Approximation of classes of analytic functions by a linear method of special form
On classes of convolutions of analytic functions in uniform and integral metrics, we find asymptotic equations for the least upper bounds of deviations of trigonometric polynomials generated by certain linear approximation method of a special form.
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508652408930304 |
|---|---|
| author | Serdyuk, A. S. Chaichenko, S. O. Сердюк, А. С. Чайченко, С. О. |
| author_facet | Serdyuk, A. S. Chaichenko, S. O. Сердюк, А. С. Чайченко, С. О. |
| author_sort | Serdyuk, A. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:34:07Z |
| description | On classes of convolutions of analytic functions in uniform and integral metrics,
we find asymptotic equations for the least upper bounds of deviations of trigonometric polynomials generated by certain linear approximation method of a special form. |
| first_indexed | 2026-03-24T02:28:37Z |
| format | Article |
| fulltext |
UDK 517.5
A. S. Serdgk (In-t matematyky NAN Ukra]ny, Ky]v),
S. O. Çajçenko (Slov’qn. ped. un-t)
NABLYÛENNQ KLASIV ANALITYÇNYX FUNKCIJ
LINIJNYM METODOM SPECIAL|NOHO VYHLQDU
On classes of convolutions of analytic functions in uniform and integral metrics, we find asymptotic
equations for the least upper bounds of deviations of trigonometric polynomials generated by certain
linear approximation method of a special form.
Najden¥ asymptotyçeskye ravenstva dlq toçn¥x verxnyx hranej otklonenyj tryhonometryçe-
skyx polynomov, poroΩdaem¥x lynejn¥m metodom pryblyΩenyq specyal\noho vyda, na klassax
svertok analytyçeskyx funkcyj v ravnomernoj y yntehral\noj metrykax.
Nexaj Lp , 1 ≤ p < ∞, — prostir 2π-periodyçnyx sumovnyx u p-mu stepeni na
periodi funkcij f, norma u qkomu vyznaça[t\sq formulog
f Lp
= f p = f t dtp
p
( )
/
0
2 1π
∫
;
L∞ — prostir 2π-periodyçnyx vymirnyx i sutt[vo obmeΩenyx funkcij f z
normog
f L∞
= f ∞ = ess sup ( )
t
f t ;
C — prostir neperervnyx 2π-periodyçnyx funkcij f, norma v qkomu zada[t\sq
rivnistg
f C = max
t
f t( ) .
Nexaj, dali, f L∈ 1 i
S f[ ] =
a0
2
+ ( cos sin ) ( ; )a kx b kx A f xk k
k
k
k
+ =
=
∞
=
∞
∑ ∑
1 0
df
— ]] rqd Fur’[. Qkwo rqd
1
2 21 ψ
β
π
β
π
( )
cos sin
k
a kx b kx
k
k k
=
∞
∑ +
+ +
=
=
cos
( )
( ; )
β π
ψ
2
1 k
A f x
k
k
=
∞
∑ –
sin
( )
( ; )
β π
ψ
2
k
A f xk
� ,
�A f xk ( ; ) = a kxk sin – b kxk cos ,
de ψ = ψ( )k — fiksovana poslidovnist\ dijsnyx çysel, β ∈R , [ rqdom Fur’[
deqko] sumovno] funkci], to ]] nazyvagt\ (ψ, β)-poxidnog funkci] f ( )⋅ i po-
znaçagt\ fβ
ψ . MnoΩynu usix funkcij z L1, wo magt\ (ψ, β)-poxidni, pozna-
çagt\ çerez Lβ
ψ
. Qkwo f L∈ β
ψ
i, krim toho, fβ
ψ ∈N , de N — deqka pidmno-
Ωyna z L1
0 = f f L: ∈{ 1 , f ⊥ }1 , to pyßut\ f L∈ β
ψ
N . Krim toho, vvaΩagt\,
© A. S. SERDGK, S. O. ÇAJÇENKO, 2011
102 ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
NABLYÛENNQ KLASIV ANALITYÇNYX FUNKCIJ LINIJNYM METODOM … 103
wo Cβ
ψ = C L∩ β
ψ , Cβ
ψ
N = C L∩ β
ψ
N . U ramkax dano] roboty rol\ N vidihra-
vatymut\ mnoΩyny
U p
0 = ϕ ϕ∈ ≤{ }Lp p
0 1: , 1 ≤ p ≤ ∞.
