Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary
We establish estimates for groups of deviations of Faber series in closed domains with piecewise-smooth boundary.
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| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3586 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509701951717376 |
|---|---|
| author | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. |
| author_facet | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. |
| author_sort | Lasuriya, R. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | We establish estimates for groups of deviations of Faber series in closed domains with piecewise-smooth boundary. |
| first_indexed | 2026-03-24T02:45:18Z |
| format | Article |
| fulltext |
UDK 517.5
R. A. Lasuryq (Abxaz. un-t, Suxum)
SYL|NAQ SUMMYRUEMOST| RQDOV FABERA
Y OCENKY SKOROSTY SXODYMOSTY
HRUPPÁ UKLONENYJ V ZAMKNUTOJ OBLASTY
S KUSOÇNO-HLADKOJ HRANYCEJ
We establish estimates of groups of deviations of the Faber series in closed domains with a piecewise
smooth boundary.
Vstanovleno ocinky hrup vidxylen\ rqdiv Fabera v zamknenyx oblastqx iz kuskovo-hladkog
meΩeg.
1. Pust\ C — rasßyrennaq kompleksnaq ploskost\, G — ohranyçennaq ob-
last\ s Ωordanovoj hranycej ψ, D — vneßnost\ v C edynyçnoho kruha: D =
= w w∈ >{ }C : 1 , D — zam¥kanye D , Γe = w w: ={ }1 — edynyçnaq ok-
ruΩnost\, z = ψ ( w ) — funkcyq Rymana, otobraΩagwaq D na C\G y nor-
myrovannaq uslovyem
lim
( )
w
w
w→∞
ψ
= α > 0,
w = Φ( )z — funkcyq, obratnaq k z = ψ( )w ,
Γ1 1+ /n = z z n: ( ) /Φ = +{ }1 1
— n-q lynyq urovnq oblasty G,
ρ1 1+ / ( )n z = min
/ζ
ζ
∈ +
−{ }
Γ1 1 n
z , z ∈ Γ,
— rasstoqnye ot toçky z ∈ Γ do n-j lynyy urovnq Γ1 1+ /n . Pust\, dalee,
C GA( ) — mnoΩestvo funkcyj f ( z ) , analytyçeskyx v G y neprer¥vn¥x v G .
Ohranyçennaq Ωordanova oblast\ G naz¥vaetsq oblast\g typa ( C ) (sm.,
naprymer, [1]), esly dlq funkcyy z = ψ ( w ) v¥polnqgtsq uslovyq: suwestvu-
gt r ∈ N, wj ∈ Γe y αj ∈ ( 0, 2 ) , j = 1, 2, … , r, takye, çto ymeet mesto ravenst-
vo
ψ ′ ( w ) = λ
α
( )w
w
wj
r
j
j
=
−
∏ −
1
1
1 ∀ w ∈ D,
hde λ ( w ) — neprer¥vnaq y otlyçnaq ot nulq na D funkcyq, modul\ nepre-
r¥vnosty kotoroj udovletvorqet uslovyg ω ( λ; t ) ≤ Kt, hde K — nekotoraq
poloΩytel\naq postoqnnaq. Kak yzvestno [1 – 4], oblasty typa ( C ) vklgçagt
v sebq krome mnohouhol\nykov y oblasty s Ωordanov¥my hranycamy, sostoqwy-
my yz koneçnoho çysla okruΩnostej yly analytyçeskyx duh.
Budem hovoryt\, çto G — oblast\ typa ( C ′ ) (sm., naprymer, [5]), esly ona
qvlqetsq oblast\g typa ( C ), a çysla αν, ν = 1, 2, … , r, yz opredelenyq
oblasty typa ( C ) udovletvorqgt uslovyqm
αν ≥
1
2
1
1
max ; max
,j r
j
=
α , ν = 1, 2, … , r.
Pust\
© R. A. LASURYQ, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 187
188 R. A. LASURYQ
S [ f ] =
k
k kc z
=
∞
∑
0
Φ ( ) (1)
— rqd Fabera funkcyy f ( z ), zadannoj na G ,
ck = ck ( f ) =
1
2π
ψ
π
π
−
−∫ f e e dtit ikt( )( ) , k = 0, 1, 2, … ,
— koπffycyent¥ Fabera funkcyy f ( z ), Sn ( f ; z ) — çastyçnaq summa rqda (1).
