (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
306 S. B. VAKARÇUK, A. N. WYTOV
Vvodq mul\tyyndeks j = ( j1 , j2 , … , jm ) , dlq kotoroho j
df= j
m
νν=∑ 1
, y pola-
haq F xj ( ) df= f x H j
m+( )=∑ νν 1
, zapys¥vaem
∆H
m F kj
^
( ) = ( ) ( )−
+ +−
= −
−∑ ∫1
1
20
m
m
ikx
m
f x H H e dxλ
λ π
π
λ π
λ j =
= ( ) ˆ( )−
−
=
∑ 1
0
m
m
i kH ikH
m
e f k eλ
λ
λ
λ
j . (10)
Tohda s uçetom (8) y (10) ymeem
∆H
m
S
F pj ( )⋅ = ∆H
m
S
f p( )⋅ . (11)
V sylu (9) y (11) poluçaem
∆nH
m
S
f p( )⋅ ≤ n fm
H
m
S p∆ ( )⋅ . (12)
Yspol\zuq formul¥ (7) y (12), zapys¥vaem
Ωm S
f nt p( , ) =
1
0
1
t
f dH
t
nH
m
S
p
p
p∫ ⋅
∆ ( )
/
≤ n f tm
m S pΩ ( , ) .
Svojstvo 3 dokazano.
4. Dlq proyzvol\n¥x çysel r, m ∈ N, t > 0 y lgboj funkcyy f ( x )8∈
∈ L Sr p( ), 1 ≤ p < ∞ , v¥polneno neravenstvo
Ωr m S
f t p+ ( , ) ≤ t f tr
m
r
S pΩ ( , )( ) .
Dokazatel\stvo. Dlq provedenyq dal\nejßyx rassuΩdenyj vospol\zuem-
sq sledugwej formuloj [10, c. 159]:
∆h
r f x( ) =
0
1
0
2
0
1 2
h h h
r
r rdu du f x u u u du∫ ∫ ∫… + + + …+( )( ) . (13)
Uçyt¥vaq, çto
∆h
r m f x+ ( ) = ∆ ∆h
m
h
r f x( )( ) ,
y yspol\zuq (13), poluçaem
∆h
r m f x+ ( ) =
0
1
0
2
0
1 2
h h h
h
m r
r rdu du f x u u u du∫ ∫ ∫… + + + …+∆ ( )( ) . (14)
Poskol\ku
∆h
m r
rf x u u u( )( )+ + + …+1 2 = ( ) ( )( )−
+ + + + …+−
=
∑ 1
0
1 2
m j
j
m
r
r
m
j
f x jh u u u ,
yz (14) ymeem
∆h
r m f x+ ( ) = ( ) ( )( )−
… + + + + …+−
=
∑ ∫ ∫ ∫1
0 0
1
0
2
0
1 2
m j
j
m h h h
r
r r
m
j
du du f x jh u u u du .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX TEORYY APPROKSYMACYY … 307
Tohda s uçetom (6) zapys¥vaem
∆h
r m f k+
^̂
( ) =
( ) ( ) ( )( )−
−{ }−
=
∑ 1
1
1
0
m j
j
m
ijkh ikh
r
r
m
j
e
ik
e f k
$$
. (15)
Netrudno proveryt\, çto v sylu (8), oçevydnoho sootnoßenyq
k er ikh r− − 1 = ( / ) sin2
2
k
khr
r
≤ hr
y formul¥ (15) poluçym
∆h
r m
S
f p
+ ⋅( ) ≤ h fr
h
m r
S p∆ ( )( )⋅ . (16)
Yz (7) y (16) ymeem trebuemoe neravenstvo
Ωr m S
f t p+ ( , ) ≤
1
0
1
t
h f dh
t
rp
h
m r
S
p
p
p∫ ⋅
∆ ( )
/
( ) ≤ t f tr
m
r
S pΩ ( , )( ) , t > 0.
Svojstvo 4 dokazano.
5. Dlq proyzvol\noj funkcyy f ( x ) ∈ S
p
, 1 ≤ p < ∞ , y lgb¥x natural\n¥x
çysel n < m ymeet mesto neravenstvo
Ωm S
f t p( , ) ≤ 2m n
n S
f t p
− Ω ( , ) ∀ t > 0 .
Dokazatel\stvo. Poskol\ku
∆h
m f x( ) = ∆ ∆h
m n
h
n f x− ( ( )) = ( ) ( )−
−
+− −
=
−
∑ 1
0
m n j
j
m n
h
n
m n
j
f x jh∆ ,
to oçevydno, çto
∆h
m f k
%%
( ) =
( ) ( )−
−
− −
=
−
∑ 1
0
m n j
j
m n
ijkh
h
n
m n
j
e f k∆
%%
.
Tohda
∆h
m
S
f p( )⋅ ≤
k
h
n
p
j
m n
m n j ijkh
p p
f k
m n
j
e
∈ =
−
− −∑ ∑ −
−
Z
∆
%%
( ) ( )
/
0
1
1 ≤
≤
j
m n
h
n
S
m n
j
f p
=
−
∑
−
⋅
0
∆ ( ) = 2m n
h
n
S
f p
− ⋅∆ ( ) . (17)
Yz (7) y (17) poluçaem neravenstvo
Ωm S
f t p( , ) ≤ 2m n
n S
f t p
− Ω ( , ) ,
svqz¥vagwee meΩdu soboj rassmatryvaem¥e xarakterystyky razlyçn¥x porqd-
kov. Svojstvo 5 dokazano.