Pry c\omu pokladatymemo L U pβ
ψ 0 = L pβ
ψ
, C U Cp pβ
ψ
β
ψ0 =( ), . Klasyfikacig 2π-
periodyçnyx funkcij za dopomohog ( ψ, β)-poxidnyx zaproponuvav O.HI.HStepa-
nec\ (dyv., napryklad, [1, s. 131]).
V [1, s. 135] pokazano, wo u vypadku, koly
ψ
βπ
( ) cosk kt
k=
∞
∑ −
1 2
[ rqdom Fur’[ deqko] sumovno] funkci] Ψβ( )t , funkci] f (x) z mnoΩyny Lβ
ψ
majΩe pry vsix x moΩut\ buty podani u vyhlqdi zhortky
f x( ) = A0 +
1
0
2
π β
ψ
π
βf x t t dt∫ −( ) ( )Ψ . (1)
U danij roboti budemo vvaΩaty, wo poslidovnosti ψ( )k > 0, k ∈N , i
zadovol\nqgt\ umovu
lim
( )
( )k
k
k
q
→∞
+
=
ψ
ψ
1
, q ∈[ )0 1; . (2)
MnoΩynu takyx poslidovnostej budemo poznaçaty çerez Dq [2]. U c\omu vy-
padku elementamy mnoΩyn Cβ
ψ
[ 2π-periodyçni funkci] f (x), qki moΩna rehu-
lqrno prodovΩyty u smuhu Im z ≤ ln1/q kompleksno] plowyny (dyv. [1,
s. 289]).
VaΩlyvym prykladom qder, poslidovnosti koefici[ntiv ψ( )k qkyx nale-
Ωat\ mnoΩyni Dq , [ qdra Puassona:
P tq, ( )β = q ktk
k=
∞
∑ −
1 2
cos
βπ
, q ∈( ; )0 1 , β ∈R . (3)
U c\omu vypadku (koly ψ( )k = qk , k ∈N , q ∈( ; )0 1 ) dlq zruçnosti pokla-
dagt\ L Lp p
q
β
ψ
β, ,= C Cp p
q
β
ψ
β, ,=( ) .
KoΩnij funkci] f iz klasu Lβ
ψ
N postavymo u vidpovidnist\ tryhonometryç-
ni polinomy U f xn−
∗
1( ; ) = U f xn−
∗
1( ; ; ; )ψ β vyhlqdu
U f xn−
∗
1( ; ) =
= A0 + λ νk
n
k k k
n
k ka kx b kx a kx b k( ) ( )( cos sin ) ( sin cos+ + − xx
k
n
){ }
=
−
∑
1
1
, (4)
de ak = a fk ( )β
ψ
, bk = b fk ( )β
ψ
, k = 1, 2, … , — koefici[nty Fur’[ funkci] fβ
ψ
,
a çysla λk
n( ) = λ ψ βk
n( )( ; ) i νk
n( ) = ν ψ βk
n( )( ; ) , k = 1, … , n – 1, n ∈N , ozna-
çagt\sq rivnostqmy
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
104 A. S. SERDGK, S. O. ÇAJÇENKO
λk
n( ) = ψ ψ ψ
βπ
( ) ( ) ( ) cosk n k n k− − − +( )2 2
2
,
k = 1 1, n − .
νk
n( ) = ψ ψ ψ
βπ
( ) ( ) ( ) sink n k n k− − + +( )2 2
2
,
Polinomy (4) zadagt\ linijnyj metod nablyΩennq, wo vyznaça[t\sq syste-
mamy çysel λk
n( )
i νk
n( )
, k = 1, … , n – 1, n ∈N . Cej metod bulo zaprovadΩeno u
[3]. U robotax [3, 4] doslidΩuvalys\ aproksymatyvni vlastyvosti vkazanoho me-
todu na klasax ( ; )ψ β -dyferencijovnyx funkcij. Zokrema, u [4] znajdeno
asymptotyçni rivnosti dlq toçnyx verxnix meΩ vidxylen\ polinomiv U f xn−
∗
1( ; )
na klasax intehraliv Puassona L p
q
β, , 1 ≤ p ≤ ∞, u rivnomirnij ta intehral\nij
metrykax.
