Best linear methods of approximation and optimal orthonormal systems of the Hardy space
We construct the best linear methods of the approximation of functions from the Hardy space Hp on compact subsets of the unit disk. We show that the Takenaka - Malmquist systems are such systems of functions that are orthonormal on the unit circle and optimal for the construction of the best linea...
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3182 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509228110708736 |
|---|---|
| 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 | 2020-03-18T19:47:45Z |
| description | We construct the best linear methods of the approximation of functions from the Hardy space Hp on compact subsets of the unit disk. We show that the Takenaka - Malmquist systems are such systems of
functions that are orthonormal on the unit circle and optimal for the construction of the best linear methods of approximation.
|
| first_indexed | 2026-03-24T02:37:46Z |
| format | Article |
| fulltext |
UDK 517.5
V. V. Savçuk (In-t matematyky NAN Ukra]ny, Ky]v)
NAJKRAWI LINIJNI METODY NABLYÛENNQ
TA OPTYMAL|NI ORTONORMOVANI SYSTEMY PROSTORU
HARDI*
We construct the best linear methods of the approximation of functions from the Hardy space Hp on
compact subsets of the unit disk. We show that the Takenaka – Malmquist systems are such systems of
functions that are orthonormal on the unit circle and optimal for the construction of the best linear
methods of approximation.
Postroen¥ nayluçßye lynejn¥e metod¥ pryblyΩenyq funkcyj prostranstva Hardy Hp na
kompaktn¥x podmnoΩestvax edynyçnoho kruha. Pokazano, çto optymal\n¥my ortonormyrovan-
n¥my na edynyçnoj okruΩnosty systemamy funkcyj dlq postroenyq nayluçßyx lynejn¥x me-
todov pryblyΩenyq qvlqgtsq system¥ Takenaky – Mal\mkvysta.
1. Poznaçennq. Postanovka zadaçi. Nexaj. 1 ≤ p ≤ ∞ i Hp — prostir Hardi
holomorfnyx v odynyçnomu kruzi D : = z z∈ <{ }C : 1 funkcij f zi skinçennog
normog
f Hp
=
sup ( ) ( ) , ,
sup ( ) , ,
/
0 1
1
1
< <
∈
∫( ) ≤ < ∞
= ∞
ρ
ρ σ
T
D
f w d w p
f z p
p p
z
de T = z z∈ ={ }C : 1 — odynyçne kolo, σ — normovana mira Lebeha na koli T ,
i UHp — odynyçna kulq prostoru Hp, tobto UHp = f H fp∈ ≤{ }: 1 .
Vidomo, wo koΩna funkciq f iz prostoru Hp ma[ majΩe skriz\ na koli T
kutovi hranyçni znaçennq, za qkymy zalyßymo te Ω same poznaçennq f, pryçomu
f L p∈ ( )T :;=
f f f dL
p
p
p
sumovna na T
T
T
: :( )
/
=
< ∞
∫ σ
1
.
Nexaj a : = ak k{ } =
∞
0 — poslidovnist\ toçok u kruzi D, sered qkyx moΩut\
buty toçky skinçenno] i navit\ neskinçenno] kratnosti. Systemog funkcij Ta-
kenaky – Mal\mkvista, porodΩenog poslidovnistg a, nazyva[t\sq [1] (§ 10.7)
systema ϕ : = ϕk k{ } =
∞
0 funkcij ϕk vyhlqdu
ϕ0( )z =
1
1
0
2
0
– a
a z−
, ϕk z( ) =
1
1 1
2
0
1–
–
– –
–
–a
a z
a
a
z a
a z
k
k
j
jj
k
j
j=
∏ , k = 1, 2, … , (1)
de pry aj = 0 poklada[mo a aj j/ = –1.
Vidomo [1] (§ 10.7), wo systema Takenaky – Mal\mkvista [ ortonormovanog
systemog na koli T, tobto
ϕ ϕk l, : =
ϕ ϕ σk l d
T
∫ = δkl , k, l = 0, 1, … ,
*
Vykonano za pidtrymky DerΩavnoho fondu fundamental\nyx doslidΩen\ Ukra]ny (hrant
#;GP/F13/0018).
© V. V. SAVÇUK, 2008
636 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
NAJKRAWI LINIJNI METODY NABLYÛENNQ TA OPTYMAL|NI … 637
de δkl — symvol Kronekera.
KoΩnomu elementu ϕk systemy ϕ postavymo u vidpovidnist\ dobutok
Blqßke Bk stepenq k. Nahada[mo, wo tak nazyvagt\sq funkci] vyhlqdu
B z0( ) = 1, B zn( ) = τ
z a
a z
j
jj
n –
–
–
10
1
=
∏ , n = 1, 2, … ,
de aj ∈D , j = 0 1, –n , i τ = 1.
Nexaj Bn — mnoΩyna vsix dobutkiv Blqßke stepenq ne bil\ße n. Bud\-
qkyj dobutok Blqßke Bn n∈B , nuli qkoho zbihagt\sq z nulqmy funkci] ϕn
vyhlqdu (1), budemo nazyvaty n-dobutkom Blqßke systemy Takenaky – Mal\m-
kvista ϕ.
Nexaj TM — mnoΩyna vsix system Takenaky – Mal\mkvista i ϕ ∈TM . Todi
dlq bud\-qko] funkci] f Hp∈ poslidovnist\ çysel
ˆ ( )f kϕ : = f k, ϕ , k = 0, 1, … ,
isnu[ j utvorg[ poslidovnist\ koefici[ntiv Fur’[ funkci] f za systemog ϕ.