4. Vsgdu v dal\nejßem na opredelennug v p.82 funkcyg Ψ ( x ) , 1 ≤ x < ∞ ,
nalahaem rqd ohranyçenyj, a ymenno: Ψ ( x ) qvlqetsq poloΩytel\noj, mono-
tonno ub¥vagwej k nulg pry vozrastanyy x y takoj, çto dlq vsex x ∈ [ 1, ∞ )
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
308 S. B. VAKARÇUK, A. N. WYTOV
suwestvuet pervaq proyzvodnaq Ψ( )( )1 x . Polahaem, çto funkcyq Ψ( )( )1 x v
toçkax oblasty opredelenyq udovletvorqet uslovyg
px x xΨ Ψ( )( ) ( )1 + ≤ 0. (18)
Prymeramy funkcyj, udovletvorqgwyx ukazann¥m trebovanyqm, qvlqgtsq,
v çastnosty, x−γ
y e x−γ
pry 1 / p ≤ γ < ∞ , a takΩe 1 1/( ln( ))x x+ .
Yzvestno [10, c. 162], çto v prostranstve C([ , ])− π π dlq funkcyy sin x ee
modul\ neprer¥vnosty m-ho porqdka v¥raΩaetsq sootnoßenyem
ωm x u(sin , ) =
2 2 0 2m m mu u usin , ; ,( / ) esly esly≤ ≤ ≥{ }π π .
Polahaem
ωm L tx
p
(sin , ) ([ , ])⋅ 0 df=
1
0
1
t
x u du
t
m
p
p
∫
ω (sin , )
/
, t > 0.
Çerez E fn S p−1( ) oboznaçym nayluçßee pryblyΩenye funkcyy f ( x ) ∈ S
p
podprostranstvom Tn−1 tryhonometryçeskyx polynomov porqdka n – 1 v met-
ryke prostranstva S
p
, t. e.
E fn S p−1( ) = inf : ( )f T T xn S n np− ∈{ }− − −1 1 1T .
Dlq proyzvol\noho mnoΩestva M ⊂ S p
polahaem
En S p−1( )M = sup ( ) : ( )E f f xn S p− ∈{ }1 M .
Teorema-1. Pust\ funkcyq Ψ ( x ) udovletvorqet sformulyrovann¥m v p.84
trebovanyqm y ohranyçenyg (18). Tohda dlq proyzvol\n¥x çysel n, m ∈ N , 1 ≤
≤ p < ∞ y 0 < t ≤ π ymegt mesto ravenstva
sup
( )
( ) ( , )( ) ( )
( )
/f x L S
f x
n S
m S
p
p
p
E f
n f t n∈
/≡
−
β β
Ψ Ψ Ω Ψ
const
1
= ωm L tx
p
(sin , ) ([ , ])⋅ −
0
1 . (19)
Pust\ B — edynyçn¥j ßar v S p; F — v¥pukloe central\no-symmetryçnoe
podmnoΩestvo v S p;
Ln
pS⊂ — n-mernoe podprostranstvo; L
n pS⊂ — pod-
prostranstvo korazmernosty n; V S p
n: → L — neprer¥vn¥j lynejn¥j opera-
tor, perevodqwyj πlement¥ prostranstva S p
v
Ln ; V S p
n
⊥ →: L — nepre-
r¥vn¥j operator lynejnoho proektyrovanyq prostranstva S p
na podprostran-
stvo
Ln . Velyçyn¥
d F Sn
p( , ) =
inf sup inf
L Ln
p
n
p
S f F g Sf g
⊂ ∈ ∈
− ,
δn
pF S( , ) =
inf inf sup
:L Ln
p p
n
p
S V S f F
Sf Vf
⊂ → ∈
− ,
b F Sn
p( , ) =
sup sup { }
L Ln
p
nS F+ +⊂ ⊂
>
1 1
0
ε
ε
B∩
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX TEORYY APPROKSYMACYY … 309
d F Sn p( , ) =
inf sup
L Ln p n
p
S f F
Sf
⊂ ∈ ∩
,
pn
pF S( , ) =
inf inf sup
:L Ln
p p
n
p
S V S f F S
f V f
⊂ → ∈
⊥
⊥
−
naz¥vagt sootvetstvenno kolmohorovskym, lynejn¥m, bernßtejnovskym, hel\-
fandovskym y proekcyonn¥m n-popereçnykamy mnoΩestva F v prostranstve
S p. MeΩdu ukazann¥my xarakterystykamy v¥polnqgtsq sledugwye sootno-
ßenyq [11]:
b F Sn
p( , ) ≤
d F S
d F S
n p
n
p
( , )
( , )
≤ δn
pF S( , ) ≤
pn
pF S( , ). (20)
Vopros¥, svqzann¥e s v¥çyslenyem n-popereçnykov nekotor¥x klassov
funkcyj v prostranstvax S
p
, 1 ≤ p < ∞ , rassmatryvalys\, naprymer, v rabo-
tax [12 – 15].
Çerez Φ ( t ) oboznaçym monotonno vozrastagwug neprer¥vnug na poluseh-
mente 0 ≤ t < ∞ funkcyg takug, çto Φ ( 0 ) = 0. Pod L S p
mβ
Ψ Ω Φ( , , ) ponyma-
em klass funkcyj f ( x ) ∈ L S p
β
Ψ( ), ( ),Ψ β -proyzvodn¥e kotor¥x udovletvorqgt
pry lgbom t ∈ ( 0, 2π ] uslovyg Ω Ψ
m S
f t p( , )β ≤ Φ ( t ) .
