Estimates for wavelet coefficients on some classes of functions
Let $ψ_m^D$ be orthogonal Daubechies wavelets that have $m$ zero moments and let $$W^k_{2, p} = \left\{f \in L_2(\mathbb{R}): ||(i \omega)^k \widehat{f}(\omega)||_p \leq 1\right\}, \;k \in \mathbb{N},$$. We prove that $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||...
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| Дата: | 2007 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3415 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509501193453568 |
|---|---|
| author | Babenko, V. F. Spector, S. A. Бабенко, В. Ф. Спектор, С. А. Бабенко, В. Ф. Спектор, С. А. |
| author_facet | Babenko, V. F. Spector, S. A. Бабенко, В. Ф. Спектор, С. А. Бабенко, В. Ф. Спектор, С. А. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:47Z |
| description | Let $ψ_m^D$ be orthogonal Daubechies wavelets that have $m$ zero moments and let $$W^k_{2, p} = \left\{f \in L_2(\mathbb{R}): ||(i \omega)^k \widehat{f}(\omega)||_p \leq 1\right\}, \;k \in \mathbb{N},$$. We prove that $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||_q}: f \in W^k_{2, p} \right\} = \frac{\frac{(2\pi)^{1/q-1/2}}{\pi^k}\left(\frac{1 - 2^{1-pk}}{pk -1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$ |
| first_indexed | 2026-03-24T02:42:06Z |
| format | Article |
| fulltext |
UDK 517.5
V. F. Babenko
(Dnepropetr. nac. un-t; Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
S. A. Spektor (Dnepropetr. nac. un-t)
OCENKY VEJVLET-KO∏FFYCYENTOV
NA NEKOTORÁX KLASSAX FUNKCYJ
Let ψ m
D be orthogonal Daubechies wavelets having m zero moments. Let also
W p
k
2, = f L i fk
p
∈ ≤{ }2 1( ) : ( ) ˆ( )R ω ω , k ∈N .
We prove that
lim sup
( , )
( )
: ,
m
m
D
m
D
q
p
kf
f W
→ ∞
′∈
ψ
ψ Ÿ 2 =
( )
( )
/ / /
/ /
2 1 2
1
2
1 1 2 1 1
1 1 2
π
π
π
p
k
pk p
q
pk
− −
−
−
−
.
Nexaj ψ m
D
— ortohonal\ni vejvlety Dobeßi, qki magt\ m nul\ovyx momentiv i
W p
k
2, = f L i fk
p
∈ ≤{ }2 1( ) : ( ) ˆ( )R ω ω , k ∈N .
Dovedeno, wo
lim sup
( , )
( )
: ,
m
m
D
m
D
q
p
kf
f W
→ ∞
′∈
ψ
ψ Ÿ 2 =
( )
( )
/ / /
/ /
2 1 2
1
2
1 1 2 1 1
1 1 2
π
π
π
p
k
pk p
q
pk
− −
−
−
−
.
Pust\ Lp = Lp ( R ) , 1 ≤ p ≤ ∞ , — prostranstvo yzmerym¥x funkcyj f : R → C
s koneçnoj normoj f p, hde
f p = f Lp ( )R
= f x dxp
p
( )
/
R
∫
1
, esly p < ∞ ,
y
f ∞ = f L∞ ( )R
= vrai sup ( )
x
f x
∈R
.
Dlq f Lp∈ ( )R y g Lq∈ ( )R , hde p q, [ ; ]∈ ∞1 , 1 1
p q
+ = 1, poloΩym ( f, g ) =
= f x g x dx( ) ( )
R∫ .
Budem rassmatryvat\ sledugwye klass¥ funkcyj f L∈ 2( )R . Dlq k ∈ N y
p ∈ ( 1, ∞ ) poloΩym
W p
k
2, = f L i fk
p
∈ ≤{ }2 1( ) : ( ) ˆ( )R ω ω ,
hde
ˆ( )f ω = 1
2π
ωf x e dxi x( ) −∫
R
— preobrazovanye Fur\e funkcyy f . Pry p = 2 poluçaem standartn¥e sobo-
levskye klass¥ W k
2 2, = f L f k∈ ≤{ }2 2
1( ) : ( )
R .