U danij roboti vstanovleno asymptotyçni pry n → ∞ rivnosti velyçyn
E L Up n
Cβ
ψ
, ; −
∗( )1 = sup ( ) ( ; )
,f L
n
C
p
f x U f x
∈
−
∗−
β
ψ
1 , (5)
E L Un
Ls
β
ψ
, ;1 1−
∗( ) = sup ( ) ( ; )
,f L
n
s
f x U f x
∈
−
∗−
β
ψ
1
1 (6)
dlq dovil\nyx 1 ≤ p, s ≤ ∞, ψ ∈Dq i β ∈R . Zaznaçymo, wo pry vstanovlenni
asymptotyçnyx rivnostej dlq velyçyn (5), (6) budut\ vykorystani rezul\taty
roboty [4].
Teorema 1. Nexaj 1 ≤ p ≤ ∞, ψ ∈Dq , 0 < q < 1, β ∈R i n ∈N . Todi pry
n → ∞ vykonu[t\sq asymptotyçna rivnist\
E C Up n
Cβ
ψ
, ; −
∗( )1 = ψ
π
( )
cos
,n
t
M
p
p
p q p
21
1 1
/
/
′
+ ′ ′
+
+ O
q
n q qp
n( )
( ) ( )( )1
1 1 2−
+
−
σ
ε
, (7)
de
Mq p, ′ =
1
2
1
1 2
2
2
−
− + ′
q
q t q pcos
, (8)
σ( )p =
1
2 1
, ,
, ,
p
p
= ∞
≤ < ∞
εn = sup
( )
( )k n
k
k
q
≥
+
−
ψ
ψ
1
, (9)
1
p
+
1
′p
= 1, a O( )1 — velyçyna, rivnomirno obmeΩena po n, q, ψ, p i β.
Dovedennq. Nexaj ψ ∈Dq , β ∈R . Todi zhidno z lemogH2 z roboty [3,
s. 302] dlq bud\-qko] funkci] f L∈ β
ψ
N , N ⊂ L1, majΩe dlq vsix x ∈R ma[
misce zobraΩennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
NABLYÛENNQ KLASIV ANALITYÇNYX FUNKCIJ LINIJNYM METODOM … 105
f x( ) – U f xn−
∗
1( ; ) =
2
20
2
π
ϕ
βππ
( ) cos ( )x t nt t dtn− −
∫ ∗Ψ –
–
1
0
2
π
ϕ
π
β( ) ( ),x t t dtn−∫ ∗Ψ , (10)
u qkomu
Ψn
k
t
n
n k kt∗
=
∞
= + +∑( )
( )
( ) cos
df ψ
ψ
2 1
i
Ψn
k n
t k n kt, ( ) ( ) cosβ ψ
βπ∗
=
∞
= + +
∑df
2
2
,
a funkciq ϕ majΩe skriz\ zbiha[t\sq z fβ
ψ .
Zrozumilo, wo qkwo f C p∈ β
ψ
, , to rivnist\ (10) vykonu[t\sq dlq vsix x ∈R .
PokaΩemo, wo dlq bud\-qko] poslidovnosti ψ ∈Dq i n ∈N vykonugt\sq
spivvidnoßennq
Ψn t∗ ( ) = ψ( ) cos ( ),n q kt r tk
k
q n
1
2 1
+ +
=
∞
∗∑ , (11)
r tq n, ( )∗ ≤
ε
ε
n
nq q( ) ( )1 1− − −
, (12)
Ψn t, ( )β
∗ = ψ
βπ
β( ) cos ( ), ,3
2
n q q kt r tn k
k n
q n
−
=
∞
∗∑ +
+
, (13)
r tq n, , ( )β
∗ ≤
ε
ε
3
31 1
n
nq q( ) ( )− − −
,
de velyçyna εn vyznaça[t\sq formulog (9). Dlq c\oho vykorysta[mo mirku-
vannq, wo zastosovuvalys\ pry dovedenni lemyH1 z roboty [2].
Vykonugçy elementarni peretvorennq, znaxodymo
Ψn t∗ ( ) = ψ
ψ
ψ
( )
( )
( )
cosn
n k
n
kt
k
1
2 1
+
+
=
∞
∑ =
= ψ
ψ
ψ
( )
( )
( )
cosn
n j
n j
kt
j
k
k
1
2
1
0
1
1
+
+ +
+
=
−
=
∞
∏∑ =
= ψ( ) cos ( ),n q kt r tk
k
q n
1
2 1
+ +
=
∞
∗∑ , (14)
de
r tq n, ( )∗ =
ψ
ψ
( )
( )
cos
n j
n j
q ktk
j
k
k
+ +
+
−
=
−
=
∞
∏∑ 1
0
1
1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
106 A. S. SERDGK, S. O. ÇAJÇENKO
Perekona[mos\ u spravedlyvosti ocinky (12). Oçevydno, wo
r tq n, ( )∗ ≤ �q qk
k
k
−
=
∞
∑
1
, �q
n j
n j
k
j
k
=
+ +
+=
−
∏df ψ
ψ
( )
( )
1
0
1
. (15)
Qkwo, napryklad, �qk – qk ≥ 0, to
�q qk
k− = �q qk
k− ≤ ( )q qn
j
k
k+ −
=
−
∏ ε
0
1
= ( )q qn
k k+ −ε .