Teorema-2. Pust\ dlq funkcyy Φ ( t ) pry nekotorom n ∈ N v¥polneno us-
lovye
Φ
Φ
( )
( )/
t
nπ 2
≥
ω
ω π
m L tn
m L
x
x
p
p
(sin , )
(sin , )
([ , ])
([ , / ])
⋅
⋅
0
0 2
, (21)
hde 0 < t < ∞ , a funkcyq Ψ ( x ) udovletvorqet trebovanyqm, yzloΩenn¥m v
formulyrovke teorem¥ 1. Tohda spravedlyv¥ ravenstva
g L S Sn
p
m
p
2 ( )( , , );β
Ψ Ω Φ = g L S Sn
p
m
p
2 1− ( )( , , );β
Ψ Ω Φ =
= E L Sn
p
m S p−1( )( , , )β
Ψ Ω Φ = ω π
πm Lx n
np
(sin , ) ( )([ , / ])⋅
−
0 2
1
2
Ψ Φ , (22)
hde gn( )⋅ — lgboj yz rassmotrenn¥x ranee n -popereçnykov. Pry πtom mno-
Ωestvo maΩorant, udovletvorqgwyx ohranyçenyg (21), ne pusto.
5. Dokazatel\stvo teorem¥-1. V [7] pokazano, çto prostranstva S
p
, 1 ≤
≤ p < ∞ , nasledugt takug vaΩnug osobennost\ hyl\bertova prostranstva, kak
mynymal\noe svojstvo çastn¥x summ rqda Fur\e. V çastnosty [15],
E fn S p−1( ) = f S fn S p− −1( ) = ˆ ( )
/
f k
p
k n
p
≥
∑
1
=
π ρ
2
2
1 2 1
=
∞
∑
/ /
( )
k n
k
p
p
f , (23)
hde
S f xn−1( , ) =
k n
ikx
f k
e
≤ −
∑
1 2
ˆ ( )
π
— çastnaq summa rqda Fur\e
S ( f , x ) ∼
k
ikx
f k
e
∈
∑
N
ˆ ( )
2π
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
310 S. B. VAKARÇUK, A. N. WYTOV
funkcyy f ( x ) ∈ S
p
;
ρk f( ) df= a f b fk k
2 2( ) ( )+ ,
a fk ( ) , b fk ( ) — koπffycyent¥ Fur\e funkcyy f ( x ) . Poskol\ku v sylu (8)
∆ Ψ
h
m
S
p
f
pβ ( )⋅ = π ρ β
p m p
k
k
p mpf kh/ ( ) / /( )( cos )2 1 1 2
1
22 1+ −
=
∞
∑ −Ψ
y
ρk f( ) = Ψ Ψ( ) ( )k fkρ β ,
to, yspol\zuq opredelenye velyçyn¥ Ωm S p( )⋅ , poluçaem
Ω Ψ
m
p
S
f t n p( , )/β ≥ π ρp m p
k n
k
p p
t n
mpnt f k kh dh/ ( ) /
/
/( ) ( ) ( cos )2 1 1 2 1
0
22 1+ − −
=
∞
−∑ ∫ −Ψ .
Rassmotrym vspomohatel\nug funkcyg
γ ( x ) df= Ψ− ∫ −p
t n
mpx xh dh( ) ( cos )
/
/
0
21 ,
hde 1 ≤ x < ∞ , y yssleduem ee povedenye. Dlq πtoho opredelym pervug pro-
yzvodnug funkcyy γ ( x ) , yspol\zuq v xode v¥kladok proceduru v¥çyslenyq
opredelennoho yntehrala po çastqm:
γ ( )( )1 x =
t
xn
xt
n
mp
1
2
−
cos
/
–
–
1
11
1
0
2
x x
px x x xh dhp
t n
mp
Ψ
Ψ Ψ+ + −∫( )
( ) ( ) ( cos )( )( )
/
/ .
V sylu (18) ymeem γ ( )( )1 x ≥ 0, 1 ≤ x < ∞ , t. e. γ ( x ) qvlqetsq neub¥vagwej
funkcyej. Sledovatel\no, min ( ) :{ }γ x n x≤ < ∞ = γ ( )n . Uçyt¥vaq dann¥j
fakt y yspol\zuq formulu (23), zapys¥vaem pryvedennug ranee ocenku snyzu
velyçyn¥ Ω Ψ
m
p
S
f t n p( , )/β v vyde
Ω Ψ
m
p
S
f t n p( , )/β ≥ π ρp m p p
t n
mp
k n
k
pn
n
t
nh dh f/ ( ) /
/
/( ) ( cos ) ( )2 1 1 2
0
22 1+ − −
=
∞
∫ ∑−
Ψ =
= Ψ−
−⋅p
m L t
p
n
p
S
n x E f
p
p( ) (sin , ) ( )([ , ])ω 0 1 .