© V. F. BABENKO, S. A. SPEKTOR, 2007
1594 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
OCENKY VEJVLET-KO∏FFYCYENTOV NA NEKOTORÁX KLASSAX FUNKCYJ 1595
Dlq funkcyy ψ( ) ( )t L∈ 2 R y çysel j k, ∈Z poloΩym
ψ j k t, ( ) = 2 22j j t k/ ( )ψ − .
Esly systema funkcyj { }, ,ψ j k j k ∈Z
obrazuet ortonormyrovann¥j bazys pro-
stranstva L2( )R , t. e. lgbug funkcyg f L∈ 2( )R moΩno predstavyt\ v vyde
summ¥ sxodqwehosq v L2( )R rqda
f ( t ) =
i j
j jf t
∈ ∈
∑ ∑ ( )
Z Z
ψ ψν ν, ,, ( ) , (1)
to funkcyq ψ ( t ) naz¥vaetsq ortohonal\n¥m vejvletom.
Mnohye prymenenyq ortohonal\n¥x vejvletov bazyrugtsq na yssledovanyy
velyçyn¥ vejvlet-koπffycyentov v predstavlenyqx typa (1) v zavysymosty kak
ot svojstv vejvleta ψ ( t ) , tak y ot hladkosty funkcyy f .
PredpoloΩyv, çto vejvlet ψ ( t ) ymeet k nulev¥x momentov, yly, çto πkvy-
valentno, ˆ ( )ψ ω ymeet nul\ kratnosty k v nule, opredelym funkcyg k tψ ( )
sootnoßenyem
k tψ ω( ) ( )( )Ÿ
= ( ) ˆ ( )i kω ψ ω− .
PoloΩym
C p qκ ψ; , ( ) = sup
( )
ˆ
,f W qp
k
f
∈ ′2
ψ
ψ
,
hde ′p =
p
p − 1
. Tohda, kak lehko vydet\, C p qκ ψ; , ( ) moΩno predstavyt\ v vyde
C p qκ ψ; , ( ) =
( )
ˆ
k p
q
ψ
ψ
Ÿ
. (2)
Otmetym, çto C p qκ ψ; , ( ) — πto toçnaq konstanta v neravenstve
ψ νj f, ,( ) ≤ C f ip q
j k
p q
q
k
pκ ψ ψ ω ω; , ( ) ˆ ˆ( )( )2
1 1− − +
′
.
Pust\ m ∈ N . Fyl\tramy Dobeßy (sm., naprymer, [1], §E16) naz¥vagt tryho-
nometryçeskye polynom¥
Hm( )ω = 2 1 2
0
2 1
−
=
−
∑/ ( )h l em
il
l
m
ω , h lm( ) ∈R ,
udovletvorqgwye ravenstvam
Hm( )ω 2 = cos sin2
1
2
2 2
ω ω
−
m
mP ,
hde
P xm−1( ) =
m k
k
xk
k
m − +
=
−
∑
1
0
1
.
Funkcyq ϕm
D, preobrazovanye Fur\e kotoroj ymeet vyd
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1596 V. F. BABENKO, S. A. SPEKTOR
ϕ ωm
D( )Ÿ
( ) = 1
2
2
1π
ωHm
l
l
( )−
=
∞
∏ ,
qvlqetsq ortohonal\noj masßtabyrugwej funkcyej. Ortohonal\n¥m vejvle-
tom Dobeßy ψm
D
naz¥vaetsq funkcyq, obraz Fur\e kotoroj ymeet vyd
ψ ωm
D( )Ÿ
( ) = e H
i
m m
D
−
+
( )
ω
ω π ϕ ω2
2 2
Ÿ
.
Vejvlet ψm
D
obladaet sledugwymy svojstvamy (sm. [2], hl.E6, [1], §E16):
1) supp ψm
D = [ – ( m – 1 ) , m ] ;
2) ψm
D
ymeet m nulev¥x momentov;
3) suwestvuet λ > 0 takaq, çto ψm
D
∈ C mλ , hde
Cα = f f d: ˆ( )
R
∫ +( ) < ∞
ω ω ωα1 , α > 0. (3)
Krome toho (sm., naprymer, [3], §E5.5),
( ) ( )ψ ωm
D Ÿ 2
=
Hm m
Dω π ϕ ω
2 2
2 2
+
( )
Ÿ
=
= 1
2 2
2
2
1 2
1π
ω π ωH Hm m
l
l
+
− −
=
∞
∏ ( ) , (4)
pryçem
Hm( )ω 2 = 1
0
2 1−
∫ −c udum
m
ω
sin . (5)
Osnovn¥m rezul\tatom dannoj stat\y qvlqetsq sledugwaq teorema.