Qkwo Ω �qk – qk < 0, to vnaslidok opuklosti funkci] t k , k = 1, 2, … ,
�q qk
k− = q qk
k− � ≤ qk – ( )q n
j
k
−
=
−
∏ ε
0
1
=
= q qk
n
k− −( )ε ≤ ( )q qn
k k+ −ε .
OtΩe, zavΩdy
�q qk
k− ≤ ( )q n
k+ ε – qk . (16)
Oskil\ky poslidovnist\ εn monotonno spada[ do nulq, to poçynagçy z de-
qkoho nomera n0 bude vykonuvatys\ nerivnist\ εn < 1 – q. OtΩe, vraxovugçy
spivvidnoßennq (15) i (16), dlq n ≥ n0 otrymu[mo
r tq n, ( )∗ ≤ ( )q qn
k
i
k+ −
=
∞
∑ ε
1
=
ε
ε
n
nq q( ) ( )1 1− − −
,
i ocinku (12) dovedeno. Spivvidnoßennq (13) [ naslidkom lemyH1 z roboty [2].
Rivnosti (11) i (13) razom z oçevydnog ocinkog
ε
ε
n
nq q( ) ( )1 1− − −
= O
q
n( )
( )
1
1 2
ε
−
dozvolqgt\ perepysaty zobraΩennq (10) u vyhlqdi
f x( ) – U f xn−
∗
1( ; ) =
2
20
2ψ
π
ϕ
βππ
( )
( ) cos ( )
n
x t nt t dtq− −
∫ P –
–
ψ
π
ϕ
π
β
( )
( ) ( ), ,
3
0
2
n
q
x t t dt
n q n−∫ P + O
n
q
n( )
( )
( )
1
1 2
ψ ε
−
, (17)
de
Pq t( ) =df 1
2
+ q ktk
k=
∞
∑
1
cos , Pq n
k
k n
t q kt, , ( ) cosβ
βπ
= +
=
∞
∑df
2
.
Zvidsy, vnaslidok invariantnosti mnoΩyny U p
0
vidnosno zsuvu arhumentu,
dlq dovil\noho 1 ≤ p ≤ ∞ ma[mo
E C Up n
Cβ
ψ
, ; −
∗( )1 =
2
20
0
2ψ
π
ϕ
βπ
ϕ
π
( )
sup ( ) cos ( )
n
t nt t d
U
q
p∈
∫ −
P tt +
+ O
n
q
x t t dt
n
U
q n
Cp
( )
( )
sup ( ) ( ), ,1
3
0
0
2ψ
π
ϕ
ϕ
π
β
∈
− +∫ P
ψψ ε( )
( )
n
q
n
1 2−
. (18)
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
NABLYÛENNQ KLASIV ANALITYÇNYX FUNKCIJ LINIJNYM METODOM … 107
Dlq zaverßennq dovedennq teoremyH1 zalyßylos\ skorystatysq rezul\tata-
my roboty [4, s. 978 – 980], zhidno z qkymy
sup ( ) cos ( )
ϕ
π
ϕ
βπ
∈
∫ −
U
q
p
t nt t dt
0
0
2
2
P = inf cos ( )
λ
βπ
λ
∈ ′
−
−
R
nt tq
p2
P =
=
cos
( )
( )
t
t
p
p q p
′
′ ′2 1π / P + O
q
n q p
( )
( ) ( )1
1 − σ , (19)
de
σ( )p =
1
2 1
, ,
, ,
p
p
= ∞
≤ < ∞
i
sup ( ) ( ), ,
ϕ
π
βϕ
∈
−∫
U
q n
Cp
x t t dt
0
0
2
P = O
q
q
n
( )1
1 −
.
TeoremuH1 dovedeno.