Otsgda ymeem ocenku sverxu
sup
( )
( ) ( , )( ) ( )
( )
/f x L S
f x
n S
m S
p
p
p
E f
n f t n∈
/≡
−
β β
Ψ Ψ Ω Ψ
const
1
≤ ωm L tx
p
(sin , ) ([ , ])⋅ −
0
1 . (24)
Rassmotrym funkcyg ˜( )f x
df= 2/ cosπ nx , prynadleΩawug klassu L S p
β
Ψ( ),
dlq kotoroj E fn S p−1( ˜ ) = 21/ p , ˜ ( )f xβ
Ψ = 2 21/ /( ) cos( )π β πΨ− +n nx n y
∆ Ψ
h
m
S
f
p
˜ ( )β ⋅ = 2 12 1 1 2m p mn nh/ / /( )( cos )+ − −Ψ . (25)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX TEORYY APPROKSYMACYY … 311
Yspol\zuq opredelenye velyçyn¥ Ωm S p( )⋅ y formulu (25), pry 0 < t ≤ π po-
luçaem
Ω Ψ( / )˜ ,f t n
S pβ = 21 1
0
/
([ , ])( ) (sin , )p
m L tn x
p
Ψ− ⋅ω .
Otsgda sleduet ocenka snyzu
sup
( )
( ) ( , )( ) ( )
( )
/f x L S
f x
n S
m S
p
p
p
E f
n f t n∈
/≡
−
β β
Ψ Ψ Ω Ψ
const
1
≥
E f
n f t n
n S
m S
p
p
−1( )
( / )
˜
( ) ˜ ,Ψ Ω Ψ
β
= ωm L tx
p
(sin , ) ([ , ])⋅ −
0
1 .
(26)
Sravnyvaq ocenky (24) y (26), poluçaem ravenstvo (19).
Teorema 1 dokazana.
6. Dokazatel\stvo teorem¥-2. Yspol\zuq ravenstvo (19), v kotorom
polahaem t = π / 2, a takΩe sootnoßenyq (20) y (23), zapys¥vaem ocenky sverxu
g L S Sn
p
m
p
2 ( )( , , );β
Ψ Ω Φ ≤ g L S Sn
p
m
p
2 1− ( )( , , );β
Ψ Ω Φ ≤
≤
p2 1n
p
m
pL S S− ( )( , , );β
Ψ Ω Φ ≤ E L Sn
p
m S p−1( )( , , )β
Ψ Ω Φ ≤
≤ ω π
πm Lx n
np
(sin , ) ( )([ , / ])⋅
−
0 2
1
2
Ψ Φ . (27)
Dlq poluçenyq ocenok snyzu, sohlasno (20), rassmotrym bernßtejnovskyj
2n-popereçnyk klassa L S p
mβ
Ψ Ω Φ( , , ) v S p
y pokaΩem prynadleΩnost\ danno-
mu klassu ßara
B̃ df=
T x T x n
nn n n S m Lp
p
( ) : (sin , ) ( )([ , / ])∈ ≤ ⋅
{ }−T ω π
π0 2
1
2
Ψ Φ .
Dlq πtoho nam potrebuetsq neravenstvo [15]
( )Tn
S pβ
Ψ ≤ Ψ−1( )n Tn S p , (28)
qvlqgweesq svoeobrazn¥m analohom neravenstva S. N. Bernßtejna dlq tryho-
nometryçeskyx polynomov v prostranstve S p
. Oçevydno, çto
∆ Ψ
h
m
n
S
T
p
( )β = 2
2
1
m
n
p mp
k n
p
T k
kh
( ) ( ) sin
/
β
Ψ
%
≤
∑
. (29)
Polahaem
(sin )t ∗ df=
sin , ; ,/ /t t tesly esly0 2 1 2≤ ≤ ≥{ }π π .
Uçyt¥vaq, çto dlq proyzvol\noho çysla h y lgboho natural\noho çysla 1 ≤
≤ k ≤ n v¥polnqetsq neravenstvo
sin
kh
2
≤ sin
nh
2
∗
,
a takΩe yspol\zuq (28), (29) y opredelenye Ωm S p( )⋅ , zapys¥vaem ocenku sverxu
Ω Ψ
m n
S
T t
p
( ) ,β( ) ≤ Ψ− ⋅1
0( ) (sin , ) ([ , ])n x Tm L tn n Sp
pω . (30)
Dlq proyzvol\noho polynoma T xn( ) ˜∈ B v sylu (21) y (30) ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
312 S. B. VAKARÇUK, A. N. WYTOV
Ω Ψ
m n
S
T t
p
( ) ,β( ) ≤
ω
ω
π
π
m L tn
m L
x
x n
p
p
(sin , )
(sin , )
([ , ])
([ , / ])
⋅
⋅
0
0 2 2
Φ ≤ Φ ( t ) ,
hde 0 < t ≤ 2π . Sledovatel\no, spravedlyvo vklgçenye
˜ ( , , )B ⊂ L S p
mβ
Ψ Ω Φ .
Yspol\zuq (20), poluçaem
g L S Sn
p
m
p
2 ( )( , , );β
Ψ Ω Φ ≥ b L S Sn
p
m
p
2 ( )( , , );β
Ψ Ω Φ ≥
≥ b Sn
p
2 ( )˜ ,B ≥ ω π
πm Lx n
np
(sin , ) ( )([ , / ])⋅
−
0 2
1
2
Ψ Φ . (31)
Sopostavlqq sootnoßenyq (27) y (31), ymeem ravenstvo (22).