Teorema!1. Pust\ k ≥ 0 — fyksyrovannoe celoe çyslo. Tohda
lim ( ); ,
m
k p q m
DC
→∞
ψ =
( ) / / /
2 1 2
1
1 1 1 1
π
π
p q
k
pk p
pk
− −−
−
.
Pry p = q = 2 πta teorema dokazana v rabote [4].
Dlq dokazatel\stva πtoj teorem¥ nam ponadobytsq sledugwaq lemma, koto-
raq pry p = 2 takΩe dokazana v [4].
Pust\ Ψ̂ = 1
2 2 2π
χ χπ π π π− −[ ] [ ]+( ), , , hde χ1 — xarakterystyçeskaq funk-
cyq yntervalaEEI.
Lemma!1. Pust\ 1 < p < ∞ , k ≥ 0 — celoe çyslo y ( )ψn — posledova-
tel\nost\ funkcyj s kompaktn¥m nosytelem, pryçem:
i) dlq nekotoroho ε, ne zavysqweho ot n, 0 < ε < π ,
ω ε
ω ψ ω ω
<
−∫ pk
n
p
d( ) ( )Ÿ → 0 pry n → ∞ ;
ii)
( ) ˆψn p
Ÿ − Ψ → 0 pry n → ∞ .
Tohda
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
OCENKY VEJVLET-KO∏FFYCYENTOV NA NEKOTORÁX KLASSAX FUNKCYJ 1597
lim ( )
n
k n p→∞
ψ Ÿ =
( ) / / /
2 1 2
1
1 1 2 1 1
π
π
p
k
pk p
pk
− −−
−
. (6)
Dokazatel\stvo. Predstavym ( )k n p
ψ Ÿ
v vyde
( )k n p
ψ Ÿ =
( ) ( ) ( )
/
i dk p
n
p
p
ω ψ ω ω−∫
Ÿ
R
1
=
ω ψ ω ω−∫
pk
n
p
p
d( ) ( )
/
Ÿ
R
1
=
=
ω ω ω ψ ω ω− − −( )
∫ pk
n
p
p
dˆ ( ) ˆ ( ) ( ) ( )
/
Ψ Ψ Ÿ
R
1
.
Prymenqq neravenstvo Mynkovskoho, poluçaem
ω ψ ω ω ω ω− −( ) +
∫ pk
n
p
p
d( ) ( ) ˆ ( ) ˆ ( )
/
Ÿ Ψ Ψ
R
1
≤
≤
ω ω ω ω ψ ω ω ω− −∫ ∫
+ −
pk p
p
pk
n
p
p
d dˆ ( ) ( ) ( ) ˆ ( )
/ /
Ψ Ψ
R R
1 1
Ÿ = I I1 2+ .
S druhoj storon¥,
ω ψ ω ω ω ω− −( ) +
∫ pk
n
p
p
d( ) ( ) ˆ ( ) ˆ ( )
/
Ÿ Ψ Ψ
R
1
≥
≥
ω ω ω ω ψ ω ω ω− −∫ ∫
− −
pk p
p
pk
n
p
p
d dˆ ( ) ( ) ( ) ˆ ( )
/ /
Ψ Ψ
R R
1 1
Ÿ = I I1 2− .
Dlq I p
1 ymeem
I p
1 = 1
2
1
2
2
2
π
ω ω
π
ω ω
π
π
π
π
+
−
−
− −∫ ∫
p
pk
p
pkd d =
( ) / / /
2 1 2
1
1 1 2 1 1
π
π
p
k
pk p
pk
− −−
−
.
PokaΩem, çto
I p
2 = ω ψ ω ω ω− −∫ pk
n
p
d( ) ( ) ˆ ( )Ÿ Ψ
R
→ 0 pry n → ∞ .