Teorema 2. Nexaj 1 ≤ p ≤ ∞ , ψ ∈Dq , 0 < q < 1, β ∈R i n ∈N . Todi pry
n → ∞ vykonu[t\sq asymptotyçna rivnist\
E C Un
Lp
β
ψ
, ;1 1−
∗( ) = ψ
π
( )
cos
,n
t
M
p
p
p q p
21 1
1 1
−
+
/
/
+
+ O
q
n q qp
n( )
( ) ( )( )
1
1 1 2−
+
−
′σ
ε
, (20)
de Mq p, , σ( )p ta εn vyznaçagt\sq za dopomohog formul (8) ta (9), a ve-
lyçyna O( )1 rivnomirno obmeΩena po n, q, ψ, p i β.
Dovedennq. Vyxodqçy iz zobraΩennq (17), dlq dovil\noho 1 ≤ p ≤ ∞ moΩe-
mo zapysaty rivnist\
E C Un
Lp
β
ψ
, ;1 1−
∗( ) =
2
2
1
0
0
2ψ
π
ϕ
βπ
ϕ
π
( )
sup ( ) cos (
n
x t nt t
U
q
∈
− −
∫ P )) dt
p
+
+ O
n
q
x t t dt
n
U
q n
p
( )
( )
sup ( ) ( ), ,1
3
1
0
0
2ψ
π
ϕ
ϕ
π
β
∈
− +∫ P
ψψ ε( )
( )
n
q
n
1 2−
. (21)
Vykorystovugçy dali rezul\taty roboty [4, s. 981, 982], zhidno z qkymy
sup ( ) cos ( )
ϕ
π
ϕ
βπ
∈
− −
∫
U
q
p
x t nt t dt
1
0
0
2
2
P =
=
cos
( )
( )
t
t
p
p q p2 1π / P + O
q
n q p
( )
( ) ( )
1
1 − ′σ
(22)
i
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
108 A. S. SERDGK, S. O. ÇAJÇENKO
sup ( ) ( ), ,
ϕ
π
βϕ
∈
−∫
U
q n
p
x t t dt
1
0
0
2
P = O
q
q
n
( )1
1 −
,
perekonu[mos\ u spravedlyvosti formuly (20).
TeoremuH2 dovedeno.
Qk neskladno perekonatysq, umovam teorem 1 i 2 zadovol\nqgt\ moduli koe-
fici[ntiv Fur’[ poliharmoniçnyx qder Puassona vyhlqdu
P m tq, ( , )β = ψ
βπ
m
k
k kt( ) cos
=
∞
∑ −
1 2
, β ∈R , m ∈N , (23)
de
ψm k( ) = q
q
j
k lk
j
j
j
m
l
j( )
!
( )
1
2
2
2
0
1
0
1−
+
=
−
=
−
∑ ∏ , q ∈( , )0 1 (24)
(dyv. [5, s. 256, 257]). Dlq koefici[ntiv ψ( )k = ψm k( ) , m = 2, 3, … , vyhlqdu
(24) moΩemo zapysaty
εn = sup
( )
( )k n
m
m
k
k
q
≥
+
−
ψ
ψ
1
=
= q
q
j
k l
k l
k l
k n
j
j l
j
sup
( )
!
( )
≥
=
−− + + +
+
−∏1
2
2
1 2
2
1
2
0
1
ll
j
j
m
j
j l
jq
j
k l
=
−
=
−
=
−
∏∑
− +
0
1
0
1
2
0
1
2
2
( )
!
( )
11
0
1 ∏∑ =
−
j
m
≤
≤ q
k l
k lk n l
m
sup
≥ =
− + +
+
−
∏ 1 2
2
1
0
2
≤
( )2 3m q
n
−
, n ∈N . (25)
Pry m = 1 ma[mo ψm k( ) = qk
i, otΩe,
εn ≡ 0 , m = 1. (26)
Iz teorem 1 ta 2 i spivvidnoßen\ (25) ta (26) oderΩu[mo nastupne tverdΩennq.
Naslidok. Nexaj 1 ≤ p ≤ ∞ i klasy C pβ
ψ
, ta Lβ
ψ
,1 porodΩugt\sq posli-
dovnostqmy ψ( )k = ψm k( ) vyhlqdu (24 ). Todi pry n → ∞ vykonugt\sq
asymptotyçni rivnosti
E C Up n
Cβ
ψ
, ; −
∗( )1 = q
q
j
n ln
j
j
j
m
l
j( )
!
