Approximation of holomorphic functions by Taylor-Abel-Poisson means
We investigate approximations of functions $f$ holomorphic in the unit disk by means $A_{\rho, r}(f)$ for $\rho \rightarrow 1_-$. In terms of an error of the approximation by these means, the constructive characteristic of classes of holomorphic functions $H_p^r \text{\;Lip\,}\alpha$ is given. The...
Saved in:
| Date: | 2007 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3385 |
| 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_ | 1860509465061621760 |
|---|---|
| author | Savchuk, V. V. Савчук, В. В. |
| author_facet | Savchuk, V. V. Савчук, В. В. |
| author_sort | Savchuk, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:51Z |
| description | We investigate approximations of functions $f$ holomorphic in the unit disk by means $A_{\rho, r}(f)$ for $\rho \rightarrow 1_-$.
In terms of an error of the approximation by these means, the constructive characteristic of classes of holomorphic functions $H_p^r \text{\;Lip\,}\alpha$ is given.
The problem of the saturation of $A_{\rho, r}(f)$ in the Hardy space $H_p$ is solved.
|
| first_indexed | 2026-03-24T02:41:32Z |
| format | Article |
| fulltext |
UDK 517.5
V. V. Savçuk (In-t matematyky NAN Ukra]ny, Ky]v)
NABLYÛENNQ HOLOMORFNYX FUNKCIJ
SEREDNIMY TEJLORA – ABELQ – PUASSONA
We investigate approximations of functions f holomorphic in the unit disk by means A frρ, ( ) for ρ →
→ 1 – . In terms of an error of the approximation by these means, the constructive characteristic of
classes of holomorphic functions Hp
r Lipα is given. The problem of the saturation of A frρ, ( ) in the
Hardy space Hp is solved.
Yssledugtsq pryblyΩenyq holomorfn¥x v edynyçnom kruhe funkcyj f srednymy A frρ, ( )
pry ρ → 1 – . V termynax pohreßnosty pryblyΩenyq πtymy srednymy pryvedena konstruktyv-
naq xarakterystyka klassov holomorfn¥x funkcyj Hp
r Lipα . Reßena zadaça o nas¥wenyy
A frρ, ( ) v prostranstve Hardy Hp .
1. Postanovka zadaç ta osnovni rezul\taty. Nexaj D = {z ∈ C : z < 1}, T 3 =
= {z ∈ C: z = 1} i Hol ( D ) — mnoΩyna usix funkcij, holomorfnyx u kruzi D.
Prostir Hardi Hp , p > 0, — ce mnoΩyna usix funkcij f ∈ Hol ( D ), dlq qkyx
f M fHp p: sup ( , )= < ∞
< <0 1ρ
ρ ,
de
M f f w d wp
p
p
( , ) : ) ( )
/
ρ ρ σ= (
∫
T
1
, 0 < p < ∞,
M f f z
z
∞ =
=( , ) : max ( )ρ
ρ
,
i σ — normovana mira Lebeha na koli T.
Vidomo, wo qkwo funkciq f naleΩyt\ Hp , 1 ≤ p ≤ ∞, to na koli T isnugt\
]] kutovi hranyçni znaçennq (za qkymy zalyßa[mo poznaçennq f ), qki naleΩat\
prostorovi Lp = Lp ( T ), pryçomu
f f f dH L
p
p
p p
= =
∫:
/
T
σ
1
.
ZvaΩagçy na ce, domovymosq dali zavΩdy pid f H p
rozumity normu funk-
ci] f ∈3Hp , a pid f Lp
— normu ]] hranyçnyx znaçen\ v Lp .
Nexaj ρ ∈ [ 0, 1 ) , r ∈ N. Rozhlqnemo linijnyj operator A ρ, r , vyznaçenyj na
Hol ( D ) , qkyj di[ za pravylom
A f z
f z
k
zr
k
k
r
k k
ρ
ρ ρ,
( )
( )( ) :
( )
!
( )= −
=
−
∑
0
1
1 , z ∈ D, f ∈3Hol ( D ) .