2. Vvedem v rassmotrenye syl\n¥e srednye stepeny q > 0 rqda (1)
H f zn q, ( ; ) =
1
0
1
n
f z S f z
k
n
k
q
q
=
∑ −
( ) ( ; )
/
. (2)
Sformulyruem vnaçale utverΩdenye, soderΩawee ocenku velyçyn¥ (2) v
toçkax hranyc¥ oblasty typa ( C ′ ).
Teorema)1. Pust\ G — oblast\ typa ( C ′ ), f ( z ) ∈ C GA( ), 1 < p ≤ 2 y
pry nekotorom γ > 2 / p – 1
lim
( ( ))
/δ
δ
γ
ω ρ
δ→ +
+
−0
1
1
z
p = ∞ , z ∈ Γ, (3)
monotonno vozrastaq, hde ω ( f; t ) = ω ( t ) — modul\ neprer¥vnosty f ( z ) na
G . Tohda dlq lgb¥x n ∈ N y q q1 0∈( , ], q
p
p
=
−1
, v toçke z ∈ Γ
H f zn q, ( ; )
1
≤ K znω ρ( ( ))/1 1+ , K = K ( q, γ ) . (4)
Dokazatel\stvo. Na osnovanyy yzvestn¥x rassuΩdenyj [1] ymeem pred-
stavlenye
ρk ( f; z ) = f ( z ) – Sk ( f; z ) =
=
1
4 2π
ζ ζ
ζ
ζ
π
π
i
D t
f f t
z
d dtk
−
∫ ∫ − 〈 〉
−
( )
( ) ( )
Γ
=
=
1
4 2
0π
ζ ζ
ζ
ζ
π
i
D t
F F t
z
d dtk∫ ∫ − 〈 〉
−
( )
( ) ( )
Γ
, (5)
hde
ζ〈 〉t = ψ ζ( )( )Φ e it− , F( )ζ = f f t( ) ( )ζ ζ− 〈− 〉 ,
Dk ( t ) — qdro Dyryxle.
PoloΩym
I ( z; t ) =
F F t
z
d
( ) ( )ζ ζ
ζ
ζ− 〈 〉
−∫
Γ
.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SYL|NAQ SUMMYRUEMOST| RQDOV FABERA … 189
Tohda, prynymaq vo vnymanye (5), naxodym
H f zn q, ( ; ) ≤
1 1
40
1
2
0
1
1
n i
D t I z t dt
k
n n
k
q q
=
−
∑ ∫
π
/
/
( ) ( ; ) +
+
1 1
40
1
2
1
1
n i
D t I z t dt
k
n
n
k
q q
=
−
∑ ∫
π
π
/
/
( ) ( ; ) =
= I z I zn
q
n
q
,
( )
,
( )( ) ( )1 2+ . (6)
Yzvestno [2, c. 38], çto
I z t( ; ) ≤ K ztω ρ( ( ))1+ . (7)
V sylu (7) ymeem
I zn
q
,
( ) ( )1 ≤
1
4
1
2
0
1
0
1
1
π n
n I z t dt
k
n n q q
=
−
∑ ∫
/
/
( ; ) ≤
≤ Kn z dt
n
t
0
1
1
/
( ( ))∫ +ω ρ ≤ K znω ρ( ( )/1 1+ . (8)
Ocenyvaq slahaemoe I zn
q
,
( ) ( )2 , predstavym qdro Dk ( ⋅ ) v vyde
Dk ( t ) =
sin
( / )
cos
kt
t
kt
2 2
1
2tg
+ .
V πtom sluçae
I zn
q
,
( ) ( )2 =
1
4
1
2 22
0
1
1π
π
n
I z t
t
kt dt
k
n
n=
−
∑ ∫
/
( ; )
( / )
sin
tg
+
+
1
1
1
2/
/
( ; ) cos
n
q q
I z t kt dt
π
∫
.
Vvedem vspomohatel\n¥e funkcyy
Φ( )( , , )1 z t n =
I z t
t
t n
t n
( ; )
( / )
, [ , ],
, [ , ] [ , ],
/
\ /
2 2
1
0 1
tg
∈
∈ −
π
π π π
Φ( )( , , )2 z t n =
1
2
1
0 1
I z t t n
t n
( ; ), [ , ],
, [ , ] [ , ],
/
\ /
∈
∈ −
π
π π π
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
190 R. A. LASURYQ
Φ( )( , , )i z t n+ 2π = Φ( )( , , )i z t n , i = 1, 2.