( )
1
2
2
2
0
1
0
1−
+
=
−
=
−
∑ ∏ ×
×
2
1
1
1
1 1
/
/
p
p
p q p p m
t
M O
mq
n q
cos
( )
( )
, ( , )
′
+ ′ ′ ′+
−
π σ
, (27)
E L Un
Lp
β
ψ
, ;1 1−
∗( ) = q
q
j
n ln
j
j
j
m
l
j( )
!
( )
1
2
2
2
0
1
0
1−
+
=
−
=
−
∑ ∏ ×
×
2
1
1
1 1
1 1
−
+ +
−
/
/
p
p
p q p p m
t
M O
mq
n q
cos
( )
( )
, ( , )π σ
, (28)
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
NABLYÛENNQ KLASIV ANALITYÇNYX FUNKCIJ LINIJNYM METODOM … 109
v qkyx
1
p
+
1
′p
= 1,
σ( , )p m =
1 1 1
2
1 1
1
, , ,
,
,
,
,
qkwo
qkwo
m p
m
m
p
= =
=
∈ { }
< ≤ ∞
N\ 1 ≤≤ ≤ ∞
p ,
velyçyny Mq p, ′ ta Mq p, oznaçagt\sq formulog (8), a O( )1 — velyçyny,
rivnomirno obmeΩeni po n, q, p, β i m.
Pry m = 1 formuly (27) ta (28) oderΩano v roboti [4, s. 977, 980, 981].
1. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj. – Kyev: Yn-t matematyky NAN Ukrayn¥,
2002. – T. I. – 427 s.
2. Stepanec A. Y., Serdgk A. S. PryblyΩenyq summamy Fur\e y nayluçßye pryblyΩenyq na
klassax analytyçeskyx funkcyj // Ukr. mat. Ωurn. – 2000. – 52, # 3. – S. 375 – 395.
3. Serdgk A. S. Pro odyn linijnyj metod nablyΩennq periodyçnyx funkcij // Problemy teo-
ri] nablyΩennq funkcij ta sumiΩni pytannq: Zb. prac\ In-tu matematyky NAN Ukra]ny. –
2004. – 1, # 1. – S. 295 – 336.
4. Serdgk A. S. NablyΩennq intehraliv Puassona odnym linijnym metodom nablyΩennq v riv-
nomirnij ta intehral\nij metrykax // Ukr. mat. Ωurn. – 2008. – 60, # 7. – S. 976 – 982.
5. Tyman M. F. Approksymacyq y svojstva peryodyçeskyx funkcyj. – Kyev: Nauk. dumka,
2009. – 376 s.
OderΩano 07.07.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
|
| id | umjimathkievua-article-2701 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:28:37Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0c/f64f9b398f9223889c2b2712aaeed20c.pdf |
| spelling | umjimathkievua-article-27012020-03-18T19:34:07Z Approximation of classes of analytic functions by a linear method of special form Наближення класів аналітичних функцій лінійним методом спеціального вигляду Serdyuk, A. S. Chaichenko, S. O. Сердюк, А. С. Чайченко, С. О. On classes of convolutions of analytic functions in uniform and integral metrics, we find asymptotic equations for the least upper bounds of deviations of trigonometric polynomials generated by certain linear approximation method of a special form. Найдены асимптотические равенства для точных верхних граней отклонений тригонометрических полиномов, порождаемых линейным методом приближения специального вида, на классах сверток аналитических функций в равномерной и интегральной метриках. Institute of Mathematics, NAS of Ukraine 2011-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2701 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 1 (2011); 102-109 Український математичний журнал; Том 63 № 1 (2011); 102-109 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2701/2158 https://umj.imath.kiev.ua/index.php/umj/article/view/2701/2159 Copyright (c) 2011 Serdyuk A. S.; Chaichenko S. O. |
| spellingShingle | Serdyuk, A. S. Chaichenko, S. O. Сердюк, А. С. Чайченко, С. О. Approximation of classes of analytic functions by a linear method of special form |
| title | Approximation of classes of analytic functions by a linear method of special form |
| title_alt | Наближення класів аналітичних функцій лінійним методом спеціального вигляду |
| title_full | Approximation of classes of analytic functions by a linear method of special form |
| title_fullStr | Approximation of classes of analytic functions by a linear method of special form |
| title_full_unstemmed | Approximation of classes of analytic functions by a linear method of special form |
| title_short | Approximation of classes of analytic functions by a linear method of special form |
| title_sort | approximation of classes of analytic functions by a linear method of special form |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2701 |
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