Znaçennq c\oho operatora na funkci] f budemo nazyvaty serednimy Tejlora –
Abelq – Puassona funkci] f . Vybir tako] nazvy poqsng[t\sq tym, wo, z odnoho
boku, operator A ρ, r budu[t\sq qk mnohoçlen Tejlora stepenq r – 1 funkci] f
v toçci ρ z , a z inßoho — qk linijna kombinaciq serednix Abelq – Puassona
funkci] f ta ]] poxidnyx do r-ho porqdku vklgçno. Zokrema, pry ρ = 0 opera-
© V. V. SAVÇUK, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9 1253
1254 V. V. SAVÇUK
tor A 0, r stavyt\ u vidpovidnist\ funkci] f ∈ Hol ( D ) çastynnu sumu porqdku
r – 1 ]] rqdu Tejlora, a pry r = 1 — zvyçajni seredni Abelq – Puassona rqdu
Tejlora.
Rozhlqnemo funkcional\ni klasy Hp
r Lipα , 0 < α ≤ 1, i KHp
r
, r ∈ Z + , qki
oznaçagt\sq takym çynom:
H f f e f O tp
r
h t
r ih r
Lp
Lip Holα α: ( ): sup ( ) ( ) ( )( ) ( )= ∈ ⋅ − ⋅ =
< ≤
D
0
, 1 ≤ p < ∞,
i
KH f f Kp
r r
H p
: ( ): ( )= ∈ ≤
Hol D , K > 0, 1 ≤ p ≤ ∞.
Pry p = ∞ pid Hp
r Lipα rozumi[mo klas holomorfnyx v D i neperervnyx v
D funkcij f, dlq qkyx max ( ) ( )( ) ( )
z
r ih rf e z f z∈ −T = O( hα
) . Zhidno z teore-
mog Hardi – Littlvuda [1] (teorema 48) (dyv. takoΩ [2, s. 78] ) klas Hp
r Lip1, 1 ≤
≤ p < ∞, zbiha[t\sq z mnoΩynog funkcij f ∈ Hol ( D ), dlq qkyx f Hr
p
( )+ ∈1
,
tobto Hp
r Lip1 = ∪ K p
rKH>
+
0
1
.
U danij roboti navedeno konstruktyvnu xarakterystyku klasiv Hp
r Lipα v
terminax operatoriv A ρ, r . TakoΩ rozv’qzano zadaçu pro nasyçennq operatora
A ρ, r qk linijnoho metodu pidsumovuvannq rqdiv Tejlora.
Nahada[mo (dyv., napryklad, [3], [4], hl. 2, i [5, c. 434]), wo metod pidsumovu-
vannq, porodΩenyj operatorom A ρ, r , [ nasyçenym u prostori Hp , qkwo isnu[
dodatna funkciq ϕ, vyznaçena na vidrizku [ 0, 1 ) , monotonna spadna do nulq i
taka, wo koΩna funkciq f ∈3Hp , dlq qko]
f A f or Hp
− =ρ ϕ ρ, ( ) ( ( )) , ρ → 1,
[ invariantnym elementom operatora A ρ, r (tobto A ρ, r
( f ) = f ) , i qkwo mnoΩyna
Φ( ) : : ( ) ( ( )),, ,A f H f A f Or H p r Hp pρ ρ ϕ ρ ρ= ∈ − = →{ }1
mistyt\ prynajmni odyn neinvariantnyj element. Pry c\omu funkciq ϕ nazyva-
[t\sq porqdkom nasyçennq, a mnoΩyna Φ( ),A r Hpρ — klasom nasyçennq.
Osnovnym rezul\tatom roboty [ nastupna teorema.
Teorema 1. Nexaj r ∈ Z +, 1 ≤ p ≤ ∞ i 0 < α ≤ 1. Holomorfna v D funk-
ciq f naleΩyt\ klasovi Hp
r Lipα todi i til\ky todi, koly
f A f Or H
r
p
− = −( )+
+
ρ
αρ, ( ) ( )1 1 , ρ → 1 – . (1)
ZauvaΩennq 1. Qkwo f Hp
r∈ , to z totoΩnosti
f z A f zr( ) ( )( ),− +ρ 1 = f z f z
f z
k
z
k
k
r
k k
k
k( ) ( )
( )
!
( )
( )( )
− − − −
−=
∑ρ ρ ρ ρ
ρ1
1
1
1
=
= f z f z f z f z z z f z z f z
k
k k
k
r
k k k k k k
k
k( ) ( ) ( ) ( ( ) ( )
( )
!( )
( ) ( ) ( ) ( )− − − )( ) − −( )
−
−=
∑ρ ρ ρ ρ ρ
ρ1
1
1
za teoremog Fatu (dyv., napryklad, [2, c. 5]) pro zbiΩnist\ f k( )( )ρ⋅ → f k( )( )⋅
vyplyva[, wo majΩe skriz\ na T A frρ, ( )+1 → f, ρ → 1.
