On the best L2 -approximations of functions by using wavelets
We obtain the exact Jackson-type inequalities for approximations in L2 (R) of functions f∈ L2 (R) with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon–Kotelnikov wavelets.
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| Дата: | 2008 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3229 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509280743981056 |
|---|---|
| author | Babenko, V. F. Zhiganova, G. S. Бабенко, В. Ф. Жиганова, С. Г. Бабенко, В. Ф. Жиганова, С. Г. |
| author_facet | Babenko, V. F. Zhiganova, G. S. Бабенко, В. Ф. Жиганова, С. Г. Бабенко, В. Ф. Жиганова, С. Г. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:39Z |
| description | We obtain the exact Jackson-type inequalities for approximations in L2 (R) of functions f∈ L2 (R)
with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon–Kotelnikov wavelets. |
| first_indexed | 2026-03-24T02:38:36Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.5
V. F. Babenko (Dnepropetr. nac. un-t,
Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
H. S. Ûyhanova (Dnepropetr. nac. un-t)
O NAYLUÇÍYX L2 -PRYBLYÛENYQX FUNKCYJ
S POMOW|G VSPLESKOV
We obtain the exact Jackson-type inequalities for approximations in L2 ( )R of functions f L∈ 2 ( )R
with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon –
Kotelnikov wavelets.
OderΩano toçni nerivnosti typu DΩeksona dlq nablyΩen\ v L2 ( )R funkcij f L∈ 2 ( )R za do-
pomohog çastynnyx sum spleskovyx rqdiv u vypadku spleskiv Mej[ra ta Íennona – Kotel\ny-
kova.
Pervoe toçnoe neravenstvo typa DΩeksona b¥lo poluçeno N. P. Kornejçukom
[1], kotor¥j dokazal, çto
E fn C( )
2π
< ω π
π
f
n C
,
2
,
hde E fn C( )
2π
— nayluçßee ravnomernoe pryblyΩenye neprer¥vnoj 2π-peryo-
dyçeskoj funkcyy f tryhonometryçeskymy polynomamy porqdka ne v¥ße n –
– 1, a ω
π
( , )f t C2
— ravnomern¥j modul\ neprer¥vnosty funkcyy f.
PozΩe N. Y. Çern¥x [2, 3] yssledoval vopros o nayluçßej konstante v nera-
venstve DΩeksona dlq nayluçßyx pryblyΩenyj E fn( ) v L2 0 2( , )π funkcyy
f t( ) ∈ L2 0 2( , )π tryhonometryçeskymy polynomamy porqdka n. Ym b¥lo polu-
çeno neuluçßaemoe neravenstvo
E fn L( ) ( , )2 0 2π < 1
2 2 0 2
ω π
π
f
n L
,
( , )
.
Dlq dokazatel\stva πtoho neravenstva N. Y. Çern¥x ustanovyl predstavlqgwee
samostoqtel\n¥j ynteres neravenstvo
E fn L( ) ( , )2 0 2π ≤ 1
2 2 2 0 2
2
0
1 2
n f u nuduL
n
ω π
π
, sin( , )
/ /
( )
∫ .
Ym Ωe b¥lo dokazano, çto dlq lgboj funkcyy f x( ), u kotoroj f xr( )( ) ∈
∈ L2 0 2( , )π , ymeet mesto neravenstvo
E fn L( ) ( , )2 0 2π ≤
1
2
1
2 2 0 2
2
0
1 2
n
n
f u nudur
r
L
n
ω
π
π
( )( )
∫ , sin
( , )
/ /
.