Poznaçymo çerez L mnoΩynu vsix neskinçennyx nyΩn\otrykutnyx matryc\
Λ : = ( ),λk n , n = 1, 2, … , k = 0, 1, … , n – 1, elementamy qkyx [ funkci] λk n, ( )⋅ ,
vyznaçeni ta neperervni v D. Dlq dano] matryci Λ ∈L i systemy ϕ ∈TM vy-
znaçymo na Hp poslidovnist\ linijnyx operatoriv Un, ,Λ ϕ pravylom
U fn, , ( )Λ ϕ = ˆ ( ) ,
–
f k k n k
k
n
ϕ λ ϕ
=
∑
0
1
, n = 1, 2, … .
Nexaj K — kompaktna pidmnoΩyna kruha D i f K : = max ( )z K f z∈ .
U danij roboti budemo doslidΩuvaty velyçynu
En pUH K( ; ; )ϕ :;=
inf sup – ( ), ,
Λ
Λ
∈ ∈L f UH
n K
p
f U fϕ , n ∈N , (2)
qka nazyva[t\sq velyçynog najkrawoho linijnoho nablyΩennq na kompakti K
po systemi ϕ ∈TM klasu Hardi. Zokrema, vkazano formulu, za qkog budugt\sq
elementy matryci Λ∗
, dlq qko] dosqha[t\sq nyΩnq meΩa v (2), ta dovedeno
[dynist\ tako] matryci. Pro taku matrycg Λ∗
kaΩut\, wo vona porodΩu[ naj-
krawyj linijnyj metod nablyΩennq klasu UHp po systemi ϕ na kompakti;;K.
Porqd iz velyçynog (2) doslidΩu[t\sq takoΩ velyçyna
L UH Kn p( ; ) :;=
inf ( ; ; )
ϕ
ϕ
∈TM
n pUH KE , n ∈N . (3)
Systemu ϕ∗ ∈TM , dlq qko] dosqha[t\sq toçna nyΩnq meΩa v (3), nazyvatyme-
mo;optymal\nog v sensi najkrawoho nablyΩennq systemog Takenaky – Mal\m-
kvista.
2. Osnovni rezul\taty. U nastupnomu tverdΩenni, qke [ klgçovym u do-
slidΩenni velyçyn (2) i (3), jdet\sq pro potoçkove nablyΩennq najkrawym
linijnym metodom, pobudovanym za systemog funkcij Takenaky – Mal\mkvista.
Poznaçymo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
638 V. V. SAVÇUK
V zp( ) =
1
1
1
1
2 1
–
, ,
, ,
/
z
p
p
p( )
≤ < ∞
= ∞
i
W zpζ, ( ) =
1
1
1
1
2
2
1
–
( – )
, ,
, .
/
ζ
ζz
p
p
p
≤ < ∞
= ∞
Teorema 1. Nexaj 1 ≤ p ≤ ∞ , ϕ ∈TM i Bn n{ } =
∞
0 — poslidovnist\ n-do-
butkiv Blqßke systemy ϕ. Todi isnu[ [dyna matrycq Λ
∗ ∈L , zaleΩna vid p
i systemy ϕ, taka, wo dlq bud\-qkoho ζ ∈D i koΩnoho natural\noho n
spravdΩugt\sq rivnosti
inf sup ( ) – ( )( ), ,
Λ
Λ
∈ ∈L f UH
n
p
f U fζ ζϕ =
= max ( ) – ( )( )
, ,f UH n
p
f U f
∈
∗ζ ζϕΛ = B Vn p( ) ( )ζ ζ . (4)
Dlq koΩnoho fiksovanoho ζ ∈D i danoho n ∈N maksymum v (4) dosqha-
[t\sq dlq funkci] f ∗ = e B Wi
n p
α
ζ, , α ∈R.
Elementy matryci Λ∗
obçyslggt\sq za formulog
λ ζk n, ( )∗ = λ ζϕk n p, , , ( )∗ : =
: = 1
1
1 1
1
1
1
12 2
1
1
1
1
2 2
2–
–
–
–
–
( – )
( – )
–
–
–
( )/
–
–
–
/a a
a
a w
w a
w
w
d wk
p
j
jj k
n jj k
n
jj k
n
pζ
ζ
ζ
ζ
ζ ζ
ζ
σ
( ) ( )
=
= +
=
∏ ∫
∏
∏T
.(5)
Naslidok 1. Za umov teoremy;1 spravdΩugt\sq rivnosti
En pUH K( ; ; )ϕ = max – ( )
, ,f UH n Kp
f U f
∈
∗Λ ϕ = B Vk p K
. (6)
Maksymum u (6) dosqha[t\sq dlq funkci] f ∗ = e B Wi
n p
α
ζ, , α ∈R, de ζ —
toçka na K taka, wo B Wn p( ) ( ),ζ ζζ = B Vn p( ) ( )ζ ζ = B Vn p K
.