V prynqt¥x oboznaçenyqx poluçaem ravenstva
I zn
q
,
( ) ( )2 =
1
4
1 1
0
1
1
π π π
π
n
z t n kt dt
k
n
=
−
−
∑ ∫
Φ( )( , , ) sin +
+
1 2
1
π π
π
−
∫
Φ( )
/
( , , ) cosz t n kt dt
q q
=
=
1
4
1
0
1
1 2
1
π n
b a
k
n
k k
q
q
=
−
∑ +
( ) ( )( ) ( )
/
Φ Φ ,
hde ak ( ϕ ), bk ( ϕ ) — koπffycyent¥ Fur\e funkcyy ϕ ( ⋅ ).
Yspol\zuq neravenstvo Mynkovskoho, ymeem
I zn
q
,
( ) ( )2 ≤
1
4
1
0
1
1
1
π n
b
k
n
k
q
q
=
−
∑
( )( )
/
Φ +
+
1
4
1
0
1
2
1
π n
a
k
n
k
q
q
=
−
∑
( )( )
/
Φ = i z i zn
q
n
q
,
( )
,
( )( ) ( )1 2+ . (9)
Uçyt¥vaq neravenstvo
2
2
tg
t
≥ t, 0 ≤ t < π,
sootnoßenye (7) y uslovye (3), v sylu teorem¥ Xausdorfa – Gnha [6, c. 153]
naxodym
i zn, ( )1 =
1
4 1
0
1
1
1
πn
bq
k
n
k
q
q
/
( )
/
( )
=
−
∑
Φ ≤
≤
1
4 2 21
1
1
π
π
n
I z t
t
dtq
n
p p
/
/
/
( ; )
( / )∫
tg
≤
≤
K
n
z
t
dtq
n
p
t
p
p
1
1
1
1
/
/
/
( ( ))
π ω ρ∫ +
=
= K
n
z t
t t
dtq
n
p
t
p
p p
p
1
1
1
1
1
1
/
/
/
( ( ))
π γ
γ
ω ρ∫ +
−
−
≤
≤ K
n
z
n
t dtq
p
n
p
n
p p
p
1
1 1
1
1
1
1
/
/
/
/
( ( ))ω ρ
γ
π
γ+
−
− −∫
≤
≤
K q
n
z n nq n
p p p p( , )
( ( ))/ /
/ /γ ω ρ γ γ
1 1 1
1 2 1
+
− − + +{ } =
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SYL|NAQ SUMMYRUEMOST| RQDOV FABERA … 191
=
K q
n
z nq n
p( , )
( ( ))/ /
/γ ω ρ1 1 1
1 1
+
− =
= K q zn( , ) ( ( ))/γ ω ρ1 1+ , q =
p
p −1
. (10)
Analohyçno
i zn
q
,
( ) ( )2 ≤ K znω ρ( ( ))/1 1+ . (11)
Prynymaq vo vnymanye (10), (11), yz (9) poluçaem
I zn
q
,
( ) ( )2 ≤ K znω ρ( ( ))/1 1+ , z ∈ Γ, (12)
K = K ( q, γ ) .
Uçyt¥vaq (12), (8), (6), a takΩe neravenstvo dlq srednyx [7]
H f zn q, ( ; )
1
≤ H f zn q, ( ; ) , 0 < q1 ≤ q ,
pryxodym k trebuemomu sootnoßenyg (4).
3. PoloΩym teper\
H f zn q,
( )( ; )λ =
k n
k k
qu f z
=
∞
∑ λ ρ( ) ( ; ) , q > 0, (13)
hde λ = ( )( )λk u , k ∈ N, — nekotoraq posledovatel\nost\ neotrycatel\n¥x
funkcyj, zadann¥x na nekotorom mnoΩestve U, ymegwaq xotq b¥ odnu pre-
del\nug toçku.