Yspol\zuq ydey N. Y. Çern¥x, Y. Y. Ybrahymov y F. H. Nasybov [4], a takΩe,
nezavysymo, V. G. Popov [5] poluçyly analohyçn¥e neravenstva dlq nayluçße-
ho pryblyΩenyq funkcyy f x( ) ∈ L2( )R cel¥my funkcyqmy πksponencyal\-
© V. F. BABENKO, H. S. ÛYHANOVA, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8 1119
1120 V. F. BABENKO, H. S. ÛYHANOVA
noho typa σ. Toçnee, v rabotax [4, 5] b¥lo dokazano neuluçßaemoe neravenstvo
E f Lσ( ) ( )2 R <
1
2 2
ω π
σ
f
L
,
( )
R
, (1)
a v rabote [5] V. G. Popov poluçyl takΩe neuluçßaem¥e neravenstva
E f Lσ( ) ( )2 R < 1
2 2 2
2
0
1 2
σ ω σ
π σ
f u uduL, sin( )
/ /
( )
∫ R , (2)
E f Lσ( ) ( )2 R <
1
2
1
2 2
2
0
1 2
σ
σ ω σ
π σ
r
r
L
f u u du( )( )
∫ , sin
( )
/ /
R
. (3)
V dal\nejßem zadaçy, svqzann¥e s toçn¥my neravenstvamy typa DΩeksona v
prostranstvax L2 0 2( , )π y L2( )R , yzuçalys\ mnohymy avtoramy (sm., naprymer,
[6 – 11]).
V nastoqwej stat\e pokazano, çto, yspol\zuq metod N. Y. Çern¥x, moΩno po-
luçyt\ toçn¥e neravenstva typa DΩeksona dlq pryblyΩenyq funkcyj f x( ) ∈
∈ L2( )R s pomow\g çastn¥x summ vspleskov¥x rqdov v sluçae vspleskov Meje-
ra y Íennona – Kotel\nykova.
Pust\ L2( )R — prostranstvo yzmerym¥x funkcyj f : R → C s koneçnoj
normoj
f 2 =
f t dt( )
/
2
1 2
R
∫
.
M¥ budem yspol\zovat\ preobrazovanye Fur\e
ˆ( )f ξ =
f t e dti t( ) – ξ
R
∫
y ravenstvo Parsevalq
f 2 = 1
2 2π
f̂ .
Napomnym, çto modulem neprer¥vnosty funkcyy f t( ) ∈ L2( )R naz¥vaetsq
funkcyq
ω δ( , )f 2 =
sup ( ) – ( ) ( )
u
Lf t u f t
≤
+
δ 2 R , δ ≥ 0.
Pryvedem neobxodym¥e svedenyq yz teoryy vspleskov (sm., naprymer, [12],
hl.E7). Funkcyq ψ ∈L2( )R naz¥vaetsq vspleskom, esly systema funkcyj
ψk l x, ( ) = 2 22k k x l/ –ψ( ), x ∈R , k, l ∈Z ,
qvlqetsq ortonormyrovann¥m bazysom v L2( )R . Ob¥çno dlq postroenyq vsples-
kov yspol\zuetsq tot yly ynoj kratnomasßtabn¥j analyz (KMA) (xotq suwest-
vugt vsplesky, ne poroΩdenn¥e nykakymy KMA).
Rassmotrym posledovatel\nost\ podprostranstv V = Vk k{ } ∈Z prostranstva
L2( )R . Ee naz¥vagt KMA, esly v¥polnqgtsq sledugwye uslovyq:
A1) Vk � Vk +1, k ∈Z;
A2 )
Vkk ∈Z∪ = L2( )R ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
O NAYLUÇÍYX L2-PRYBLYÛENYQX FUNKCYJ S POMOW|G VSPLESKOV 1121
A3)
Vkk ∈Z∩ = 0{ };
A4 ) f x( ) ∈ Vk , esly y tol\ko esly f x( )2 ∈ Vk +1;
A5) suwestvuet funkcyq ϕ ∈ L2( )R takaq, çto systema ϕ( – )x k{ k ∈ }Z
obrazuet ortonormyrovann¥j bazys v V0. Funkcyg ϕ naz¥vagt masßtabnoj
funkcyej KMA V.
Yz svojstv A4) y A5) KMA sleduet, çto dlq proyzvol\noho k ∈Z systema
ϕ{ }k =
ϕk l k l, ,{ } ∈Z , hde ϕk l x, ( ) = 2 2k / ϕ 2k x l–( ), obrazuet ortonormyrovann¥j
bazys v Vk .
Dlq lgboho k ∈Z çerez Wk oboznaçym ortohonal\noe dopolnenye pro-
stranstva Vk do prostranstva Vk +1:
V Wk k⊕ = Vk +1, k ∈Z.