Spravdi, z rivnostej (4) vyplyva[, wo
En pUH K( ; ; )ϕ ≤ sup – ( )
, ,
f UH
n K
p
f U f
∈
∗Λ ϕ ≤ B Vn p K
. (7)
Z inßoho boku, oskil\ky funkciq B Vn p [ neperervnog na K, znajdet\sq toçka
ζ ∈K , dlq qko] B Vn p( ) ( )ζ ζ = B Vn p K
. U cij toçci z uraxuvannqm (4) spravd-
Ωugt\sq spivvidnoßennq
En pUH K( ; ; )ϕ ≥
inf sup ( ) – ( )( ), ,
Λ
Λ
∈ ∈L f UH
n
p
f U fζ ζϕ =
= f ∗( )ζ = B Vn p( ) ( )ζ ζ = B Vn p K
,
qki v po[dnanni z (7) i dovodqt\ (6).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
NAJKRAWI LINIJNI METODY NABLYÛENNQ TA OPTYMAL|NI … 639
ZauvaΩennq. 1. Dlq toho wob dlq systemy ϕ, porodΩeno] poslidovnistg
toçok ak k{ } =
∞
0 , spravdΩuvalosq spivvidnoßennq lim ( )n nB z→∞ = 0 rivnomirno
po z na bud\-qkij zamknenij pidmnoΩyni kruha D, neobxidno i dostatn\o, wob
1
0
– ak
k
( ) = ∞
=
∞
∑ . (8)
Za ci[] umovy systema ϕ [ povnog u prostori H2 (dyv., napryklad, [1], hl. 10,
[2], § 1).
2. Vykorystovugçy intehral\nu formulu Koßi, neskladno pokazaty, wo
λ ϕk n z, , , ( )2
∗ = 1 dlq bud\-qkoho z ∈D , qkog b ne bula systema ϕ ∈TM .
OtΩe, pry p = 2 operator U
n, ,Λ∗ ϕ stavyt\ u vidpovidnist\ funkci] f ças-
tynnu sumu porqdku n ]] rqdu Fur’[ za systemog ϕ, tobto
U f
n, ,
( )Λ∗ ϕ = S fn, ( )ϕ = ˆ ( )
–
f k k
k
n
ϕ ϕ
=
∑
0
1
.
Pry c\omu rivnist\ (6) nabyra[ vyhlqdu
En UH K( ; ; )2 ϕ = max – ( ),
f UH
n K
f S f
∈ 2
ϕ = B Vn K2 .
3. Za umov teoremy;1 λk n la, ( ) = 1 i B an l( ) = 0, qkwo k ≤ l ≤ n – 1. Tomu
U f an l, , ( )( )Λ ϕ = ˆ ( ) ( )f k ak
k
l
lϕ ϕ
=
∑
0
= f al( ), l = 0 1, –n ,
tobto znaçennq operatora U f
n, ,
( )Λ∗ ϕ interpolggt\ funkcig f u toçkax al ,
l = 0 1, –n , z vidpovidnog ]x kratnistg.
4. Dlq bud\-qko] funkci] f H∈ 2 spravdΩu[t\sq rivnist\ (dyv. formu-
lu;(27) v dovedenni teoremy;1)
ˆ ( ) ( )
–
f k zk
k
n
ϕ ϕ
=
∑
0
1
=
f z B z f w
B w
zw
d wn
n( ) – ( ) ( )
( )
–
( )
1
T
∫ σ ∀ ∈z D . (9)
V roboti [3] (teorema;3) stverdΩu[t\sq, wo dlq dano] funkci] f H∈ 2 sered
mnoΩyny vsix znaçen\ vyrazu, wo sto]t\ u pravij çastyni rivnosti (9), koly v
n\omu Bn probihatyme vsg mnoΩynu Bn, mistytymet\sq j racional\nyj drib
najkrawoho nablyΩennq v metryci prostoru H2 funkci] f mnoΩynog racio-
nal\nyx drobiv porqdku n, usi polgsy qkyx leΩat\ zovni kruha D. Po[dnavßy
cej fakt z rivnistg (9), moΩemo stverdΩuvaty, wo dlq koΩno] funkci] f H∈ 2
isnu[ svoq optymal\na systema ϕ v tomu rozuminni, wo çastynna suma rqdu
Fur’[ po cij systemi bude racional\nog funkci[g najkrawoho nablyΩennq v
metryci H2.
Naslidok 2. Nexaj 1 ≤ p ≤ ∞, K — kompaktna pidmnoΩyna kruha D. Todi
dlq bud\-qkoho natural\noho n spravdΩu[t\sq rivnist\
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
640 V. V. SAVÇUK
L UH Kn p( ; ) =
inf
\ –B
n p K
n n n
B V
∈B B 1
.
Dlq koΩnoho fiksovanoho natural\noho n isnugt\ optymal\na systema
ϕ∗ ∈TM i [dyna matrycq Λ
∗ ∈L , elementy qko] znaxodqt\sq za pravy-
lom;(5), taki, wo
L UH Kn p( ; ) = max – ( )
,, ,f UH n Kp
f U f
∈
∗ ∗Λ ϕ = B Vn p K
∗ , (10)
de Bn
∗
— n-dobutok Blqßke systemy ϕ∗.
Ob©runtuvannq potrebu[ lyße rivnist\ (10). Nexaj mn =
infB n p Kn n
B V∈B i
Bn
j
j{ } ≥1
, Bn
j
n∈B , — poslidovnist\ dobutkiv Blqßke taka, wo B Vn
j
p K
→ mn , j
→ ∞. Oskil\ky poslidovnist\ Bn
j
j{ } ≥1
rivnomirno obmeΩena v zamknenomu kru-
zi D , to za pryncypom kompaktnosti z ci[] poslidovnosti moΩna vydilyty pid-
poslidovnist\, qka bude rivnomirno zbihatysq v seredyni D do deqko] holomor-
fno] funkci], napryklad Bn
∗
. OtΩe, mn = B Vn p K
∗
. Za dopomohog teoremy
Ruße moΩna pokazaty, wo funkciq Bn
∗
ma[ ne bil\ße n nuliv v D, qki pozna-
çymo aν
∗
, ν = 0 1, –k , 1 ≤ k ≤ n. Vykorystovugçy rezul\tat z [4], moΩna po-
kazaty, wo vsi aν
∗
leΩat\ v opuklij obolonci kompakta K. Cej fakt dozvolq[
vydilyty z poslidovnosti Bn
j
j{ } ≥1
pidposlidovnist\, qka bude zbihatysq do Bn
∗
rivnomirno i na T. Tomu funkciq Bn
∗
[ dobutkom Blqßke stepenq ne bil\ße n
i ma[ vyhlqd B zn
∗( ) = λ νν z a
k
–
– ∗
= ( )∏ 0
1
/ 1 – a zν
∗( ) , λ = 1.