Zafyksyruem ε ∈ ( 0; π ) . Razbyvaq ynterval yntehryrovanyq na dve çasty,
ymeem
I p
2 =
ω ψ ω ω ω ω ψ ω ω ω
ω ε ω ε
−
<
−
>
− + −∫ ∫pk
n
p pk
n
p
d d( ) ( ) ˆ ( ) ( ) ( ) ˆ ( )Ÿ ŸΨ Ψ =
= I I11 12+ .
Rassmotrym I11. Uçyt¥vaq uslovye i) y to, çto
ˆ ( )Ψ ω = 0 dlq ω ∈ ( – π;
π ) , poluçaem, çto I11 → 0 pry n → ∞ . Krome toho, v sylu uslovyq ii)
I12 ≤
ε ψ ω ω ω
ω ε
−
>
−∫kp
n
p
d( ) ( ) ˆ ( )Ÿ Ψ → 0 pry n → ∞ .
Takym obrazom,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1598 V. F. BABENKO, S. A. SPEKTOR
( )k n p
ψ Ÿ → I
pk
p
p
k
pk p
1
1 1 2 1 1
2 1 2
1
= −
−
− −( ) / / /
π
π
pry n → ∞ .
Lemma dokazana.
Otmetym takΩe çastn¥j sluçaj lemm¥. Pry k = 0 poluçaem
lim ( )
n
n q→∞
ψ Ÿ = ( ) / /2 1 1 2π q − . (7)
Dokazatel\stvo teorem¥!1. Neobxodymo proveryt\ v¥polnenye uslovyj
i) y ii) lemm¥EE1 dlq ortohonal\n¥x vejvletov Dobeßy. Pry πtom budem ys-
pol\zovat\ sootnoßenyq (4), (5) y
cm =
0
2 1
1π
ω ω∫ −
−
sin m d =
Γ
Γ
( )
( )
/m
m
+ 1 2
π
∼ m
π
. (8)
Dlq dokazatel\stva uslovyq i) v¥berem 0 < ε < 1, a takΩe otmetym, çto
Hm( )ω ≤ 1 dlq lgboho ω. V rezul\tate poluçym
ω ψ ω ω
ω ε
−
<
∫ pk
m
D p
d( ) ( )Ÿ ≤ 1
2 2
2
π
ω ω π ω
ω ε
+
−
<
∫
p
pk
m
p
H d
/
≤
≤
c
t dt dm
p
pk m
p
2
2
0
2
2 1
2
π
ω ω
ω
ω ε
− −
<
∫∫
/ / /
sin .
Krome toho, tak kak
ω ω
2 2
1
−
sin ≤ 1, budem ymet\
c
t dt dm
p
pk m
p
2
2
0
2
2 1
2
π
ω ω
ω
ω ε
− −
<
∫∫
/ / /
sin =
= 2
2
1 2
2
0
2
2 1
2
− +( )
−
−
<
∫∫p k m
p pk
m
p
c
t dt d/
/ / /
sin
π
ω ω
ω
ω ε
≤
≤ 2
2 2 2
1 2
2
2 1
2
− +( )
−
−
<
∫p k m
p pk
m
pc
d/
/ /
sin
π
ω ω ω ω
ω ε
≤
≤ 2
2 2
1 2
2 2
2 1
2
− +( )
− +
−
<
∫p k m
p pk p
m
pc
d/
/ / /
sin
π
ω ω ω
ω ε
≤
≤ 2
2
1 2
2
2 2− +( ) −
<
∫p k m
p
mp pkc
d/
/
/sin
π
ω ω
ω ε
.
Pry m > k
2
2
1 2
2
2 2− +( ) −
<
∫p k m
p
mp pkc
d/
/
/sin
π
ω ω
ω ε
≤ 2 2
2
1 2
2
− +( ) −
p k m
p
pm pkc/
/
sin
π
ε ε ≤
≤ 2
2
1 2 1
2 1
− +( )+
− +
p k m
p pm pkc/
/
π
ε .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
OCENKY VEJVLET-KO∏FFYCYENTOV NA NEKOTORÁX KLASSAX FUNKCYJ 1599
Poskol\ku cm ∼ m
π
, poslednee v¥raΩenye stremytsq k nulg pry m → ∞ .
Sootnoßenye i) dokazano.
Dlq dokazatel\stva uslovyq ii) poloΩym I = [ – 2 π; 2 π ] y
Iδ = [ , ) ( , ) ( , ) ( , ]− − + − − − + − + −2 2 2 2π π δ π δ π δ π δ π δ π δ π∪ ∪ ∪ .