ZauvaΩennq 2. Qk bude vydno z dovedennq teoremy 1 (dyv. spivvidnoßen-
nq3(8)), dlq velyçyny
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
NABLYÛENNQ HOLOMORFNYX FUNKCIJ SEREDNIMY … 1255
A KH H f A f f KHr p
r
p r H p
r
p
ρ ρ, ,( , ) : sup ( ) := − ∈
, K > 0,
vykonugt\sq rivnosti
A KH Hr p
r
pρ, ( , ) = f z A f zr* , *
( ) ( )( )− ρ =
K
r
r
!
( )1 − ρ ∀ ∈z D ,
v qkyx
f z Q z e K
r
zr
i
r
∗ −= +( ) : ( )
!1
α
,
α — dovil\ne dijsne çyslo i Qr−1 — bud\-qkyj alhebra]çnyj mnohoçlen stepe-
nq r – 1.
U vypadku, koly r = 0, tverdΩennq teoremy 1 nabyra[ vyhlqdu
f H f f Op Hp
∈ ⇔ ⋅ − ⋅ = −( )Lipα ρ ρ α( ) ( ) ( )1 , ρ → 1 – . (2)
Uperße tverdΩennq (2) dovedeno pry p = ∞ v [1 ]. Dlq zahal\noho vypadku,
koly p ≥ 1, ce tverdΩennq ta bibliohrafig moΩna znajty v [5, c. 111].
Rozv’qzok zadaçi pro nasyçennq metodu pidsumovuvannq rqdiv Tejlora, po-
rodΩenoho operatorom A ρ, r
, mistyt\sq v nastupnomu tverdΩenni.
Teorema 2. Nexaj 1 ≤ p ≤ ∞ i r ∈ N. Operator A rρ, porodΩu[ linijnyj
metod pidsumovuvannq rqdiv Tejlora, qkyj [ nasyçennym v H p z porqdkom nasy-
çennq ( 1 – ρ )
r
ta klasom nasyçennq Hp
r−1 1Lip .
2. Dovedennq. Navedemo spoçatku vlastyvosti operatora A ρ, r , qki docil\no
sformulgvaty u vyhlqdi okremyx tverdΩen\.
Lema 1. Nexaj r ∈ N, 1 ≤ p ≤ ∞ i 0 ≤ ρ < 1. Qkwo funkciq f Hp
r∈ ,
to dlq bud\-qkoho z ∈ D i majΩe koΩnoho z ∈ T
f z A f z z
r
f z dr
r
r r( ) ( )( )
( )!
( )( ),
( )− =
−
−∫ −
ρ
ρ
ζ ζ ζ
1
1
1
1
. (3)
Pry r = 2 rivnist\ (3) dovedeno v [6] (dyv. takoΩ [7, c. 421]) za umovy
∂
∂
=
−
2
2
1
1θ
ρ
ρ
θRe ( )f e Oi
,
qka, oçevydno, [ slabßog, niΩ umova ′′ ∈f Hp.
Dovedennq rivnosti (3) u vypadku, koly z ∈ D, [ elementarnym; vono ©runtu-
[t\sq na formuli Tejlora. Tomu vvaΩa[mo, wo z ∈ T, a pid f rozumi[mo hra-
nyçni znaçennq funkci].
Nexaj � — mnoΩyna toçok z na T , v qkyx dlq koΩnoho k = 0, … , r
isnugt\ radial\ni hranyçni znaçennq f zk( )( ) = lim ( )( )
R
kf Rz→ −1 .
Zafiksu[mo z ∈ � i rozhlqnemo na vidrizku [ρ, 1] funkcig g( )ζ : = f z( )ζ .