Yz uslovyj A1) – A3) sleduet razloΩenye prostranstva L2( )R
⊕
∈k
kW
Z
= L2( )R ,
pryçem v sylu uslovyq A4) f x( ) ∈ Wk , esly y tol\ko esly f x( )2 ∈ Wk +1.
Pust\ funkcyq ψ takova, çto systema ψ{ }0 = ψ0,l{ = ψ( – )⋅ } ∈
l
l Z
obrazuet
ortonormyrovann¥j bazys prostranstva W0 . Tohda dlq lgboho k ∈Z systema
ψ{ }k = ψk l,{ = 2 22k k
l
l/ ( – )ψ ⋅ }
∈Z
obrazuet ortonormyrovann¥j bazys prostran-
stva Wk , a systema ψ{ } = ψk l k l, ,{ } ∈Z — ortonormyrovann¥j bazys L2( )R , t.Ee.
ψ qvlqetsq vspleskom. V πtom sluçae hovorqt, çto ortohonal\n¥j vsplesk ψ
poroΩdaetsq KMA V.
Pust\ 0 < ε ≤ 1 / 3 y neprer¥vnaq funkcyq θ ξ( ) = θ ξε( ), ξ ∈R , udovletvo-
rqet sledugwym uslovyqm:
1) 0 ≤ θ ξ( ) ≤ 1, θ ξ(– ) = θ ξ( ), ξ ∈R ;
2) θ ξ( ) = 1 pry ξ ≤ ( – )1 ε π y θ ξ( ) = 0 pry ξ ≥ ( )1 + ε π ;
3) θ π ξ2( – ) + θ π ξ2( )+ = 1 pry ξ ∈ 0, π[ ].
Opredelym funkcyg ϕ εM, ∈ L2( )R ravenstvom
%
ϕ ξεM, ( ) = θ ξ( ), ξ ∈R .
Pust\ zadan KMA V ε = Vk k{ } ∈Z, kotor¥j poroΩdaetsq masßtabnoj funkcyej
ϕ εM,
. ∏tot KMA poroΩdaet ortonormyrovann¥e vsplesky Mejera, kotor¥e op-
redelqgtsq ravenstvom
%
ψ ξεM, ( ) = e
i
ξ
θ ξ θ ξ2 2 2
2
– ( ) , ξ ∈R .
Otmetym, çto
%
supp ,ψ εM = –( )1 2+[ ε π , –( – )1 ε π] ∪ ( – )1 ε π[ , ( )1 2+ ]ε π , tak çto
%
ψ ε
k j
M t,
, ( ) = 0 dlq lgboho t ∈ –( – )1 2ε π k[ , ( – )1 2ε π k ].
Dlq lgboho k ∈Z y KMA V ε
poloΩym
E f Vk
ε( , )2 = inf – :f h h Vk2 ∈{ }.
Perexodq k yzloΩenyg osnovn¥x rezul\tatov stat\y, v pervug oçered\ ustano-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1122 V. F. BABENKO, H. S. ÛYHANOVA
vym spravedlyvost\ sledugwej teorem¥.
Teorema 1. Dlq lgboj funkcyy f L∈ 2( )R , neπkvyvalentnoj nulg, y lg-
boho n ∈Z v¥polnqetsq neravenstvo
E f Vn
ε( , )–1 2 < 1
2
2 1 2 11
2
2
0
1
2 1
1 2
n nf u u du
n
–
( – )
/
( – ) , sin ( – )ε π ω π ε
ε
( ) ( )
∫ . (4)
Pry πtom konstantu
1
2
v pravoj çasty neravenstva (4) umen\ßyt\ nel\zq.