DokaΩem, çto
( ) ( ) ˆ ( )
\
/
ψ ω ω ω
δ
m
D p
I I
p
dŸ −
∫ Ψ
1
→ 0 pry m → ∞ .
Ymeem
( ) ( ) ˆ ( )
\
/
ψ ω ω ω
δ
m
D p
I I
p
dŸ −
∫ Ψ
1
≤
( ) ( )
\
/
ψ ω
π
ω π ω
δ
m
D
m
p
I I
p
H dŸ − +
∫ 1
2 2
1
+
+ ˆ ( )
\
/
Ψ ω
π
ω π ω
δ
− +
∫ 1
2 2
1
H dm
p
I I
p
. (9)
Dlq fyksyrovannoho δ posledovatel\nost\
1
2 2π
ω πHm +
ravnomerno
sxodytsq k
ˆ ( )Ψ ω v I I\ δ pry m → ∞ , poπtomu vtoroe slahaemoe v pravoj
çasty neravenstva (9) stremytsq k nulg pry m → ∞ .
Pervoe slahaemoe v pravoj çasty neravenstva (9) moΩno perepysat\ v vyde
1
2 2
1 2
1
1
1
π
ω π ω ω
δ
H H dm
p
I I l
m
l
p p
+
−
∫ ∏
≥
− −
\
/
( ) . (10)
Uçyt¥vaq tot fakt, çto Hm
pω π
2
+
≤ 1, a takΩe ustanovlenn¥e v rabote
[4] dlq ω δ∈ I I\ ocenky
Hm
ω
4
≥ 1
2 2 4
2 1 1 2
− −
{ }
−
cm
mπ π δsin
/
,
Hm
ω
8
≥ 1
4
2 1 2
−
cm
mπ
/
y
l
m
l p
H
≥
− −∏
1
32( )ω ≥ 1 2 2
1
2 1 2 2−( )− − −m m( ) ,
yntehral (10) pry vsex dostatoçno bol\ßyx m moΩno ocenyt\ sledugwym ob-
razom:
1
2 2
1 2
1
1
1
π
ω π ω ω
δ
H H dm
p
I I l
m
l
p p
+
−
∫ ∏
≥
− −
\
/
( ) ≤
≤ 1
2
4 1 1
2 2 4
1
4
1 2
2 1 1 2 2 1 2
2
1
2 1 2 2
π
π π π δ π( ) sin
/ /
( )
p
m
m
m
m
mc c m− − −
{ }
−
−( )
−
− − − .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1600 V. F. BABENKO, S. A. SPEKTOR
Qsno, çto pravaq çast\ poluçennoho neravenstva stremytsq k nulg pry m → ∞ .
Poskol\ku Iδ = 6 δ , to
( ) ( ) ˆ ( )ψ ω ω ω
δ
m
D p
I
dŸ −∫ Ψ ≤ 1
2
2
π δ
p
I
/
≤ 1
2
6
2
π
δ
p/
. (11)
Teper\ ubedymsq, çto dlq vsex p > 1
( ) ( )ψ ω ω
ω π
m
D p
dŸ
≥
∫
2
→ 0 pry m → ∞ .
Yz yzvestn¥x rezul\tatov o rehulqrnosty vejvletov Dobeßy (sm., naprymer, [5],
§E2.2.4) sleduet, çto najdutsq poloΩytel\n¥e konstant¥ C y C̃ takye, çto
dlq vsex ω takyx, çto ω > 2 π , v¥polnqetsq neravenstvo
( ) ( )ψ ωm
D Ÿ ≤ ˜ logC C mω − .
Tohda
( ) ( )ψ ω ω
ω π
m
D p
dŸ
>
∫
2
≤ ˜ ( ) logC dC p mω ω
ω π
−
>
∫
2
= ˜ ( ) ( log )C dC p mω ω
ω π
− −
>
∫ 2
2
≤
≤ ( ) ( )( log )2 2 2
2
π ω ω
ω π
− − −
>
∫C p m d .
Poslednee v¥raΩenye stremytsq k nulg pry m → + ∞ .
Takym obrazom, predpoloΩenyq lemm¥EE1 dlq ortohonal\n¥x vejvletov
Dobeßy v¥polnqgtsq.