Zhidno z teoremog Pryvalova – Rissa (dyv., napryklad, [2, c. 42]) poxidni f k( )
dlq koΩnoho k = 0, … , r – 1 [ neperervnymy v D i absolgtno neperervnymy
na T. Dali, oskil\ky f Hr
p
( ) ∈ , to zhidno z teoremog Fej[ra – Rissa (dyv., na-
pryklad, [2, c. 46])
g d f z d fr p r p r
H
p
p
( ) ( ) ( )( ) ( )ζ ζ ζ ζ π
0
1
0
1
∫ ∫= ≤ < ∞ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1256 V. V. SAVÇUK
OtΩe, funkciq g [ neperervnog razom zi svo]my poxidnymy do porqdku r – 1
vklgçno na vidrizku [ρ, 1], a ]] poxidna r-ho porqdku [ sumovnog na [ρ, 1) . Ci
fakty dozvolqgt\ zastosuvaty do funkci] g formulu Tejlora iz zalyßkovym
çlenom v intehral\nij formi
g
g
k r
g d
k
k
r
k r r( )
( )
!
( )
( )!
( )( )
( )
( )1 1 1
1
1
0
1 1
1= − +
−
−
=
−
−∑ ∫ρ ρ ζ ζ ζ
ρ
.
Povertagçys\ do funkci] f, z uraxuvannqm toho, wo g k( )( )ζ = z f zk k( )( )ζ i
σ ( � ) = 1, zvidsy robymo vysnovok, wo rivnist\ (3) vykonu[t\sq v koΩnij toçci
�, a otΩe, majΩe skriz\ na T.
U nastupnyx dvox tverdΩennqx jdet\sq pro komutatyvnist\ operatora A ρ, r
pry riznyx parametrax ρ1 i ρ2 ta pro vyhlqd nerivnosti typu Bernßtejna dlq
poxidnyx vywyx porqdkiv A r
r
ρ,
( )
.
Lema 2. Nexaj r ∈ N. Dlq bud\-qko] funkci] f ∈ Hol ( D ) i bud\-qkyx ρ1,
ρ2 ∈ [ 0, 1 )
A A f A A fr r r rρ ρ ρ ρ1 2 2 1, , , ,( ) ( )( ) = ( ).
Dovedennq. Nexaj funkciq f ∈ Hol ( D ) i
f z f z( ) ˆ=
=
∞
∑ ν
ν
ν 0
,
ˆ :
( )
!
( )
f
f
ν
ν
ν
= 0
,
— ]] rqd Tejlora. Zafiksu[mo ρ1, ρ2 ∈ [ 0, 2 ) i poklademo
ϕ ρ ν
ρ
ν
ν
ν ν
k
k k
k
kz
z f z
k k
f z( ) :
( )
!
ˆ
( )
= =
=
∞
−∑1
1 , z ∈ D,
de
ν ν
νk k k
=
−
:
!
!( )!
.
Lehko baçyty, wo dlq bud\-qkoho z ∈ D i bud\-qkoho m ∈ Z +
z z
m k m
f z
m
k
m
k m
k mϕ ρ ν ν
ρ ρ
ν
ν
ν ν ν
( )
max( , )
( )
!
ˆ2
1 2=
=
∞
− −∑ .
Na osnovi ci[] formuly ta linijnosti operatora A ρ, r dlq bud\-qkoho z ∈ D
ma[mo rivnist\
A A f zr rρ ρ2 1, , ( ) ( )( ) = A
f
k
zr
k
k
r
k k
ρ
ρ ρ
2
1
0
1
11,
( )( )
!
( ) ( ) ( )
⋅ ⋅ −
=
−
∑ =
= A zr k
k
k
r
ρ ϕ ρ
2
1 1
0
1
, ( ) ( ) ( )⋅ −
=
−
∑ =
k
r
r k
kA z
=
−
∑ −
0
1
12
1ρ ϕ ρ, ( )( )( ) =
=
z z
m
m
k
m
m
r
k
r
m kϕ ρ
ρ ρ
( )( )
!
( ) ( )2
0
1
0
1
2 11 1
=
−
=
−
∑∑ − − =
=
ν ν
ρ ρ ρ ρ
ν
ν
ν ν ν
k m
f z
k mm
r
k
r
k m m m
− −
=
∞
=
−
=
−
− −∑∑∑
max( , )
ˆ ( ) ( )
0
1
0
1
1 2 1 21 1 . (4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
NABLYÛENNQ HOLOMORFNYX FUNKCIJ SEREDNIMY … 1257
Analohiçno moΩna dovesty, wo i dlq A A f zr rρ ρ1 2, , ( ) ( )( ) vykonu[t\sq taka Ω riv-
nist\, tomu lema 2 [ pravyl\nog.