Dokazatel\stvo. Proyzvol\nug funkcyg f L∈ 2( )R moΩno predstavyt\
v vyde summ¥ sxodqwehosq v L2( )R rqda
f x( ) =
c xk j k j
M
jk
, ,
, ( )ψ ε
∈∈
∑∑
ZZ
,
hde ck j, =
f x x dxk j
M( ) ( ),
,ψ ε
R∫ . Nayluçßee pryblyΩenye funkcyy f prostran-
stvom Vn –1 realyzugt çastn¥e summ¥ S f xn
ε ( ) =
jk
n
k j k j
Mc x∈= ∞∑∑ Z–
–
, ,
, ( )
1 ψ ε
, tak
çto
E f Vn
ε , –1 2
2( ) = f S fn– ε
2
2
=
ck j k j
M
jk n
, ,
,ψ ε
∈=
∞
∑∑
Z 2
2
=
= ck j
jk n
,
2
∈=
∞
∑∑
Z
= 1
2
2
2
π
ψ εck j k j
M
jk n
, ,
, .
%
Z∈=
∞
∑∑
Poskol\ku
%
ψ ε
k j
M t,
, ( ) prynymaet nulev¥e znaçenyq dlq lgboho t ∈ –( – )1 2ε π k[ ,
( – )1 2ε π k ], to
E f Vn
ε , –1 2
2( ) =
1
2
2
2 1
π
ψ ξ ξε
ε π ε
c dk j k j
M
jk nn
, ,
,
( – )
( )
^
Z∈=
∞
≥
∑∑∫ . (5)
Ocenym modul\ neprer¥vnosty funkcyy f L∈ 2( )R :
ω( , )f u 2
2 ≥ f u f( ) – ( )⋅+ ⋅ 2
2 = 1
2
1
2
2
π
ψ εc ek j k j
M j u
jk
, ,
, ( )( ) –
%
ZZ
⋅ ( ) =⋅
∈∈
∑∑
= 1
2
1
2
2
π
ψ ξ ξε ξc e dk j k j
M
jk
i u
, ,
, ( ) –
^
ZZR ∈∈
∑∑∫ =
= 1 1
2
π
ψ ξ ξ ξεc u dk j k j
M
jk
, ,
, ( ) ( – cos )
^
ZZR ∈∈
∑∑∫ ≥
≥
1 1
2
2 1
π
ψ ξ ξ ξε
ξ ε π
c u dk j k j
M
jkn
, ,
,
( – )
( ) ( – cos )
^
ZZ ∈∈≥
∑∑∫ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
O NAYLUÇÍYX L2-PRYBLYÛENYQX FUNKCYJ S POMOW|G VSPLESKOV 1123
=
1
2
2 1
π
ψ ξ ξε
ξ π ε
c dk j k j
M
jkn
, ,
,
( – )
( )
^
ZZ ∈∈≥
∑∑∫ –
–
1
2
2 1
π
ψ ξ ξ ξε
ξ π ε
c udk j k j
M
jkn
, ,
,
( – )
( ) cos
^
ZZ ∈∈≥
∑∑∫ ≥
≥
1
2
2 1
π
ψ ξ ξε
ξ π ε
c dk j k j
M
jk nn
, ,
,
( – )
( )
^
Z∈≥≥
∑∑∫ –
–
1
2
2 1
π
ψ ξ ξ ξε
ξ π ε
c u dk j k j
M
jkn
, ,
,
( – )
( ) cos
^
ZZ ∈∈≥
∑∑∫ .
Uçyt¥vaq ravenstvo (5), poluçaem
ω( , )f u 2
2 ≥ 2 1 2
2
E f Vn
ε , –( ) –
1
2
2 1
π
ψ ξ ξ ξε
ξ π ε
c u dk j k j
M
jkn
, ,
,
( – )
( ) cos
^
ZZ ∈∈≥
∑∑∫
yly
E f Vn
ε , –1 2
2( ) ≤ 1
2 2
2ω( , )f u +
1
2
2
2 1
π
ψ ξ ξ ξε
ξ π ε
c u dk j k j
M
jkn
, ,
,
( – )
( ) cos
^
ZZ ∈∈≥
∑∑∫ .