V zaverßenye dokazatel\stva teorem¥ 2 pokaΩem, çto mnoΩestvo maΩorant
Φ ( t ) , udovletvorqgwyx uslovyg (21), ne pusto. Dlq πtoho ubedymsq v tom, çto
funkcyq
˜ ( )Φ t df= t pα / , hde
α df= 2 12
0 2
mp
m L
px
p
/
([ , / ])(sin , )ω π⋅ −− , (32)
udovletvorqet uslovyg (21) pry lgbom n. Uçyt¥vaq pryvedenn¥j v p. 4 vyd
modulq neprer¥vnosty m-ho porqdka ωm x t(sin , ), perepys¥vaem formulu (32)
v vyde
α =
π
π
2 2
11 2
0
2 1
+
−
−∫mp
mpt
dt/
/
sin (33)
y ocenyvaem velyçynu α sverxu y snyzu. Yspol\zuq sootnoßenye sin( )/t 2 >
> 2 t /π , hde 0 < t < π / 2, yz (33) poluçaem neravenstvo α < mp . Dlq ocen-
ky snyzu velyçyn¥ α pokaΩem v¥polnenye neravenstva
2
4
sin
πu
< uπ/4 ,
hde 0 < u < 1 . Polahaq G ( u )
df= u uπ π/ sin( )/4 2 4− , vklgçaem v rassmotre-
nye dlq udobstva rassuΩdenyj toçky u = 0 y u = 1. Oçevydno, çto v dosta-
toçno maloj okrestnosty nulq
G ( u ) df= u O uπ π/ /( )4 1 41 −( )− > 0.
Dokazatel\stvo provedem metodom ot protyvnoho, polahaq, çto na yntervale ( 0,
1 ) suwestvuet nekotoraq toçka η, v kotoroj G ( u ) menqet znak. Poskol\ku
G ( 0 ) = G ( 1 ) = 0, sohlasno teoreme Rollq pervaq proyzvodnaq
G u( )( )1 =
π π ππ
4 2 4
4 1
3 2u
u/
/ cos− −
dolΩna ymet\ ne menee dvux razlyçn¥x nulej na mnoΩestve ( 0, 1 ) , a tak kak
G( )( )1 1 = 0, v sylu ukazannoj teorem¥ vtoraq proyzvodnaq
G u( )( )2 =
π π π π π
2
7 2
4 2
2 4 4
1
4/
/sin
u
u
− −
−
takΩe dolΩna ymet\ na yntervale ( 0, 1 ) ne menee dvux razlyçn¥x nulej. Od-
nako πto nevozmoΩno v sylu toho, çto G u( )( )2 , kak raznost\ v¥pukloj vverx y
v¥pukloj vnyz funkcyj, moΩet ymet\ na ( 0, 1 ) ne bolee odnoho nulq. Polu-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX TEORYY APPROKSYMACYY … 313
çennoe protyvoreçye dokaz¥vaet trebuemoe neravenstvo. Polahaq v nem u =
= 2t /π , hde 0 < t < π / 2, ymeem
2
2
sin
t
<
2 4t
π
π
/
.
Yspol\zuq dannoe sootnoßenye, yz (33) poluçaem α > π mp / 4. Takym obrazom,
π mp / 4 < α < mp . (34)
Dlq funkcyy
˜ ( )Φ t uslovye (21) prymet vyd
2tn
π
α
≥
π τ τ
τ τ
π2
2
2
0
0
2tn
d
d
mptn
mp
(sin )
(sin )
/
/
/
∗∫
∫
. (35)
V¥polnqq v sootnoßenyy (35) zamenu v = 2tn / π, poluçaem neravenstvo
vα+1 ≥
(sin )
(sin )
/
/
/
/
τ τ
τ τ
π
π
2
2
0
2
0
2
∗∫
∫
mp
mp
d
d
v
, (36)
hde 0 < v < ∞ , kotoroe nam y trebuetsq dokazat\.
Oboznaçym
Q ( v ) df= v
v
α
π
π
τ τ
τ τ
+ ∗
−
∫
∫
1 0
2
0
2
2
2
(sin )
(sin )
/
/
/
/
mp
mp
d
d
(37)
y rassmotrym πtu funkcyg na otrezke [ 0, 1 ] , predvarytel\no doopredelyv ee v
toçke v = 0 po neprer¥vnosty sprava nulev¥m znaçenyem. Yz (37) y (34) sle-
duet, çto v dostatoçno maloj okrestnosty nulq Q ( v ) = v vα α+ −−1 1( )( )O mp >
> 0. RassuΩdaq ot protyvnoho, pokaΩem, çto Q ( v ) > 0 na yntervale 0 < v <
< 1 . PredpoloΩym, çto suwestvuet nekotoraq toçka ξ ∈ ( 0, 1 ) , v kotoroj
Q ( v ) menqet znak. Poskol\ku Q ( 0 ) = Q ( 1 ) = 0, v sylu teorem¥ Rollq per-
vaq proyzvodnaq Q
( )( )1 v dolΩna ymet\ na yntervale ( 0, 1 ) , po krajnej mere,
dva razlyçn¥x nulq. Uçyt¥vaq formulu (33), zapys¥vaem
Q
( )( )1 v =
( ) sinα πα+ −
1 2
4
v
v mp
. (38)
Yz (38) oçevydno, çto Q( )( )1 0 = Q( )( )1 1 = 0. Sledovatel\no, vtoraq proyzvodnaq
Q( )( )2 v = 2 1
2
2 3 1
2 3
mp
mpmp
mp
G/
/( ) ˜ ( )− −
−+ −
π α α
π
αv v , (39)
hde
˜ ( )G v df=
v
v v1
2
2 4
−
−
α π π
sin sin
mp
,
dolΩna ymet\ na yntervale ( 0, 1 ) ne menee trex razlyçn¥x nulej. Oçevydno,
çto kolyçestvo peremen znaka funkcyy (39) opredelqetsq v¥raΩenyem, zapy-
sann¥m v fyhurn¥x skobkax. Yssleduem dlq πtoho povedenye funkcyy
˜ ( )G v ,
v¥çyslyv ee pervug proyzvodnug
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
314 S. B. VAKARÇUK, A. N. WYTOV
˜ ( )( )G 1 v =
π π πα
2 2 4
1
2
v
v v
v−
−
cos ( )sin
mp
F ,
hde
F( )v df=
1
2 4 2
1 2 1
4+
− − −
tg ctg
tg(π π α π
π
v v v
v
mp
( )
)/ .