Lema 3. Nexaj r ∈ N, 1 ≤ p ≤ ∞ i ρ ∈ [0, 1). Dlq bud\-qko] funkci] r
∈ Hp
d
dz
A f C
fr
r r
H
r
H
r
p
p
ρ ρ, ( )( )
( )
⋅ ≤
−1
,
de Cr — stala, wo zaleΩyt\ til\ky vid r.
Dovedennq. Dlq k = 0, … , r – 1 ma[mo rivnist\
d
dz
z f z
r d
dz
z
d
dz
f z
r
r
k k
r r
r
k k( ) ( )( ) ( )ρ
ν
ρ
ν
ν
ν
ν
ν( ) =
=
−
−∑
0
=
=
r k
k r
z f z
r k
r
k r k
ν ν
ρ ρ
ν
ν ν ν
− += −
− + +∑ !
( )!
( )( )
.
OtΩe,
d
dz
A f d
dz
z f z
k
r
r r
H k
r r
r
k k
H
k
p p
ρ ρ ρ
,
( )( )( ) ( )
( )
!
⋅ ≤ ( ) −
=
−
∑
0
1 1
≤
≤
r
k r
f
r k
r
k
r
k
H
k
pν ν
ρ ρ
ν
ν
− +
⋅ −
= −=
−
+∑∑ 1
1
0
1
( )!
( ) ( )( )
. (5)
Ocinku f k
Hp
( )( )+ ⋅ν ρ lehko otrymaty z formuly Koßi
f k
Hp
( )( )+ ⋅ν ρ ≤ ( )!
( )
k f
d w
w zH kp
+
− + +∫ν σ
ρ ν1 1
T
≤
2
1
( )!
( )
k f H
k
p
+
− +
ν
ρ ν .
ProdovΩugçy dali ocinku (5), z uraxuvannqm poperedn\o] nerivnosti ostatoçno
oderΩu[mo
d
dz
A f
r
r r
H p
ρ, ( )( )⋅ ≤ f
r k
k rH
r k
r
k
r
p
2
1
10
1
ν
ν
ν ρν
ν
+
− + −= −=
−
∑∑ ( )!
( )!( )
≤ C
f
r
H
r
p
( )1 − ρ
,
de
C
r k
k rr
r k
r
k
r
=
+
− += −=
−
∑∑2
0
1
ν
ν
νν
( )!
( )!
.
U nastupnomu tverdΩenni jdet\sq pro zobraΩennq operatora A ρ, r , u
vyhlqdi linijnoho metodu pidsumovuvannq rqdiv Tejlora i, vidtak, u vyhlqdi
zhortky z qdrom metodu.
Lema 4. Nexaj r ∈ N, 0 ≤ ρ < 1 i funkciq f ∈ Hol ( D ). Todi
A f zrρ, ( )( ) =
ˆ ( ) ˆ
,f z f z
r
r
r
ν
ν
ν
ν
ν
ν
νλ ρ
=
−
=
∞
∑ ∑+
0
1
∀ ∈z D , (6)
de
λ ρν, ( )r :3=
ν
ρ ρν
kk
r
k k
−
=
−
−∑
0
1
1( ) =
( )
!
1
0
1 −
=
−
∑ ρ
ρ
ρν
k
k
r k
kk
d
d
, ν = ∞r, , r ∈ N.
Zokrema, qkwo f ∈ H1, to dlq bud\-qkoho z ∈ D
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1258 V. V. SAVÇUK
A f z f w K w z d wr rρ ρ σ, ,( )( ) ( ) ( , ) ( )= ∫
T
, (7)
de
K w z z w z wr
r
r
r
ρ
ν ν
ν
ν
ν
ν νλ ρ, ,( , ) : ( )= +
=
−
=
∞
∑ ∑
0
1
.
Dovedennq. Zrozumilo, wo dostatn\o dovesty rivnist\ (7), z qko] lehko ot-
rymaty (6).
OtΩe, z formuly Koßi
f z f w z w d w( ) ( ) ( )=
∫ ∑
=
∞
T
ν ν
ν
σ
0
vyplyva[
A f zrρ, ( )( ) = f w
k
w z d w
k
r k
k
r
r
z
k k( )
!
( ) ( )
T
∫ ∑ ∑∑
=
−
=
∞
=
−
=
+
−1
1
0
1
0
1∂
∂ζ
ζ ρ σ
νν
ν ν
ζ ρ
.