(6)
UmnoΩym pravug y levug çasty neravenstva (6) na sin (2 1n( – ε π) u) y proyn-
tehryruem poluçennoe neravenstvo po u na otrezke 0 1
2 1
,
( – )n ε
. V rezul\tate
poluçym
E f V u dun
n
n
ε
ε
π ε, sin ( – )–
( – )
1 2
2
0
1
2 1
2 1( ) ( )∫ ≤
≤ 1
2
2 12
2
0
1
2 1
ω π ε
ε
f u u du
n
n, sin ( – )
( – )
( ) ( )∫ +
+
1
2
2 1
0
1
2 1 2
2 1
π
ψ ξ ξ π ε ξ
ε
ε
ξ π ε
n
n
c u u d duk j k j
M
jk
n
( – )
, ,
,
( – )
( ) cos sin ( – )∫ ∑∑∫
∈∈≥
( )
^
ZZ
. (7)
Otmetym, çto
E f V u dun
n
n
ε
ε
π, sin–
( – )
1 2
2
0
1
2 1
2( ) ( )∫ = E f V u dun
n
n
ε
ε
π, sin–
( – )
1 2
2
0
1
2 1
2( ) ( )∫ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1124 V. F. BABENKO, H. S. ÛYHANOVA
= E f V un n
n
nε ε
π ε
π, –
( – )
cos–
( – )
1 2
2
0
1
2 11
2 1
2( )
= 1
2 11 1 2
2
n nE f V– –( – )
,
π ε
ε( ) .
PokaΩem, çto vtoroe slahaemoe v pravoj çasty (7) otrycatel\no. Ymeem
1
2
2 1
0
1
2 1 2
2 1
π
ψ ξ ξ ε π ξ
ε
ε
ξ π ε
n
n
c u u d duk j k j
M
jk
n
( – )
, ,
,
( – )
( ) cos sin ( – )∫ ∑∑∫
∈∈≥
( )
^
ZZ
=
=
1
2
2 1
2 1
2
0
1
2 1
π
ψ ξ ξ π ε ξ
ξ π ε
ε
ε
≥ ∈∈
∫ ∑∑ ∫ ( )
n
n
c u u dudk j k j
M
jk
n
( – )
, ,
,
( – )
( ) cos sin ( – )
^
ZZ
.
Rassmotrym
0
1
2 1
2
n
u u dun
( – )
cos sin
ε
ξ π∫ ( ) =
= 1
2
2 1
0
1
2 1n
n u
( – )
sin ( – )
ε
ε π ξ∫ +( )[ + sin ( – ) –2 1n u duε π ξ( ) ] =
= 1
2
1
2 1
2 1–
( – )
cos ( – )n
n u
π ε ξ
π ε ξ
+
+( ) –
– 1
2 1
2 1
0
1
2 1
n
n u
n
π ε ξ
π ε ξ ε
( – )–
cos ( – ) –
( – )( )
=
= 1
2
2 1
2 1 2 1
2 1
2 1
1
2 2 2 2
1
2 2 2 2
n
n n
n
n
+ +
+
π ε
ε π ξ
ξ
ε
π ε
ε π ξ
( – )
( – ) –
cos
( – )
( – )
( – ) –
=
=
2 1
2 1
1
2 12 2 2 2
n
n n
π ε
ε π ξ
ξ
ε
( – )
( – ) –
cos
( – )
+
.
Poskol\ku 2 1n π ε( – ) 1
2 1
+
cos
( – )
ξ
εn > 0 dlq poçty vsex ξ, pry vsex ξ ≥
≥ 2 1n π ε( – )
cos sin ( – )
( – )
ξ π ε
ε
u u du
n
n
0
1
2 1
2 1∫ ( ) < 0.
Sledovatel\no,
1
2
2 1
2 1
2
0
1
2 1
π
ψ ξ ξ π ε ξ
ξ π ε
ε
ε
≥ ∈∈
∫ ∑∑∫ ( )
n
n
c u u d duk j k j
M
jk
n
( – )
, ,
,
( – )
( ) cos sin ( – )
^
ZZ
< 0,
t.Ee. otrycatel\nost\ vtoroho slahaemoho v pravoj çasty (7) dokazana.
Teper\ yz (7) poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
O NAYLUÇÍYX L2-PRYBLYÛENYQX FUNKCYJ S POMOW|G VSPLESKOV 1125
1
2 11 1 2
2
n nE f V– –( – )
,
ε π
ε( ) < 1
2
2 1
0
1
2 1
2
2
n
f u u dun
( – )
( , ) sin ( – )
ε
ω π ε∫ ( ) .