Teorema)2. Pust\ v¥polnqgtsq vse uslovyq teorem¥I1 y λ = ( )( )λk u —
nekotoraq posledovatel\nost\ neotrycatel\n¥x funkcyj takaq, çto pry
kaΩdom fyksyrovannom u ∈ U posledovatel\nost\ çysel λk u( ) ne vozras-
taet otnosytel\no k ∈ N. Tohda dlq lgboho n ∈ N v toçke z ∈ Γ, oprede-
lqemoj ravenstvom (3),
H f zn q,
( )( ; )λ ≤ K n u z u zn
q
n
k n
k
q
kλ ω ρ λ ω ρ( ) ( ( )) ( ) ( ( ))/ /1 1 1 1+
=
∞
++
∑ , (14)
q =
p
p −1
, K = K ( q, γ ) .
Dokazatel\stvo. Predstavlqq velyçynu (13) v vyde
H f zn q,
( )( ; )λ =
i k n
n
k k
q
i
i
u f z
=
∞
=
−
∑ ∑
+
0 2
2 11
λ ρ( ) ( ; ) ,
a takΩe uçyt¥vaq, çto v sylu (4)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
192 R. A. LASURYQ
1 2 1
n
f z
k n
n
k
q
=
−
∑ ρ ( ; ) ≤ 2 1 1K zq q
nω ρ( )/ ( )+ ,
poluçaem
H f zn q,
( )( ; )λ ≤
i
n
k n
n
k
q
i
i
i
u f z
=
∞
=
−
∑ ∑
+
0
2
2
2 11
λ ρ( ) ( ; ) ≤
≤ K u n z
i
n
i q
ni i
=
∞
+∑
0
2 1 1 2
2λ ω ρ( ) ( ( ))
/
=
= K n u z u n zn
q
n
i
n
i q
ni iλ ω ρ λ ω ρ( ) ( ( )) ( ) ( ( ))/ /1 1
1
2 1 1 2
2+
=
∞
++
∑ ≤
≤ K n u zn
q
nλ ω ρ( ) ( ( ))/1 1+
+
+
i k n
n
n
q
n
i
i
i iu z
=
∞
=
−
+∑ ∑
−
1 2
2 1
2 1 1 2
1
λ ω ρ( ) ( ( ))
/
≤
≤ K n u zn
q
nλ ω ρ( ) ( ( ))/1 1+
+
+
i k n
n
k
q
k
i
i
u z
=
∞
=
−
+∑ ∑
−
1 2
2 1
1 1
1
λ ω ρ( ) ( ( ))/ =
= K n u z u zn
q
n
k n
k
q
kλ ω ρ λ ω ρ( ) ( ( )) ( ) ( ( ))/ /1 1 1 1+
=
∞
++
∑ ,
K = K ( q, γ ) .
Polahaq v (14) n = 1, naxodym
H f zq1,
( )( ; )λ ≤ K u z
k
k
q
k
=
∞
+∑
1
1 1λ ω ρ( ) ( ( ))/ , (15)
q =
p
p −1
.
Ysxodq yz neravenstva (15), moΩno poluçyt\ ocenky dlq dostatoçno ßyro-
koho spektra syl\n¥x srednyx λ-metodov summyrovanyq rqdov, v çastnosty dlq
syl\n¥x srednyx Abelq, Fejera, loharyfmyçeskyx, Valle Pussena y dr.
4. Ustanovym ocenky skorosty sxodymosty hrupp uklonenyj, opredelqem¥x
ravenstvamy
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SYL|NAQ SUMMYRUEMOST| RQDOV FABERA … 193
D f zn
( )( ; )λ =
1
1
2
n
u f z
k n
n
k k+ =
∑ λ ρ( ) ( ; ) , (16)
G f zn
( )( ; )λ =
k n
k ku f z
=
∞
∑ λ ρ( ) ( ; ) , (17)
n ∈ N .
Teorema)3. Pust\ G — oblast\ typa ( C ′ ) y posledovatel\nost\ λ =
= ( )( )λk u , k ∈ N, u ∈ U, takaq, çto pry kaΩdom fyksyrovannom u ∈ U çysla
λk u( ) ne vozrastagt. Tohda dlq lgboj f ( z ) ∈ C GA( ) pry vsex n ∈ N y z ∈ Γ
v¥polnqetsq neravenstvo
D f zn
( )( ; )λ ≤ K
u
n
z
t
dtn
n
tλ ω ρπ
( ) ( ( ))
/1
1
2∫ + , (18)
hde K — poloΩytel\naq postoqnnaq, ne zavysqwaq ot n y f ∈ C GA( ).