Dopovnymo nabir a0
∗, … , ak –1
∗
dovil\nog neskinçennog poslidovnistg
aj j k{ } ≥
toçok z D i pobudu[mo po utvorenij poslidovnosti systemu Takenaky –
Mal\mkvista ϕ∗
. Vyberemo funkcig f ∗ = ˜
,B Wn pζ , v qkij ζ — toçka, dlq qko]
˜ ( ) ( )B Vn pζ ζ = B̃ Vn p K
i
˜ ( )B zn :;= B z
a
a
z a
a zn
j
jj k
n
j
j
∗
=
∏( )
– –
–
–1
1
, (11)
pryçomu qkwo k = n, to druhyj mnoΩnyk v (11) poklada[t\sq rivnym odynyci.
Dlq velyçyn L UH Kn p( ; ) ta f
K
∗
na pidstavi poperednix mirkuvan\ ta na-
slidku 1 spravdΩugt\sq spivvidnoßennq
mn ≤ L UH Kn p( ; ) ≤
En pUH K; ; ϕ∗( ) = f
K
∗ = B̃ Vn p K
. (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
NAJKRAWI LINIJNI METODY NABLYÛENNQ TA OPTYMAL|NI … 641
Ale
˜ ( )B zn ≤ B zn
∗( ) ∀ ∈z D . Tomu skriz\ v (12) vykonugt\sq rivnosti. Os-
kil\ky
˜ \ –Bn n n∈B B 1, to mn = inf B Vn p K{ : Bn n∈B \
Bn –1}. Çysla aj , j =
= k n, – 1, v (11) moΩna vybraty dovil\nym çynom. Napryklad, viz\memo aj = 0.
V takomu vypadku, qkwo prypustyty, wo k < n, otryma[mo nerivnist\
B̃ Vn p K
;≤ r B Vn k
n p K
– ∗ < B Vn p K
∗ = mn , v qkij r — radius najmenßoho kruha,
wo mistyt\ kompakt K. Taka nerivnist\ [ supereçlyvog, tomu k = n i B̃ Bn n= ∗
,
tobto Bn
∗
[ dobutkom Blqßke stepenq toçno n, a dovil\na systema ϕ∗ ∈TM ,
pobudovana za poslidovnistg toçok, perßi n z qkyx zbihagt\sq z toçkamy
a0
∗, … , an –1
∗ , [ optymal\nog dlq danoho n.
ZauvaΩennq 5. Nexaj Ap = A Kp( ) — zvuΩennq odynyçno] kuli UHp na
kompakt K , C K( ) — prostir neperervnyx funkcij na K z normog ⋅ K i
D An p( ; C K( )) — velyçyna, qka oznaça[ popereçnyk porqdku n za Kolmohoro-
vym, abo Hel\fandom, abo linijnyj popereçnyk klasu Ap u prostori C K( )
(oznaçennq dyv., napryklad, u [5] ). U roboti [5] pokazano, wo D An ∞( ; C K( )) =
=
inf \ –B n Kn n n
B∈B B 1
. Zistavlqgçy cg rivnist\ z rivnistg (10), baçymo, wo pry
p = ∞ i danomu n optymal\na systema ϕ∗
[ optymal\nog i v sensi popereçnyka
za Kolmohorovym, tobto n-vymirnyj pidprostir z bazysom ϕ0
∗{ , … , ϕn –1
∗ } [ naj-
krawym nablyΩugçym pidprostorom dlq klasu A∞ . Pry c\omu linijnyj ope-
rator U
n, ,Λ∗ ∗ϕ [ operatorom, znaçennq qkoho najkrawe nablyΩagt\ klas A∞
sered znaçen\ usix linijnyx operatoriv Ln na C K( ) ranhu n.
Zaznaçymo, wo u druhij çastyni naslidku 2 stverdΩu[t\sq isnuvannq svo[]
optymal\no] systemy Takenaky – Mal\mkvista dlq koΩnoho natural\noho n.
Taka obstavyna vyklykana heometryçnog budovog kompaktu K. U nastupnomu
tverdΩenni navodyt\sq pryklad kompaktu K, dlq qkoho isnu[ systema Takena-
ky – Mal\mkvista, optymal\na dlq vsix natural\nyx n.