Neravenstvo (4) dokazano. UtverΩdenye o toçnosty konstant¥
1
2
v nera-
venstve (4) sleduet yz toçnosty konstant¥ v neravenstve (2) pry σ = (1 – ε π) 2n
.
Teorema dokazana.
Teorema 2. Dlq lgboho n ∈Z y lgboj funkcyy f L∈ 2( )R , neπkvyvalent-
noj nulg, v¥polnqetsq neravenstvo
E f Vn
ε , –1 2( ) < 1
2
1
1 2 2
ω
ε
f n,
( – )
. (8)
Pry lgbom fyksyrovannom n ∈Z konstantu
1
2
v pravoj çasty neravenst-
va nel\zq zamenyt\ men\ßej.
Dokazatel\stvo. Poskol\ku ω( , )f u 2 — neub¥vagwaq funkcyq, yz nera-
venstva (4) poluçaem
E f Vn
ε , –1 2( ) ≤ 1
2
1
2 1
2 1 2 1
2
2
1
0
1
2 1
1 2
ω
ε
ε π π ε
ε
f u dun
n n
n
,
( – )
( – ) ( – )–
( – )
/
( )
∫ =
= 1
2
1
1 2 2
ω
ε
f n,
( – )
.
Neravenstvo (8) dokazano. Neuluçßaemost\ konstant¥
1
2
v pravoj çasty
neravenstva (8) sleduet yz neuluçßaemosty konstant¥ v neravenstve (1) pry σ =
= (1 – ε π) 2n
.
Teorema dokazana.
Teper\ pryvedem neravenstvo, ocenyvagwee E f Vn
ε , –1 2( ) çerez modul\ ne-
prer¥vnosty r-j proyzvodnoj funkcyy f.
Teorema 3. Pust\ n ∈Z y r ∈N . Tohda dlq lgboj funkcyy f L∈ 2( )R , u
kotoroj (r – 1)-q proyzvodnaq lokal\no absolgtno neprer¥vna y f Lr( ) ( )∈ 2 R ,
v¥polnqetsq neravenstvo
E f Vn
ε , –1 2( ) <
< 1
2
2 1 2 11
0
1
2 1
2
2n r n r
n
f u( – ) ( – ) ,
– –
( – )
( )ε π ε π ω
ε
( ) ( )
∫ sin ( – )
/
2 1
1 2
n u duε π( )
. (9)
Dlq lgboho n ∈Z konstantu
1
2
2 1n r
( – )
–
ε π( ) umen\ßyt\ nel\zq.
Dokazatel\stvo. Otmetym, çto v uslovyqx teorem¥
$
f r( )( )ξ = ( ) ˆ( )i frξ ξ . (10)
Uçyt¥vaq sootnoßenyq (5) y (10), poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1126 V. F. BABENKO, H. S. ÛYHANOVA
E f Vr
n
ε ( )
–, 1 2( ) =
1
2
2
2 1
1 2
π
ξ ψ ξ ξε
ξ π ε
( ) ( ), ,
,
( – )
/
i c dr
j
k j k j
M
k nn ∈=
∞
≥
∑∑∫
Z
^
=
= 1
2
2
2
2 1
1 2
π
ξ ψ ξ ξε
ξ π ε
i c dr
j
k j k j
M
k nn ∈=
∞
≥
∑∑∫
Z
^
, ,
,
( – )
/
( ) ≥
≥ E f Vn
n rε π ε, ( – )–1 2
2 1( ) ( ) .
Otsgda y yz neravenstva (4) sleduet neravenstvo
E f Vn
ε , –1 2( ) <
< 1
2
2 1 2 11
0
1
2 1
2
2n r n r
n
f u( – ) ( – ) ,
– –
( – )
( )ε π ε π ω
ε
( ) ( )
∫ sin ( – )
/
2 1
1 2
n u duε π( )
.
UtverΩdenye o toçnosty konstant¥
1
2
2 1n r
( – )
–
ε π( ) v neravenstve (9) sleduet
yz toçnosty neravenstva (3).