Zameçanye. Polahaq
V f zm
m2 ( ; ) =
1
1
2
m
S f z
k m
m
k+ =
∑ ( ; ),
vydym, çto V f zm
m2 ( ; ) , m ∈ N, est\ summa Valle Pussena V f zn p
n
− ( ; ), v kotoroj
n = 2m, p = m. Tohda v uslovyqx teorem¥I3, v sylu (18), pry λk u( ) ≡ 1 dlq
lgboj f ( z ) ∈ C GA( ) y z ∈ Γ
f z V f zm
m( ) ( ; )− 2 ≤
K
m
z
t
dt
m
t
1
1
2
/
( ( ))
π ω ρ∫ +
. (19)
Ocenka (19) ranee ustanovlena v [5].
Dokazatel\stvo teorem¥)3. Kak y preΩde, dlq velyçyn¥ (16) ymeem
predstavlenye
D f zn
( )( ; )λ =
1
1 4
2
2
0n
u
i
D t
F F t
z
d dt
k n
n
k
k+
− 〈 〉
−
=
∑ ∫ ∫λ
π
ζ ζ
ζ
ζ
π
( )
( )
( ) ( )
Γ
≤
≤
1
1 4
2
2
0
1
n
u
i
D t I z t dt
k n
n
k
n
k+ =
∑ ∫λ
π
( )
( ) ( ; )
/
+
+
1
1 4
2
2
1n
u
i
D t I z t dt
k n
n
k
n
k+ =
∑ ∫λ
π
π
( )
( ) ( ; )
/
=
= D f z D f zn n,
( )
,
( )( ; ) ( ; )1 2
λ λ+ . (20)
Yspol\zuq ocenku (7), naxodym
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
194 R. A. LASURYQ
D f zn,
( )( ; )1
λ ≤
K
n
u n z dt
k n
n
k
n
t+ =
+∑ ∫1
2
0
1
1λ ω ρ( ) ( ( ))
/
≤
≤ K u zn nλ ω ρ( ) ( ( ))/1 1+ . (21)
Prymenqq preobrazovanye Abelq
k n
n
k u k t
=
∑ +
2
1 2λ ( ) sin ( )/ ≤
K u
t
nλ ( )
, (22)
s uçetom (7) poluçaem
D f zn,
( )( ; )2
λ =
1
4 1 2 2
1 22
1
2
π
λ
π
( )
( ; )
( / )
( ) sin( )
/
/
n
I z t
t
u k t dt
n k n
n
k+
+∫ ∑
=sin
≤
≤
K u
n
z u k t dtn
n
t
k n
n
k
λ ω ρ λ
π
( )
( ( )) ( ) sin ( )
/
/+
+∫ ∑+
=1
1 2
1
1
2
≤
≤
K u
n
z
t
dtn
n
tλ ω ρπ
( ) ( ( ))
/1
1
2∫ +
. (23)
Sohlasno (21) y (23) yz (20) naxodym
D f zn
( )( ; )λ ≤ K u z
n
z
t
dtn n
n
tλ ω ρ ω ρπ
( ) ( ( ))
( ( ))
/
/
1 1
1
1
2
1
+
++
∫ . (24)
Zameçaq, çto
1
1
1
2n
z
t
dt
n
t
/
( ( ))
π ω ρ∫ + ≥
1
1 1
1
2
n
z t dtn
n
ω ρ
π
( ( ))/
/
+
−∫ ≥
≥
π
π
ω ρ−
+
1
1 1( ( ))/n z ,
yz (24) okonçatel\no v¥vodym
D f zn
( )( ; )λ ≤
K u
n
z
t
dtn
n
tλ ω ρπ
( ) ( ( ))
/1
1
2∫ + .
Vvedem v rassmotrenye sledugwug velyçynu:
Ωk ( z ) = sup
( ( ))
/m k m
t
m
z
t
dt
≥
+∫1
1
1
2
π ω ρ
. (25)
Ysxodq yz teorem¥I3, dokaΩem spravedlyvost\ takoho utverΩdenyq.