S uçetom (34) ymeem
lim ( )
v
v
→ +0
F = mp – α > 0,
(40)
lim ( )
v
v
→ −1 0
F =
2
4 2
1
π
π α πmp − − −
< 0.
V sylu (40) y monotonnoho vozrastanyq funkcyj tg ctg( ) ( )/ /π πv v2 4 y
tg( ) ( )/ /π πv v4 netrudno vydet\, çto F ( v ) menqet znak s plgsa na mynus pry
vozrastanyy v ∈ ( 0, 1 ) . Sledovatel\no, neotrycatel\naq na otrezke [ 0, 1 ]
funkcyq
˜ ( )G v qvlqetsq monotonno vozrastagwej pry 0 ≤ v ≤ λ y monoton-
no ub¥vagwej pry λ < v ≤ 1 , hde F ( λ ) = 0. No tohda zapysannaq v fyhur-
n¥x skobkax formul¥ (39) funkcyq dolΩna ymet\ ne bolee dvux razlyçn¥x
nulej na yntervale ( 0, 1 ) . Yz poluçennoho protyvoreçyq sleduet, çto pry 0 <
< v < 1 v¥polnqetsq neravenstvo Q( )v > 0.
Pust\ v ∈ [ 1, 2 ] . Yz sootnoßenyj (34) y (38) ymeem
Q( )( )1 2 = ( )( )/α α+ −1 2 2 2mp > ( )( )/ /α π+ −1 2 24 2mp mp > 0. (41)
Na polusehmente 1 ≤ v < 2 zapyßem formulu (38) v vyde
Q( )( )1 v = ( ) ( )/α α+ 1 2 2mp v vY ,
hde
Y ( )v df= 2
4
2− −−
mp
mp
/ v
vα π
sin , (42)
y yssleduem povedenye funkcyy (42). Pry πtom
Y ( )( )1 v = –
π π π
α4 4 4
1
v
v v
vcos sin
−mp
X( ),
hde
X( )v df= mp −
4
4
α
π
π
v
v
tg .
Oçevydno, çto X ( v ) qvlqetsq monotonno ub¥vagwej funkcyej, dlq kotoroj v
sylu (34) X ( 1 ) = 4 4( )/ /mpπ α π− < 0. Sledovatel\no, Y
( )( )1 0t > pry 1 ≤ v <
< 2 . Poskol\ku Y ( )1 0= , to Y ( )v ≥ 0 , kohda 1 ≤ v < 2 , a πto, s uçetom
(41), ravnosyl\no neravenstvu Q( )( )1 0v ≥ dlq 1 ≤ v ≤ 2 . Znaçyt, Q ( v ) qv-
lqetsq neub¥vagwej funkcyej na otrezke [ 1, 2 ] . Uçyt¥vaq, çto Q( )1 0= ,
poluçaem trebuemoe neravenstvo Q( )v ≥ 0 pry 1 ≤ v ≤ 2 .
Pust\ teper\ 2 ≤ v < ∞ . Tohda
Q( )v =
v
v
α
π
π
τ τ π
τ τ
+ −
+ −∫
∫
1 0
0
2
2 2 2
2
(sin ) ( )
(sin )
/ /
/
/
mp
mp
d
d
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
O NEKOTORÁX ∏KSTREMAL|NÁX ZADAÇAX TEORYY APPROKSYMACYY … 315
S uçetom (33), (34) ymeem
Q
( )( )1 v = ( )( )/α α+ −1 2 2v mp > 0. (43)
Poskol\ku Q( )2 0≥ , v sylu (43) poluçaem Q( )v > 0 pry 2 < v < ∞ .
Teorema 2 dokazana.
Sledstvye-1. Pust\ funkcyq Ψ ( x ) udovletvorqet trebovanyqm teore-
m¥81, a velyçyna α opredelqetsq sootnoßenyem (32). Tohda ymegt mesto
sledugwye ravenstva:
g L S Sn
p
m
p
2 β
Ψ Ω Φ( , , ˜ );( ) = g L S Sn
p
m
p
2 1− ( )β
Ψ Ω Φ( , , ˜ ); =
= E L Sn
p
m
S p− ( )1 β
Ψ Ω Φ( , , ˜ ) = ω π
π
α
m L
p
x
n
n
p
(sin , ) ( )([ , / ])
/
⋅
−
0 2
1
2
Ψ ,
hde 1 ≤ p < ∞ ; gn( )⋅ — lgboj yz rassmotrenn¥x v teoreme 2 n-popereçny-
kov.
7. Vopros¥ naxoΩdenyq toçn¥x verxnyx hranej modulej koπffycyentov
Fur\e na razlyçn¥x klassax funkcyj dejstvytel\noho peremennoho yzuçalys\
mnohymy matematykamy (sm., naprymer, [15]). Analohyçnaq zadaça, s naßej
toçky zrenyq, predstavlqet opredelenn¥j ynteres y v rassmatryvaemom zdes\
sluçae.