Yspol\zuq (2), (6) y (7), ymeem
lim ( ); ,
m
k p q m
DC
→∞
ψ =
lim
( )
( )m
k m
D
p
m
D
q
→∞
′
ψ
ψ
Ÿ
Ÿ
=
( )
( )
/ / /
/ /
2 1 2
1
2
1 1 2 1 1
1 1 2
π
π
π
p
k
pk p
q
pk
− −
−
−
−
,
lim ( ); ,
m
k p q m
DC
→∞
ψ =
( ) / / /
2 1 2
1
1 1 1 1
π
π
p q
k
pk p
pk
− −−
−
.
Teorema dokazana.
1. Novykov Y. Q., Steçkyn S. B. Osnovn¥e teoryy vspleskov // Uspexy mat. nauk. – 1998. – 53,
# 6. – S.E53 – 128.
2. Dobeßy Y. Desqt\ lekcyj po vejvletam. – YΩevsk: NYC „Rehulqrnaq y xaotyçeskaq
dynamyka”, 2001. – 463 s.
3. Strang G., Nguyen T. Wavelets and filter banks. – Wellesley: Cambridge Press, 1996. – 520 p.
4. Ehrich S. On the estimation of wavelet coefficients // Adv. Comput. Math. – 2000. – 13. –
P. 105 – 129.
5. Louis A. K., Maab P., Rieder A. Wavelets theory and applications. – Chichester etc.: John Wiley &
Sons Ltd, 1997. – 323 p.
Poluçeno 19.06.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
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| id | umjimathkievua-article-3415 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:42:06Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8e/7d7f67dcbc6edbe5706452b8f6906f8e.pdf |
| spelling | umjimathkievua-article-34152020-03-18T19:53:47Z Estimates for wavelet coefficients on some classes of functions Оценки вейвлет-коэффициентов на некоторых классах функций Babenko, V. F. Spector, S. A. Бабенко, В. Ф. Спектор, С. А. Бабенко, В. Ф. Спектор, С. А. Let $ψ_m^D$ be orthogonal Daubechies wavelets that have $m$ zero moments and let $$W^k_{2, p} = \left\{f \in L_2(\mathbb{R}): ||(i \omega)^k \widehat{f}(\omega)||_p \leq 1\right\}, \;k \in \mathbb{N},$$. We prove that $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||_q}: f \in W^k_{2, p} \right\} = \frac{\frac{(2\pi)^{1/q-1/2}}{\pi^k}\left(\frac{1 - 2^{1-pk}}{pk -1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$ Нехай $ψ_m^D$ — ортогональні вейвлети Добеші, які мають $m$ нульових моментів i $$W^k_{2, p} = \left\{f \in L_2(\mathbb{R}): ||(i \omega)^k \widehat{f}(\omega)||_p \leq 1\right\}, \;k \in \mathbb{N},$$. Доведено, що $$\lim_{m\rightarrow\infty}\sup\left\{\frac{|\psi^D_m, f|}{||(\psi^D_m)^{\wedge}||_q}: f \in W^k_{2, p} \right\} = \frac{\frac{(2\pi)^{1/q-1/2}}{\pi^k}\left(\frac{1 - 2^{1-pk}}{pk -1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}.$$ Institute of Mathematics, NAS of Ukraine 2007-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3415 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 12 (2007); 1594–1600 Український математичний журнал; Том 59 № 12 (2007); 1594–1600 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3415/3573 https://umj.imath.kiev.ua/index.php/umj/article/view/3415/3574 Copyright (c) 2007 Babenko V. F.; Spector S. A. |
| spellingShingle | Babenko, V. F. Spector, S. A. Бабенко, В. Ф. Спектор, С. А. Бабенко, В. Ф. Спектор, С. А. Estimates for wavelet coefficients on some classes of functions |
| title | Estimates for wavelet coefficients on some classes of functions |
| title_alt | Оценки вейвлет-коэффициентов на некоторых классах функций |
| title_full | Estimates for wavelet coefficients on some classes of functions |
| title_fullStr | Estimates for wavelet coefficients on some classes of functions |
| title_full_unstemmed | Estimates for wavelet coefficients on some classes of functions |
| title_short | Estimates for wavelet coefficients on some classes of functions |
| title_sort | estimates for wavelet coefficients on some classes of functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3415 |
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