Naslidok 3. Nexaj 1 ≤ p ≤ ∞, 0 ≤ ρ < 1 i K = z ∈{ D : z ≤ }ρ . Todi:
1) dlq koΩnoho natural\noho n spravdΩu[t\sq rivnist\
L UH Kn p( ; ) = ρ ρn
pV ( );
2) dlq vsix natural\nyx n optymal\nog ortonormovanog systemog Take-
naky – Mal\mkvista [ systema ϕ∗ = zk
k{ } =
∞
0
;
3) najkrawyj linijnyj metod nablyΩennq [ [dynym i porodΩugt\sq çyslo-
vog matryceg Λ∗ = ( ),λk n
∗
, elementy qko] obçyslggt\sq za formulog
λk n,
∗ = 1 1
2
2 2
0
1
2 2– (– )
/
–
– /
– –
( – )ρ
ν
ρ ρν
ν
ν( )
( )
=
∑p
n k
n k
p
, n = 1, ∞ , k = 0 1, –n ,
(13)
de
2
0
1
/ p
= ,
2 2 2 1 2 1/
:
/ ( / – ) ( / – )
!
p p p p
ν
ν
ν
= … +
, ν = …1 2, , .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
642 V. V. SAVÇUK
Spravdi, v [5] pokazano, wo
inf \ –B n Kn n n
B∈B B 1
= ρn
, pryçomu [dynym, z
toçnistg do mnoΩnyka eiα
, α ∈R, dobutkom Blqßke stepenq n, dlq qkoho
dosqha[t\sq toçna nyΩnq meΩa, [ funkciq B zn
∗( ) = zn
. Z uraxuvannqm c\oho
faktu, oçevydno] rivnosti
inf \ –B n p Kn n
B V∈B B 1
=
inf ( )\ –B n K pn n
B V∈B B 1
ρ ta
rivnosti (10) perekonu[mosq v pravyl\nosti tverdΩen\ 1 i 2. Formula (13) vy-
plyva[ z formuly (5), v qkij pokladeno ak = 0, k = 0 1, –n (detal\ni vykladky
dyv. u [6]).
2. Ekstremal\ni vlastyvosti ta najkrawe nablyΩennq qdra Koßi. Re-
zul\taty c\oho punktu [ etapamy dovedennq teoremy;1. Prote vony ne pozbavle-
ni j samostijnoho interesu.
Qdrom Koßi dlq kruha D nazyva[t\sq funkciq C, vyznaçena v C
2
: =
: = C C× takym çynom:
C z w( , ) = 1
1 – zw
.
Vidomo, wo funkciq C [ tvirnym qdrom prostoru Hp, 1 ≤ p ≤ ∞, tobto v
bud\-qkij toçci z ∈D
f z( ) = f C z, ( , )⋅ ∀ ∈f Hp . (14)
Nexaj n ∈ +Z , p–1 + q–1 = 1,
Hp
n
,ϕ : = f H f k k np∈ = ={ }: ˆ ( ) , , –ϕ 0 0 1
i Hp,ϕ
0 = Hp.
Lehko baçyty, wo qdro
C z wn, ( , )ϕ : = C z w z wk
k
n
k( , ) – ( ) ( )
–
ϕ ϕ
=
∑
0
1
(15)
ma[ vlastyvist\ vidtvorennq po vidnoßenng do funkcij prostoru Hp
n
,ϕ , tobto v
bud\-qkij toçci z ∈D
f z( ) = f C zn, ( , ),ϕ ⋅ ∀ ∈f Hp
n
,ϕ . (16)
Zrozumilo, wo funkci] C i Cn,ϕ u vkazanomu sensi ne [dyni tvirni qdra dlq
prostoriv Hp i Hp
n
,ϕ vidpovidno.
Naßog najblyΩçog metog [ pobudova inßyx tvirnyx qder dlq zaznaçenyx
prostoriv tak, wob ci qdra maly pevni ekstremal\ni vlastyvosti.
Nexaj 1 ≤ p ≤ ∞, n ∈ +Z i Bn — n-dobutok Blqßke systemy ϕ. Oznaçymo v
D2
funkcig Cp n, ,ϕ pravylom
C z wp n, , ( , )ϕ : = B z B w zw
z
zwn n
p
p
( ) ( )( – )
( – )
–
/
– /
1
1
1
2
2 1 2
2 . (17)
Teorema 2. Nexaj 1 ≤ p ≤ ∞, n ∈ +Z i ϕ ∈TM . Qkwo funkciq f nale-
Ωyt\ prostoru Hp
n
,ϕ , to dlq koΩnoho z ∈D
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
NAJKRAWI LINIJNI METODY NABLYÛENNQ TA OPTYMAL|NI … 643
f z( ) = f w C z w d wp n( ) ( , ) ( ), ,ϕ σ
T
∫ . (18)
Dovedennq. Zafiksu[mo z ∈D i rozhlqnemo funkcig g, oznaçenu v D
pravylom
g w( ) : = ( – ) ( )( – )– / / –1 12 1 2 2 1z f w zwp p .
Nexaj ak k{ } =
∞
0 — poslidovnist\ toçok, qka porodΩu[ systemu ϕ, i
R w0( ) =
1 0
2
0
–
–
a
w a
, R wk ( ) =
1 12
0
1–
–
– –
–
–a
w a
a
a
a w
w a
k
k
j
jj
k
j
j=
∏ , k ∈N .
Oskil\ky ϕk w( ) = wR wk ( ) ∀ w ∈T , to za teoremog pro lyßky ma[mo formulu
ˆ ( ) ( )f f aϕ 0 0= ,
ˆ ( )f kϕ = 1
2πi
f w R w dwk( ) ( )
T
∫ = res
w a
k
k
f w R w
==
∑
νν
( ) ( )
0
=
=
1
1
1
1
0 ( – )!
lim ( – ) ( ) ( )
–
–k
d
dw
w a f w R w
w a
k
k
k
k
k
ν
ν
ν ν
ν
ν
ν
→=
( )
∑ , k ∈N ,
de kν — kratnist\ mnoΩnyka ( – )1 a wν / ( – )w aν u vyrazi R wk ( ) , pryçomu
k
k
νν=∑ 0
1–
= k.