PredloΩenye-1. Pust\ v¥polnen¥ uslovyq teorem¥ 2. Tohda ymegt mes-
to sledugwye ravenstva:
sup ˆ ( )
( ) ( , , )
( )
f x L S
f x
p
m
f n
∈
/≡
β
Ψ Ω Φ
const
= sup ˆ ( )
( ) ( , , )
( )
f x L S
f x
p
m
f n
∈
/≡
−
β
Ψ Ω Φ
const
=
= 2
2
1
0 2
1− −⋅
/
([ , / ])(sin , ) ( )p
m Lx n
np
ω π
π Ψ Φ . (44)
Dokazatel\stvo. Yz (23) y toho fakta, çto
ˆ ( )f n = ˆ ( )f n− , dlq f ( x )8∈
∈ S
p
poluçaem
E fn S p−1( ) ≥ 21/ ˆ ( )p f n .
Yz (22) y dannoho neravenstva sleduet ocenka sverxu
sup ˆ ( )
( ) ( , , )
( )
f x L S
f x
p
m
f n
∈
/≡
β
Ψ Ω Φ
const
≤ 2 1
1
−
− ( )/ ( , , )p
n
p
m
S
E L S
pβ
Ψ Ω Φ ≤
≤ 2
2
1
0 2
1− −⋅
/
([ , / ])(sin , ) ( )p
m Lx n
np
ω π
π Ψ Φ . (45)
Dlq poluçenyq ocenky snyzu rassmotrym funkcyg
f x∗( ) df= 2
2 2 2
1
0 2
1− −
−
⋅
+
/
([ , / ])(sin , ) ( )p
m L
inx inx
x n
n
e e
p
ω π
π ππ Ψ Φ .
Poskol\ku
f S p∗ = ω π
πm Lx n
np
(sin , ) ( )([ , / ])⋅
−
0 2
1
2
Ψ Φ ,
f x∗( ) prynadleΩyt ßaru B̃ yz teorem¥ 2, a znaçyt, y klassu L S p
mβ
Ψ Ω Φ( , , ) .
Sledovatel\no,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
316 S. B. VAKARÇUK, A. N. WYTOV
sup ˆ ( )
( ) ( , , )
( )
f x L S
f x
p
m
f n
∈
/≡
β
Ψ Ω Φ
const
≥ ˆ ( )f n∗ = 2
2
1
0 2
1− −⋅
/
([ , / ])(sin , ) ( )p
m Lx n
np
ω π
π Ψ Φ . (46)
Sopostavlqq sootnoßenyq (45) y (46), poluçaem formulu (44), çto y zaverßaet
dokazatel\stvo predloΩenyq 1.
Sledstvye-2. Pust\ v¥polnen¥ uslovyq sledstvyq 1. Tohda dlq lgboho
natural\noho çysla n y 1 ≤ p < ∞ spravedlyv¥ ravenstva
sup ˆ ( )
( ) ( , , ˜ )
( )
f x L S
f x
p
m
f n
∈
/≡
β
Ψ Ω Φ
const
= sup ˆ ( )
( ) ( , , ˜ )
( )
f x L S
f x
p
m
f n
∈
/≡
−
β
Ψ Ω Φ
const
=
= 2 1
0 2
1− + −⋅
( )/
([ , / ])
/
(sin , ) ( )α
π
α
ω πp
m L
p
x n
np
Ψ .
1. StoroΩenko ∏. A., Krotov V. H., Osval\d P. Prqm¥e y obratn¥e teorem¥ typa DΩeksona
v8prostranstvax Lp , 0 1< <p // Mat. sb. – 1975. – 98, # 3. – S. 395 – 415.
2. Runovskyj K. V. O pryblyΩenyy semejstvamy lynejn¥x polynomyal\n¥x operatorov
v8prostranstvax Lp , 0 1< <p // Tam Ωe. – 1994. – 185, # 8. – S. 81 – 102.
3. Runovskyj K. V. Ob odnoj ocenke dlq yntehral\noho modulq hladkosty // Yzv. vuzov. Mate-
matyka. – 1992. – # 1. – S.878 – 80.
4. Ivanov K. G. New estimates of errors of quadrature formulae, formulae of numerical differention
and interpolation // Anal. Math. – 1980. – 6, # 4. – P. 281 – 303.
5. Ivanov K. G. On a new characteristic of functions. I // Serdyka Bælh. Mat. spysanye. – 1982. –
8, # 3. – S.8262 – 279.
6. Vakarçuk S. B. O nayluçßyx polynomyal\n¥x pryblyΩenyqx v L2 nekotor¥x klassov 2π-
peryodyçeskyx funkcyj y toçn¥x znaçenyqx yx n-popereçnykov // Mat. zametky. – 2001. –
70, # 3. – S. 334 – 345.
7. Stepanec A. Y. Approksymacyonn¥e xarakterystyky prostranstv S p
ϕ // Ukr. mat. Ωurn. –
2001. – 53, # 3. – S.8392 – 416.
8. Stepanec A. Y., Serdgk A. S. Prqm¥e y obratn¥e teorem¥ teoryy pryblyΩenyq funkcyj
v prostranstve S p
// Tam Ωe. – 2002. – 54, # 1. – S.8106 – 124.
9. Stepanec A. Y. Klassyfykacyq y pryblyΩenye peryodyçeskyx funkcyj. – Kyev: Nauk.
dumka, 1987. – 2688s.