Cq rivnist\ z uraxuvannqm toho, wo za formulog Tejlora
1 1
0
1
0
1
0
1
k
w z z w
k
r k
k
r
z
k k
r
!
( )
=
−
=
−
= =
−
∑ ∑ ∑
− =∂
∂ζ
ζ ρν ν
ν ζ ρ
ν ν
ν
,
i dovodyt\ (7).
Perejdemo teper do dovedennq teorem.
Dovedennq teoremy 1. Neobxidnist\. Za formulog (3) dlq bud\-qkoho
R ∈ ( 0, 1 ), ma[mo rivnist\
M R f A f f R A f Rp r r Lp
, ( ), ,−( ) = ( ⋅) − ( )( ⋅)+ +ρ ρ1 1 = 1 11
1
r
f R dr r
Lp
!
( )( )( )+ ⋅ −∫ ζ ζ ζ
ρ
.
Dali, ocinggçy pravu çastynu ci[] rivnosti za intehral\nog nerivnistg Minkov-
s\koho, z uraxuvannqm toho, wo funkciq M fp
r( )⋅ +, ( )1
[ nespadnog, oderΩu[mo
M R f A fp r, ( ),−( )+ρ 1 ≤
1 1
1
1
r
f R dr
L
r
p!
( ) ( )( )
ρ
ζ ζ ζ∫ + ⋅ − ≤
≤
1 11
1
r
M f dp
r r
!
, ( )( )ζ ζ ζ
ρ
+( ) −∫ . (8)
Za teoremog Hardi – Littlvuda [8] (dyv., napryklad, [2, c. 78]), qkwo f ∈
∈ Hp
r Lipα , to
C M ff p
r: sup , ( )= −( ) ( ) < ∞
∈
− +
ζ
αζ ζ
D
1 1 1
.
Z uraxuvannqm c\oho faktu z ocinky (8) oderΩu[mo spivvidnoßennq
f A fr Hp
− +ρ, ( )1 = sup , ( ),
0 1
1
< <
+−( )
R
p rM R f A fρ ≤
1 11
1
r
M f dp
r r
!
( , )( )( )ζ ζ ζ
ρ
+∫ − =
=
1 1 11
1
1 1
r
M f dp
r r
!
( , )( ) ( )( )ζ ζ ζ ζ
ρ
α α+ − + −∫ − − ≤
C
r
df r
!
( )1 1
1
− + −∫ ζ ζα
ρ
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
NABLYÛENNQ HOLOMORFNYX FUNKCIJ SEREDNIMY … 1259
=
C
r r
f
r
!
( )1 −
+
+ρ
α
α
∀ ∈[ )ρ 0 1, ,
qki i dovodqt\ neobxidnist\ umov teoremy.
Dostatnist\. Poklademo ρn = 1 – 2− n
, n ∈ 3Z + , i A zn( ) :3= A f zn( )( ) :3=
:3=3 A f z
n rρ , ( )( )+1 . PokaΩemo, wo
A A Ok
r
k
r
L
k
p
( ) ( ) ( )( ) ( ) ( )+
−
+ −⋅ − ⋅ =1
1
1 12ρ ρ α
, k ∈ Z , ∀ ∈[ )ρ 0 1, .
Dali domovymos\ ce zapysuvaty tak:
A w A w Ok
r
k
r
p
k( ) ( ) ( )( ) ( ) ( )+
−
+ −− =1
1
1 12ρ ρ α
, k ∈ N, ∀ ∈[ )ρ 0 1, , w ∈3T. (9)
Spravdi, vykorystovugçy lemy 2, 3 ta umovu (1), oderΩu[mo
A Ak
r
k
r
Lp
( ) ( )( ) ( )+
−
+⋅ − ⋅1
1
1ρ ρ ≤ A Ak
r
k
r
Hp
( ) ( )+
−
+−1
1
1
=
= A f A f A f A fk
r
k k
r
k Hp
( ) ( )( ) ( )+
− −
+−( ) − −( )1
1 1
1
≤
≤ A f A fk
r
k Hp
( ) ( )+
−−( )1
1 3+3 A f A fk
r
k Hp
−
+ −( )1
1( ) ( ) ≤
≤ C
f A fk H
k
r
p
1
1
11
−
−
−
+
( )
( )ρ
3+3 C
f A fk H
k
r
p
1
1
11
−
− −
+
( )
( )ρ
≤
≤ C k
r
k
r2
1
1
1
1
( )
( )
−
−
−
+
+
ρ
ρ
α
3+3 C k
r
k
r2
1
1
1
1
( )
( )
−
−
+
−
+
ρ
ρ
α
=
= ( ) ( )C Cr r k
2 2
1 12 2 2+ − − −α
≤ C k
3
12 ( )−α
, k ∈3N, ∀ ∈[ )ρ 0 1, ,
de C1, C2, C3 — rizni konstanty, wo zaleΩat\ vid f i r.