Qkwo funkciq f naleΩyt\ prostoru Hp
n
,ϕ , to z poperedn\o] formuly po-
slidovno dlq koΩnoho k = 0, 1, … , n – 1 otrymu[mo znaçennq f ak( ) = 0. Tomu
g ak( ) = 0, k = 0 1, –n . OtΩe, obçyslggçy koefici[nty Fur’[ funkci] g po
systemi ϕ tak, qk ce zrobleno dlq funkci] f, znaxodymo, wo i ˆ ( )g kϕ = 0, k =
= 0 1, –n , tobto g Hp
n∈ ,ϕ .
Tomu zhidno z (16) dlq funkci] g v toçci w = z ma[mo rivnist\
f z( ) = g z( ) =
T
∫ ( – ) ( )( – ) ( , ) ( )– / / –
,1 12 1 2 2 1z f w zw C z w d wp p
n ϕ σ .
Na zaverßennq dovedennq teoremy zalyßa[t\sq skorystatysq totoΩnistg [2]:
dlq koΩnoho fiksovanoho z ∈D
C z wn, ( , )ϕ = B z B w
zwn n( ) ( )
–
1
1
∀ ∈w T .
Nexaj
B z q n( , , , )ϕ : =
inf ( , ) – ( ) : ( )( ) ,
,C z g g LL p
n
q
⋅ ⋅ ∈{ }⊥
T
Tϕ , (19)
de
Lp
n
,
, ( )ϕ
⊥
T = inf ( ) : , ,g L f g f Hq p
n∈ = ∀ ∈{ }T 0 ϕ .
U nastupnomu tverdΩenni da[t\sq toçne znaçennq velyçyny B z q n( , , , )ϕ ta
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
644 V. V. SAVÇUK
vkazu[t\sq ekstremal\na funkciq gz , dlq qko] dosqha[t\sq toçna nyΩnq meΩa
v (19).
Teorema 3. Nexaj 1 ≤ p ≤ ∞, 1/ p + 1/q = 1, n ∈ +Z , ϕ ∈TM i Bn — n-do-
butok Blqßke systemy ϕ. Todi dlq koΩnoho z ∈D
B z q n( , , , )ϕ = B z V zn p( ) ( ), (20)
a ekstremal\nog v (19) [ [dyna funkciq
gz ( )⋅ = C z( , )⋅ – C zp n, , ( , )ϕ ⋅ . (21)
Dovedennq. Zafiksu[mo z ∈D . Dlq ocinky zverxu velyçyny B z q n( , , , )ϕ
viz\memo funkcig gz , vyznaçenu pravylom (21), i pokaΩemo, wo gz( )⋅ ∈ Lq
n
,
, ( )ϕ
⊥
T .
Nasampered zauvaΩymo, wo sup ( )
w zg w∈D
< ∞ i tym paçe gz( )⋅ ∈ Lq( )T .
Dali, zhidno z (14) i (18) dlq bud\-qko] funkci] f Hp
n∈ ,ϕ spravdΩugt\sq riv-
nosti
f gz, = f C z, ( , )⋅ – f C zp n, ( , ), ,ϕ ⋅ = f z( ) – f z( ) = 0. (22)
OtΩe, gz ∈ Lq
n
,
, ( )ϕ
⊥
T .
Teper dlq velyçyny B z q n( , , , )ϕ lehko otrymaty ocinku zverxu:
B z q n( , , , )ϕ ≤ C z gz Lq
( , ) – ( ) ( )⋅ ⋅
T
= C zp n Lq
, , ( )
( , )ϕ ⋅
T
= B z V zn p( ) ( ). (23)
Z inßoho boku, za teoremog dvo]stosti (dyv., napryklad, [7, s. 137] ) i formu-
log (14)
B z q n( , , , )ϕ = sup , ( , ) : ,,f C z f H fp
n
Hp
⋅ ∈ ≤{ }ϕ 1 =
= sup ( ) : ,,f z f H fp
n
Hp
∈ ≤{ }ϕ 1 = : A z p n( , , , )ϕ . (24)
Dlq ocinky znyzu velyçyny A z p n( , , , )ϕ viz\memo funkcig f ∗
, vyznaçenu
pravylom
f w∗( ) = B w
z
zw
n
p
( )
–
( – )
/
1
1
2
2
1
.
Lehko baçyty, wo f
Hp
∗ = 1. Okrim c\oho, dlq vsix k = 0, 1, … , n – 1 za inteh-
ral\nog formulog Koßi spravdΩu[t\sq rivnist\
ˆ ( )f kϕ
∗ = B w
z
zw
a
wa
a
a
w a
a w
d wn
p
k
k
j
jj
k
j
j
( )
–
( – )
–
–
– –
–
( )
/ –
1
1
1
1 1
2
2
1 2
0
1
∫ ∏
=T
σ =
= 1 1
1
1
1 1
2 1 2
1
2– –
–
– ( – )
( )
–
/
–
/z a
w a
a w zw
d w
wa
p
k
j
jj k
n
p
k
( ) ∫ ∏
=
λ σ
T
=
= 1 1
1
1
1
2 1 2
1
2– –
–
– ( – )
/
–
/z a
a a
a a za
p
k
k j
j kj k
n
k
p( )
=
∏λ = 0, λ = 1. (25)
OtΩe, f Hp
n∗ ∈ ,ϕ i tomu
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
NAJKRAWI LINIJNI METODY NABLYÛENNQ TA OPTYMAL|NI … 645
A z p n( , , , )ϕ ≥ f z∗( ) = B z V zn p( ) ( ). (26)
Zistavyvßy spivvidnoßennq (23) – (26), perekonu[mos\ u pravyl\nosti riv-
nosti (20).