10. Dzqd¥k V. K. Vvedenye v teoryg ravnomernoho pryblyΩenyq funkcyj polynomamy. – M.:
Nauka, 1977. – 5128s.
11. Tyxomyrov V. M. Teoryq pryblyΩenyj // Ytohy nauky y texnyky. Sovr. problem¥ matema-
tyky. Fundam. napravlenyq / VYNYTY. – 1987. – 14. – S.8103 – 260.
12. Vojcexovskyj V. R. Popereçnyky deqkyx klasiv z prostoru S p
// Ekstremal\ni zadaçi
teori] funkcij ta sumiΩni pytannq: Pr. In-tu matematyky NAN Ukra]ny. – 2003. – 46. –
S.817 – 26.
13. Serdgk A. S. Popereçnyky v prostori S p
klasiv funkcij, wo oznaçagt\sq modulqmy ne-
perervnosti ]x ψ-poxidnyx // Tam Ωe. – S.8229 – 248.
14. Vakarçuk S. B. O nekotor¥x πkstremal\n¥x zadaçax teoryy approksymacyy v prostranstvax
S p
( )1 ≤ < ∞p // VoroneΩ. zym. mat. ßkola „Sovremenn¥e metod¥ teoryy funkcyj y
smeΩn¥e problem¥” (VoroneΩ, 26 qnv. – 82 fevr. 2003 h.). – VoroneΩ: VoroneΩ. un-t, 2003.
– S.847 – 48.
15. Vakarçuk S. B. Neravenstva typa DΩeksona y toçn¥e znaçenyq popereçnykov klassov funk-
cyj v prostranstvax S p , 1 ≤ < ∞p // Ukr. mat. Ωurn. – 2004. – 56, # 5. – S. 595 – 605.
Poluçeno 29.06.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
|
| id | umjimathkievua-article-3455 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:42:53Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8e/acdc87e383d9623701c8f09209414d8e.pdf |
| spelling | umjimathkievua-article-34552020-03-18T19:55:07Z On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ O некоторых экстремальных задачах теории аппроксимации функций в пространствах $S^p,\quad 1 \leq p < \infty$ Vakarchuk, S. B. Shchitov, A. N. Вакарчук, С. Б. Щитов, А. Н. Вакарчук, С. Б. Щитов, А. Н. We consider and study properties of the smoothness characteristics $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, of functions $f(x)$ that belong to the space $S^p,\quad 1 \leq p < \infty$, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$ are calculated. Розглянуто та досліджено властивості гладкісних характеристик $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, функцій $f(x)$, що належать уведеному O. I. Степанцем простору $S^p,\quad 1 \leq p < \infty$. Одержано точні нерівності типу Джексона та обчислено точні значення поперечників класів функцій, визначених за допомогою $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$. Institute of Mathematics, NAS of Ukraine 2006-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3455 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 3 (2006); 303-316 Український математичний журнал; Том 58 № 3 (2006); 303-316 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3455/3647 https://umj.imath.kiev.ua/index.php/umj/article/view/3455/3648 Copyright (c) 2006 Vakarchuk S. B.; Shchitov A. N. |
| spellingShingle | Vakarchuk, S. B. Shchitov, A. N. Вакарчук, С. Б. Щитов, А. Н. Вакарчук, С. Б. Щитов, А. Н. On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ |
| title | On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ |
| title_alt | O некоторых экстремальных задачах теории аппроксимации функций в пространствах $S^p,\quad 1 \leq p < \infty$ |
| title_full | On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ |
| title_fullStr | On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ |
| title_full_unstemmed | On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ |
| title_short | On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$ |
| title_sort | on some extremal problems in the theory of approximation of functions in the spaces $s^p,\quad 1 \leq p < \infty$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3455 |
| work_keys_str_mv | AT vakarchuksb onsomeextremalproblemsinthetheoryofapproximationoffunctionsinthespacesspquad1leqpltinfty AT shchitovan onsomeextremalproblemsinthetheoryofapproximationoffunctionsinthespacesspquad1leqpltinfty AT vakarčuksb onsomeextremalproblemsinthetheoryofapproximationoffunctionsinthespacesspquad1leqpltinfty AT ŝitovan onsomeextremalproblemsinthetheoryofapproximationoffunctionsinthespacesspquad1leqpltinfty AT vakarčuksb onsomeextremalproblemsinthetheoryofapproximationoffunctionsinthespacesspquad1leqpltinfty AT ŝitovan onsomeextremalproblemsinthetheoryofapproximationoffunctionsinthespacesspquad1leqpltinfty AT vakarchuksb onekotoryhékstremalʹnyhzadačahteoriiapproksimaciifunkcijvprostranstvahspquad1leqpltinfty AT shchitovan onekotoryhékstremalʹnyhzadačahteoriiapproksimaciifunkcijvprostranstvahspquad1leqpltinfty AT vakarčuksb onekotoryhékstremalʹnyhzadačahteoriiapproksimaciifunkcijvprostranstvahspquad1leqpltinfty AT ŝitovan onekotoryhékstremalʹnyhzadačahteoriiapproksimaciifunkcijvprostranstvahspquad1leqpltinfty AT vakarčuksb onekotoryhékstremalʹnyhzadačahteoriiapproksimaciifunkcijvprostranstvahspquad1leqpltinfty AT ŝitovan onekotoryhékstremalʹnyhzadačahteoriiapproksimaciifunkcijvprostranstvahspquad1leqpltinfty |