Dali, za formulog Koßi ma[mo rivnist\
f z A zr r( ) ( )( ) ( )+ +−1 1
ρ =
( )!
( ) ( )
( )
r
i
f A
d
z
R
r
+ −( )
−=
+∫1
2 2π
ζ ζ ζ
ζρ
ζ
,
z R< , 0 < R < 1,
z qko] pry R → 1 vyplyva[ ocinka
f Ar
N r
r
N Lp
( )
,
( )( ) ( )+
+
+⋅ − ⋅1
1
1ρ ρρ ≤ C
f A fN H
N
r
p
4 11
−
− +
( )
( )ρ
≤
≤ C N5
11( ) ( )− − −ρ α
= C N
5
12 ( )−α
,
tobto
f w A w Or
N N
r
N p
N( ) ( ) ( )( ) ( ) ( )+ + −− =1 1 12ρ ρ α
. (10)
Teper, pidsumovugçy rivnosti (9), v qkyx pokladeno ρ = ρ N, po k vid 1 do N ,
N ∈ N, i dodagçy do nyx rivnist\ (10), z uraxuvannqm toho, wo A wN0( )ρ =
=3 S f wr N−1( )( )ρ , oderΩu[mo
f w S f wr
N r N
( )( ) ( )( )+
−−1
1ρ ρ = Op
k
k
N
2 1
1
( )−
=
∑
α
=
= Op
N( )( )2 1−α
= Op
N
1
1 1( )−
−ρ α , N ∈ N.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1260 V. V. SAVÇUK
Zvidsy, oskil\ky 1 – ρN ≤ 1 – ρ ≤ 2 ( 1 – ρN
) dlq ρ ∈ [ ρ
N – 1
, ρN
]
, vyplyva[ ocinka
M f O O Op
r( ),
( )
( )
( )
( )ρ
ρ ρα α
+
− −=
−
+ =
−
1
1 1
1
1
1
1
1
.
OtΩe, za teoremog Hardi – Littlvuda [1] (teorema 48) (dyv. takoΩ [2, c. 78])
f Hr
p
( ) ∈ 0Lipα , tobto f Hp
r∈ Lipα .
Dovedennq teoremy 2. PokaΩemo, wo mnoΩyna holomorfnyx v D funk-
cij, dlq qkyx vykonu[t\sq spivvidnoßennq f A fr Lp
− ρ, ( ) = o r( )1 −( )ρ , ρ →
→ 1–
, zbiha[t\sq z mnoΩynog invariantnyx elementiv operatora A ρ, r , a takog,
qk vydno z rivnosti (3), [ mnoΩyna alhebra]çnyx mnohoçleniv stepenq ne bil\ße
r – 1.
Dijsno, zhidno z (6) i (7) ma[mo rivnist\
T
∫ −( )f w A f w w d wr( ) ( )( ) ( ),ρ
ν σ =
0 0 1
1 1
0
1
, , ,
( ) ˆ , , ,
ν
ν
ρ ρ νν
ν
= −
−
−
= ∞
=
−
−∑
r
k
f r
k
r
k k
z qko] vyplyva[ ocinka
1 1
0
1
−
−
≤ −
=
−
−∑
ν
ρ ρν
ν ρk
f f A f
k
r
k k
r Lp
( ) ˆ ( ), ∀ ∈[ )ρ 0 1, , ν ≥ r. (11)
Oskil\ky
k
k k
k=
−∑
− = − +( ) =
0
1 1 1
ν
ν νν
ρ ρ ρ ρ( ) ( ) ,
to dlq vsix ν ≥ r z nerivnosti (11) vyplyva[
ˆ ˆ lim
( )
( )f
r
f
kr
k r
k k
ν ν
ρ
ν
νν
ρ
ν
ρ ρ
=
−
−
→ =
−∑
1
1
1
1 =
=
ˆ lim
( )
( )f
kr
k
r
k k
ν
ρ
ν
ρ
ν
ρ ρ
→ =
−
−
−
−
−
∑
1 0
1
1
1
1 1 ≤
≤ lim
( )
( ),ρ ρρ→ −
− =
1
1
1
0r r L
f A f
p
, ν ≥ r,
tobto funkciq f [ alhebra]çnym mnohoçlenom stepenq ne bil\ße r – 1.