Dali, oskil\ky dlq vsix znaçen\ parametra p ∈ ∞[ ]1, znajdet\sq ekstremal\-
na funkciq f ∗
, dlq qko] dosqha[t\sq toçna verxnq meΩa v (24), ekstremal\na
funkciq gz [ [dynog (dyv. dovedennq teorem;1.2 i 1.3 v [7], hl. VI).
3. Dovedennq teoremy 1. Vyberemo funkcig gζ , oznaçenu pravylom (21), i
rozhlqnemo matrycg Λ∗
, v qkij
λ ζk n, ( )∗ = 1
ϕ ζ
ϕ ζ
k
k g
( )
, = 1 –
1
ϕ ζ
ϕ ζ σϕ
k
k n pw C w d w
( )
( ) ( , ) ( ), ,
T
∫ , k = 0 1, –n .
Funkci] λ ζk n, ( )∗
oznaçeni korektno, oskil\ky v toçkax aj z uraxuvannqm ]x
kratnosti vykonugt\sq rivnosti ϕk ja( ) = C a wn p j, , ( , )ϕ = 0, j = 0 1, –k , k =
= 1 1, –n . ZauvaΩymo, wo take vyznaçennq λ ζk n, ( )∗
pislq oçevydnyx peretvo-
ren\ zbiha[t\sq z formulog (5).
Nexaj f Hp∈ i S fn( ) : = ˆ ( )
–
f k kk
n
ϕ ϕ=∑ 0
1
. Todi funkciq f – S fn( ) naleΩyt\
prostoru Hp
n
,ϕ i zhidno z (22)
f g, ζ = f S f gn– ( ), ζ + S f gn( ), ζ = S f gn( ), ζ =
= 1
0
1
ϕ ζ
ϕ ϕ ζζ ϕ
kk
n
k kg f k
( )
, ˆ ( ) ( )
–
=
∑ = U f
n, ,
( )( )Λ∗ ϕ ζ .
Zvidsy z uraxuvannqm (14) vyplyva[ totoΩnist\
f ( )ζ – U f
n, ,
( )( )Λ∗ ϕ ζ = f ( )ζ – f g, ζ =
= f C, ( , )ζ ⋅ – f g, ζ = f Cn p, ( , ), ,ϕ ζ ⋅ . (27)
OtΩe,
inf sup ( ) – ( )( ), ,
Λ
Λ
∈ ∈L f UH
n
p
f U fζ ζϕ ≤ sup ( ) – ( )( )
, ,
f UH
n
p
f U f
∈
∗ζ ζϕΛ =
= sup , ( , ), ,
f UH
n p
p
f C
∈
⋅ϕ ζ . (28)
Za nerivnistg Hel\dera z uraxuvannqm rivnosti (23) ma[mo ocinku
sup , ( , ), ,
f UH
n p
p
f C
∈
⋅ϕ ζ ≤ sup ( , ), , ( )
f UH
H n p L
p
p p
f C
∈
⋅ϕ ζ
T
= B Vn p( ) ( )ζ ζ , (29)
de 1/ p + 1/q = 1 i Bn — n-dobutok Blqßke systemy ϕ.
Funkciq f ∗ = B Wn pζ, , qk ce pokazano v dovedenni teoremy;3 (dyv. (25)), na-
leΩyt\ Hp
n
,ϕ i f UHp
∗ ∈ . Tomu dlq ne] U fn, , ( )Λ ϕ = 0, qkog b ne bula matrycq
Λ, i vnaslidok c\oho
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
646 V. V. SAVÇUK
inf sup ( ) – ( )( ), ,
Λ
Λ
∈ ∈L f UH
n
p
f U fζ ζϕ ≥ f ∗( )ζ = B Vn p( ) ( )ζ ζ . (30)
Ob’[dnavßy spivvidnoßennq (28) – (30), otryma[mo rivnist\ (4).
Dovedemo teper [dynist\ matryci Λ∗
, qka porodΩu[ najkrawyj linijnyj me-
tod nablyΩennq. Nexaj funkciq g naleΩyt\ L p
n
,
, ( )ϕ
⊥
T . Lehko baçyty, wo
funkcig g majΩe skriz\ na T moΩna podaty u vyhlqdi g = ˆ ( )
–
g k kk
n
ϕ ϕ=∑ 0
1
+ h ,
de ˆ ( )g kϕ = g k, ϕ i h — kutovi hranyçni znaçennq deqko] funkci] z Hp,ϕ
1
. Ot-
Ωe, dlq bud\-qko] funkci] f UHp∈
f C g, ( , ) –ζ ⋅( ) = f ( )ζ – f g, = f ( )ζ – f g k
k
n
k, ˆ ( )
–
=
∑
0
1
ϕ ϕ =
= f ( )ζ –
k
n
g k f k
=
∑
0
1–
ˆ ( ) ˆ ( )ϕ ϕ .