Teorema dokazana.
Pust\ teper\ zadan KMA V 0 = Vk k{ } ∈Z
, kotor¥j poroΩdaetsq masßtabnoj
funkcyej
ϕs t( ) =
sin π
π
t
t
,
$
ϕ ξs( ) = χ ξπ π(– , )( ).
∏tot KMA poroΩdaet ortonormyrovann¥e vsplesky Íennona – Kotel\nykova
ψs t( ) = 2 2 1ϕs t( – ) – ϕs t( – / )1 2 .
Otmetym, çto
Ò
suppψs = – ; –2π π[ ] ∪ π π; 2[ ], tak çto
%
ψk j
s t, ( ) = 0 dlq lgboho t,
t < π2k
.
PoloΩym
E f Vn
0
1 2
, –( ) =
f ck j k j
s
jk
n
– , ,
–
–
ψ
∈= ∞
∑∑
Z
1
2
.
Netrudno proveryt\, çto dlq velyçyn¥ E f Vn
0
1 2
, –( ) ymegt mesto neraven-
stva, analohyçn¥e neravenstvam (4), (8) y (9) s ε = 0. Dlq πtoho dostatoçno po-
vtoryt\ s oçevydn¥my yzmenenyqmy dokazatel\stva teoremE1 – 3. Vproçem πty
analohy mohut b¥t\ poluçen¥ (pry ε → 0) y neposredstvenno yz teoremE1 – 3.
Takym obrazom, yz teoremE1 – 3 poluçaem takye sledstvyq.
Sledstvye 1. Dlq lgboj funkcyy f L∈ 2( )R , neπkvyvalentnoj nulg, y
lgboho n ∈Z v¥polnqetsq neravenstvo
E f Vn
0
1 2
, –( ) < 1
2
2 21
2
2
0
1
2
1 2
n nf u u du
n
–
/
( , ) sin( )π ω π∫
. (11)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
O NAYLUÇÍYX L2-PRYBLYÛENYQX FUNKCYJ S POMOW|G VSPLESKOV 1127
Pry πtom konstantu
1
2
v pravoj çasty neravenstva (11) umen\ßyt\
nel\zq.
Sledstvye 2. Dlq lgboho n ∈Z y lgboj funkcyy f L∈ 2( )R , neπkvyva-
lentnoj nulg, v¥polnqetsq neravenstvo
E f Vn
0
1 2
, –( ) < 1
2
1
2 2
ω f n,
. (12)
Pry lgbom fyksyrovannom n ∈Z konstantu
1
2
v pravoj çasty neraven-
stva (12) umen\ßyt\ nel\zq.
Sledstvye 3. Pust\ n ∈Z y r ∈N . Tohda dlq lgboj funkcyy f ∈
∈ L2( )R , u kotoroj ( – )r 1 -q proyzvodnaq lokal\no absolgtno neprer¥vna y
f r( ) ∈ L2( )R , v¥polnqetsq neravenstvo
E f Vn
0
1 2
, –( ) < 1
2
2 2 21
2
2
0
1
2
1 2
n r n r nf u u du
n
π π ω π( ) ( )
∫
– – ( )
/
, sin( ) . (13)
Dlq lgboho n ∈Z konstantu
1
2
2n r
π( )–
v pravoj çasty neravenstva (13)
umen\ßyt\ nel\zq.
1. Kornejçuk N. P. O toçnoj konstante v neravenstve DΩeksona dlq neprer¥vn¥x peryody-
çeskyx funkcyj // Dokl. AN SSSR. – 1962. – 145, # 3. – S. 514 – 515.
2. Çern¥x N. Y. O neravenstve DΩeksona v L2 // Tr. Mat. yn-ta AN SSSR. – 1967. – 88. –
S.E71 – 74.
3. Çern¥x N. Y. O nayluçßem pryblyΩenyy peryodyçeskyx funkcyj tryhonometryçeskymy
polynomamy v L2 // Mat. zametky. – 1967. – 2, #5. – S. 513 – 522.