OtΩe, operator A ρ, r [ nasyçenym z porqdkom nasyçennq ( 1 – ρ )
r
, a zhidno z
teoremog 1 joho klas nasyçennq Φ ( A ρ, r
) zbiha[t\sq z klasom Hp
r−1 1Lip .
1. Hardy G., Littlewood J. E. Some properties of fractional integrals. II // Math. Z. – 1931. – 34. –
P. 403 – 439.
2. Duren P. Theory of Hp spaces. – New York: Acad. Press, 1970. – 258 p.
3. Havrylgk V. T., Stepanec A. Y. Vopros¥ nas¥wenyq lynejn¥x metodov // Ukr. mat. Ωurn.
– 1991. – 43, # 3. – S. 291 – 308.
4. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj : V 2 ç. – Kyev: Yn-t matematyky NAN Ukra-
yn¥, 2002. – Ç. I. – 427 s.
5. Butzer P., Nessel J. R. Fourier analysis and approximation. – Basel: Birkhäuser, 1971. – 553 p.
6. Zygmund A. Smooth functions // Duke Math. J. – 1945. – 12. – P. 47 – 76.
7. Zyhmund A. Tryhonometryçeskye rqd¥ : V 2 t. – M.: Myr, 1965. – T. 1. – 615 s.
OderΩano 23.08.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
|
| id | umjimathkievua-article-3385 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:32Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5e/899d50f3166094ef015605f04e43fc5e.pdf |
| spelling | umjimathkievua-article-33852020-03-18T19:52:51Z Approximation of holomorphic functions by Taylor-Abel-Poisson means Наближення голоморфних функцій середніми Тейлора - Абеля - Пуассона Savchuk, V. V. Савчук, В. В. We investigate approximations of functions $f$ holomorphic in the unit disk by means $A_{\rho, r}(f)$ for $\rho \rightarrow 1_-$. In terms of an error of the approximation by these means, the constructive characteristic of classes of holomorphic functions $H_p^r \text{\;Lip\,}\alpha$ is given. The problem of the saturation of $A_{\rho, r}(f)$ in the Hardy space $H_p$ is solved. Исследуются приближения голоморфных в единичном круге функций $f$ средними $A_{\rho, r}(f)$ при $\rho \rightarrow 1_-$. В терминах погрешности приближения этими средними приведена конструктивная характеристика классов голоморфных функций $H_p^r \text{\;Lip\,}\alpha$. Решена задача о насыщении $A_{\rho, r}(f)$ в пространстве Гарди $H_p$. Institute of Mathematics, NAS of Ukraine 2007-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3385 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 9 (2007); 1253–1260 Український математичний журнал; Том 59 № 9 (2007); 1253–1260 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3385/3513 https://umj.imath.kiev.ua/index.php/umj/article/view/3385/3514 Copyright (c) 2007 Savchuk V. V. |
| spellingShingle | Savchuk, V. V. Савчук, В. В. Approximation of holomorphic functions by Taylor-Abel-Poisson means |
| title | Approximation of holomorphic functions by Taylor-Abel-Poisson means |
| title_alt | Наближення голоморфних функцій середніми Тейлора - Абеля - Пуассона |
| title_full | Approximation of holomorphic functions by Taylor-Abel-Poisson means |
| title_fullStr | Approximation of holomorphic functions by Taylor-Abel-Poisson means |
| title_full_unstemmed | Approximation of holomorphic functions by Taylor-Abel-Poisson means |
| title_short | Approximation of holomorphic functions by Taylor-Abel-Poisson means |
| title_sort | approximation of holomorphic functions by taylor-abel-poisson means |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3385 |
| work_keys_str_mv | AT savchukvv approximationofholomorphicfunctionsbytaylorabelpoissonmeans AT savčukvv approximationofholomorphicfunctionsbytaylorabelpoissonmeans AT savchukvv nabližennâgolomorfnihfunkcíjserednímitejloraabelâpuassona AT savčukvv nabližennâgolomorfnihfunkcíjserednímitejloraabelâpuassona |