Zvidsy za teoremog dvo]stosti ma[mo spivvidnoßennq
sup ( ) – ( ) ˆ ( ) ( ),
–
f UH
k n k
k
n
p
f f k
∈
∗
=
∑ζ λ ζ ϕ ζϕ
0
1
≥
inf sup ( ) – ˆ ( ) ˆ ( )
,
, ( )
–
g L f UH k
n
p
n
p
f g k f k
∈ ∈ =
⊥ ∑
ϕ
ζ ϕ ϕ
T 0
1
≥
≥ sup ( )
,f UHp
n
f
∈ ϕ
ζ = inf ( , ) –
,
, ( )
( )
g L
L
p
n q
C g
∈ ⊥
⋅
ϕ
ζ
T
T
, ζ ∈D .
Zhidno z dovedenym vywe u cyx spivvidnoßennqx vykonugt\sq lyße rivnosti, a
zhidno z teoremog;3 funkciq gζ [ [dynog funkci[g z Lp
n
,
, ( )ϕ
⊥
T , dlq qko]
C( , )ζ ⋅ – g Lqζ ( )T
=
inf ( , )
,
, ( )g Lp
n C∈ ⊥ ⋅
ϕ
ζ
T
– g Lq ( )T
. OtΩe, funkciq gζ bude [dy-
nog ekstremal\nog funkci[g i v tomu sensi, wo
inf sup ( ) – ˆ ( ) ˆ ( )
,
, ( )
–
g L f UH k
n
p
n
p
f g k f k
∈ ∈ =
⊥ ∑
ϕ
ζ ϕ ϕ
T 0
1
= sup ( ) – ˆ ( ) ˆ ( ),
–
f UH k
n
p
f g k f k
∈ =
∑ζ ζ ϕ ϕ
0
1
.
1. Uolß DΩ. L. Ynterpolqcyq y approksymacyq racyonal\n¥my funkcyqmy v kompleksnoj
oblasty. – M.: Yzd-vo ynostr. lyt., 1961. – 508 s.
2. DΩrbaßqn M. M. RazloΩenyq po systemam racyonal\n¥x funkcyj s fyksyrovann¥my
polgsamy // Yzv. AN ArmSSR. Ser. mat. – 1967. – 2, # 1. – S. 3 – 51.
3. Eroxyn V. D. O nayluçßem pryblyΩenyy analytyçeskyx funkcyj posredstvom racyonal\-
n¥x drobej so svobodn¥my polgsamy // Dokl. AN SSSR. – 1959. – 128, # 1. – S. 29 – 32.
4. Walsh J. L. Note on the location of zeros of extremal polynomials in the non-euclidean plane //
Acad. Serbe Sci. Publ. Inst. Math. – 1952. – 4. – P. 157 – 160.
5. Fisher S. D., Miccelli C. A. The n-widths of sets of analytic functions // Duke Math. J. – 1980. –
47, # 4. – P. 789 – 801.
6. Savçuk V. V. Najkrawi linijni metody nablyΩennq funkcij klasu Xardi Hp // Ukr. mat.
Ωurn. – 2003. – 55, # 7. – S. 919 – 925.
7. Harnett DΩ. Ohranyçenn¥e analytyçeskye funkcyy. – M.: Myr, 1984. – 469 s.
OderΩano 13.08.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
|
| id | umjimathkievua-article-3182 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:46Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c0/ed396530f2c0c42b0728dd279a8e26c0.pdf |
| spelling | umjimathkievua-article-31822020-03-18T19:47:45Z Best linear methods of approximation and optimal orthonormal systems of the Hardy space Найкращі лінійні методи наближення та оптимальні ортонормовані системи простору Гарді Savchuk, V. V. Савчук, В. В. We construct the best linear methods of the approximation of functions from the Hardy space Hp on compact subsets of the unit disk. We show that the Takenaka - Malmquist systems are such systems of functions that are orthonormal on the unit circle and optimal for the construction of the best linear methods of approximation. Построены наилучшие линейные методы приближения функций пространства Гарди Hp на компактных подмножествах единичного круга. Показано, что оптимальными ортонормирован-ными на единичной окружности системами функций для построения наилучших линейных методов приближения являются системы Такенаки - Мальмквиста. Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3182 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 636–646 Український математичний журнал; Том 60 № 5 (2008); 636–646 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3182/3111 https://umj.imath.kiev.ua/index.php/umj/article/view/3182/3112 Copyright (c) 2008 Savchuk V. V. |
| spellingShingle | Savchuk, V. V. Савчук, В. В. Best linear methods of approximation and optimal orthonormal systems of the Hardy space |
| title | Best linear methods of approximation and optimal orthonormal systems of the Hardy space |
| title_alt | Найкращі лінійні методи наближення та оптимальні ортонормовані системи простору Гарді |
| title_full | Best linear methods of approximation and optimal orthonormal systems of the Hardy space |
| title_fullStr | Best linear methods of approximation and optimal orthonormal systems of the Hardy space |
| title_full_unstemmed | Best linear methods of approximation and optimal orthonormal systems of the Hardy space |
| title_short | Best linear methods of approximation and optimal orthonormal systems of the Hardy space |
| title_sort | best linear methods of approximation and optimal orthonormal systems of the hardy space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3182 |
| work_keys_str_mv | AT savchukvv bestlinearmethodsofapproximationandoptimalorthonormalsystemsofthehardyspace AT savčukvv bestlinearmethodsofapproximationandoptimalorthonormalsystemsofthehardyspace AT savchukvv najkraŝílíníjnímetodinabližennâtaoptimalʹníortonormovanísistemiprostorugardí AT savčukvv najkraŝílíníjnímetodinabližennâtaoptimalʹníortonormovanísistemiprostorugardí |