4. Ybrahymov Y. Y., Nasybov F. H. Ocenka nayluçßeho pryblyΩenyq summyruemoj funkcyy
na dejstvytel\noj osy cel¥my funkcyqmy koneçnoj stepeny // Dokl. AN SSSR. – 1970. –
194, # 5. – S. 1113 – 1116.
5. Popov V. G. O nayluçßyx srednekvadratyçeskyx pryblyΩenyqx cel¥my funkcyqmy πks-
ponencyal\noho typa // Yzv. vuzov. Ser. mat. – 1972. – # 6. – S. 65 – 73.
6. Lyhun A. A. Toçn¥e neravenstva typa DΩeksona dlq peryodyçeskyx funkcyj v prostrans-
tve L2 // Mat. zametky. – 1988. – 43, # 6. – S. 757 – 769.
7. Gdyn V. A. Dyofantov¥e pryblyΩenyq v πkstremal\n¥x zadaçax // Dokl. AN SSSR. – 1980.
– 251, # 1. – S. 54 – 57.
8. Babaenko A. H. O toçnoj konstante v neravenstve DΩeksona v L2 // Mat. zametky. – 1986. –
39, # 5. – S. 651 – 664.
9. Arestov V. V., Chernykh N. I. On the L2-approximation of periodic function by trigonometric
polynomials // Approxim. and Funct. Spaces: Proc. Conf. (Gdansk, 1970). – Amsterdam: North-
Holland, 1981. – P. 5 – 43.
10. Volçkov V. V. O toçn¥x konstantax v neravenstve DΩeksona v prostranstve L2 // Ukr. mat.
Ωurn. – 1995. – 47, # 1. – S. 108 – 110.
11. Vakarçuk S. B. O nayluçßyx polynomyal\n¥x pryblyΩenyqx 2π-peryodyçeskyx funkcyj
y toçn¥x znaçenyqx n-popereçnykov funkcyonal\n¥x klassov v prostranstve L2 // Ukr.
mat. Ωurn. – 2002. – 54, # 12. – S. 1603 – 1615.
12. Kaßyn B. S., Saakqn A. A. Ortohonal\n¥e rqd¥. – M.: AFC, 1999. – 550 s.
Poluçeno 02.08.06,
posle dorabotky – 12.03.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
|
| id | umjimathkievua-article-3229 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:36Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c7/7336610283730d0d946eb2c51f5f96c7.pdf |
| spelling | umjimathkievua-article-32292020-03-18T19:48:39Z On the best L2 -approximations of functions by using wavelets О наилучших L2 -приближениях функций с помощью всплесков Babenko, V. F. Zhiganova, G. S. Бабенко, В. Ф. Жиганова, С. Г. Бабенко, В. Ф. Жиганова, С. Г. We obtain the exact Jackson-type inequalities for approximations in L2 (R) of functions f&isin; L2 (R) with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon&#8211;Kotelnikov wavelets. Одержано точні нерівності типу Джексона для наближень в L2 (R) функцій f&isin; L2 (R) за допомогою частинних сум сплескових рядів у випадку сплесків Мейєра та Шеннона&#8211;Котельникова. Institute of Mathematics, NAS of Ukraine 2008-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3229 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 8 (2008); 1119 – 1127 Український математичний журнал; Том 60 № 8 (2008); 1119 – 1127 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3229/3204 https://umj.imath.kiev.ua/index.php/umj/article/view/3229/3205 Copyright (c) 2008 Babenko V. F.; Zhiganova G. S. |
| spellingShingle | Babenko, V. F. Zhiganova, G. S. Бабенко, В. Ф. Жиганова, С. Г. Бабенко, В. Ф. Жиганова, С. Г. On the best L2 -approximations of functions by using wavelets |
| title | On the best L2 -approximations of functions by using wavelets |
| title_alt | О наилучших L2 -приближениях функций с помощью всплесков |
| title_full | On the best L2 -approximations of functions by using wavelets |
| title_fullStr | On the best L2 -approximations of functions by using wavelets |
| title_full_unstemmed | On the best L2 -approximations of functions by using wavelets |
| title_short | On the best L2 -approximations of functions by using wavelets |
| title_sort | on the best l2 -approximations of functions by using wavelets |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3229 |
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