Teorema)4. Pust\ G — oblast\ typa ( C ′ ) y posledovatel\nost\ λ =
= ( λk ( u ) ) pry kaΩdom fyksyrovannom u ∈ U ne vozrastaet. Tohda dlq
lgboj f ( z ) ∈ C GA( ) pry vsex n ∈ N y z ∈ Γ v¥polnqetsq neravenstvo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SYL|NAQ SUMMYRUEMOST| RQDOV FABERA … 195
G f zn
( )( ; )λ ≤ K n u z u zn n
k n
k kλ λ( ) ( ) ( ) ( )Ω Ω+
=
∞
∑ , (26)
hde K — poloΩytel\naq postoqnnaq, ne zavysqwaq ot z ∈ Γ, n ∈ N, u ∈ U y
f ∈ C GA( ), a G f zn
( )( ; )λ
— velyçyna, opredelqemaq ravenstvom (17).
Dokazatel\stvo. Zametym, çto pry kaΩdom fyksyrovannom z ∈ Γ vely-
çyna Ωk z( ) ne vozrastaet po k ∈ N. Dalee, yspol\zuq sootnoßenye (18), ymeem
G f zn
( )( ; )λ =
i k n
n
k k
i
i
u f z
=
∞
=
−
∑ ∑
+
0 2
2 11
λ ρ( ) ( ; ) ≤
≤
i k n
n
k k
i
i
u f z
=
∞
=
−
∑ ∑
+
0 2
2 11
λ ρ( ) ( ; ) ≤
≤ K n
u
n
z
t
dt
i
i n
i
n
ti
i=
∞
+∑ ∫
0
2
1 2
1
22
2
( )
( ) ( ( ))
/
λ ω ρπ
=
= K u
z
t
dt
i
n
n
t
i
i=
∞
+∑ ∫
0
2
1 2
1
2λ ω ρπ
( )
( ( ))
/
=
= K u
z
t
dtn
n
tλ ω ρπ
( )
( ( ))
/1
1
2∫ +
+
+ 2 2 1
21
2
1
1 2
1
2
i
n
i
i
n
t
i
i
u n
n
z
t
dt
=
∞
− +∑ ∫
λ ω ρπ
( )
( ( ))
/
≤
≤ K u
z
t
dtn
n
t
1
1
1
2λ ω ρπ
( )
( ( ))
/
∫ +
+
+
i k n
n
n i
n
t
i
i
i
i
u
n
z
t
dt
=
∞
=
−
+∑ ∑ ∫
−
1 2
2 1
2
1 2
1
2
1
1
2
λ ω ρπ
( )
( ( ))
/
.
Uçyt¥vaq opredelenye (25) velyçyn¥ Ωk ( z ) , poluçaem
G f zn
( )( ; )λ ≤ K λ ω ρπ
n
n
tu
z
t
dt( )
( ( ))
/1
1
2∫ +
+
+
i k n
n
n
m n m
t
i
i
i
i
u
m
z
t
dt
=
∞
=
−
≥
+∑ ∑ ∫
−
1 2
2 1
2
2 1
1
2
1
1λ ω ρπ
( ) sup
( ( ))
/
=
= K u
z
t
dt u zn
n
t
i k n
n
n n
i
i
i iλ ω ρ λ
π
( )
( ( ))
( ) ( )
/1
1
2
1 2
2 1
2 2
1
∫ ∑ ∑+
=
∞
=
−
+
−
Ω ≤
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
196 R. A. LASURYQ
≤ K n u
m
z
t
dtn
m n m
tλ ω ρπ
( ) sup
( ( ))
/≥
+∫
1
1
1
2 +
+
k n
k ku z
=
∞
∑
λ ( ) ( )Ω .
TeoremaI4 dokazana.
Polahaq v (26) n = 1, ymeem
k
k ku f z
=
∞
∑
1
λ ρ( ) ( ; ) ≤ K u z
k
k k
=
∞
∑
1
λ ( ) ( )Ω . (27)
Pust\
U f z( )( ; )λ = U f z u( )( ; ; )λ =
k
k ku S f z
=
∞
∑
1
λ ( ) ( ; )
y dopolnytel\no v¥polneno uslovye
k
k u
=
∞
∑
1
λ ( ) = 1 ∀ u ∈ U .
Tohda v uslovyqx teorem¥I4 s uçetom ocenky (27) poluçaem neravenstvo
f z U f z( ) ( ; )( )− λ ≤ K u z
k
k k
=
∞
∑
1
λ ( ) ( )Ω , z ∈ Γ . (28)
Na osnovanyy sootnoßenyq (28) moΩno poluçyt\ ocenky uklonenyj nekotor¥x
lynejn¥x srednyx summ Fabera, v tom çysle poroΩdaem¥x beskoneçn¥my prq-
mouhol\n¥my matrycamy λ = λk
n( )( ) , k, n ∈ N , neotrycatel\n¥x çysel. Pola-
haq, naprymer, λk
r( ) = ( )1 1− −r rk , 0 < r < 1, poluçaem ocenku uklonenyq sred-
nyx Abelq
A f zr( ; ) = ( ) ( ; )1
1
1−
=
∞
−∑r r S f z
k
k
k ,
pry
λk
n( ) = ( )ln ( )k n + −1 1, 1 ≤ k ≤ n , λk
n( ) = 0, k > n ,
— ocenku uklonenyq loharyfmyçeskyx srednyx
L f zn( ; ) = 1
1
1
1ln( )
( ; )
n k
S f zk
k
n
+ =
∑
y t. d.
1. Dzqd¥k V. K. Vvedenye v teoryg ravnomernoho pryblyΩenyq funkcyj polynomamy. – M.:
Nauka, 1966. – 672 s.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SYL|NAQ SUMMYRUEMOST| RQDOV FABERA … 197
2. Dzqd¥k V. K., Alybekov H. A. Summyrovanye rqdov Fabera lynejn¥my metodamy Ryssa y
Fejera v oblastqx s kusoçno-hladkoj hranycej. – Kyev, 1989. – 54 s. – (Preprynt / AN
USSR. Yn-t matematyky; 89.41).
3. Markußevyç A. Y. Teoryq analytyçeskyx funkcyj: V 2 t. – M.: Nauka, 1968. – T. 2. – 624 s.
4. Lebedev N. A., Íyrokov N. A. O ravnomernom pryblyΩenyy funkcyj na zamknut¥x
mnoΩestvax, ymegwyx koneçnoe çyslo uhlov¥x toçek s nenulev¥my vneßnymy uhlamy //
Yzv. AN ArmSSR. – 1971. – 6, # 47. – S. 311 – 341.
5. Alybekov H. A., Trofymenko V. Y. Summyrovanye rqdov Fabera metodamy ValleIPussena,
Rohozynskoho y DΩeksona v oblastqx s kusoçno-hladkoj hranycej // Yssledovanyq po
teoryy pryblyΩenyq funkcyj. – Kyev: Yn-t matematyky AN USSR, 1991. – S. 4 – 12.
6. Zyhmund A. Tryhonometryçeskye rqd¥: V 2 t. – M.: Myr, 1965. – T. 2. – 537 s.
7. Xardy H., Lyttlvud D., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. – 456 s.
Poluçeno 17.09.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
|
| id | umjimathkievua-article-3586 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:18Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/43/401b219996eafdd81982ae09067f2743.pdf |
| spelling | umjimathkievua-article-35862020-03-18T19:59:22Z Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary Сильная суммируемость рядов Фабера и оценки скорости сходимости группы уклонений в замкнутой области с кусочно-гладкой границей Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. We establish estimates for groups of deviations of Faber series in closed domains with piecewise-smooth boundary. Встановлено оцінки груп відхилень pядів Фабера в замкнених областях iз кусково-гладкою межею. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3586 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 187–197 Український математичний журнал; Том 57 № 2 (2005); 187–197 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3586/3904 https://umj.imath.kiev.ua/index.php/umj/article/view/3586/3905 Copyright (c) 2005 Lasuriya R. A. |
| spellingShingle | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary |
| title | Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary |
| title_alt | Сильная суммируемость рядов Фабера и оценки скорости сходимости группы уклонений в замкнутой области с кусочно-гладкой границей |
| title_full | Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary |
| title_fullStr | Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary |
| title_full_unstemmed | Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary |
| title_short | Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary |
| title_sort | strong summability of faber series and estimates for the rate of convergence of a group of deviations in a closed domain with piecewise-smooth boundary |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3586 |
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