Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics
We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces C and Lp .
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| Datum: | 2008 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509262480932864 |
|---|---|
| author | Serdyuk, A. S. Сердюк, А. С. |
| author_facet | Serdyuk, A. S. Сердюк, А. С. |
| author_sort | Serdyuk, A. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:48:23Z |
| description | We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces C and Lp . |
| first_indexed | 2026-03-24T02:38:18Z |
| format | Article |
| fulltext |
UDK 517.5
A. S. Serdgk (In-t matematyky NAN Ukra]ny, Ky]v)
NABLYÛENNQ INTEHRALIV PUASSONA
ODNYM LINIJNYM METODOM NABLYÛENNQ
V RIVNOMIRNIJ TA INTEHRAL|NYX METRYKAX
We find asymptotic equalities for the least upper bounds of approximations of classes of the Poisson
integrals of periodic functions by a certain linear approximation method of special form in metrics of the
spaces C and L p .
Najden¥ asymptotyçeskye ravenstva dlq toçn¥x verxnyx hranej pryblyΩenyj klassov ynteh-
ralov Puassona peryodyçeskyx funkcyj nekotor¥m lynejn¥m metodom pryblyΩenyq specy-
al\noho vyda v metrykax prostranstv C y L p .
Nexaj L p , 1 ≤ p < ∞, — prostir 2π-periodyçnyx sumovnyx u p-mu stepeni
funkcij f z normog
f p = f Lp
= f t dtp
p
( )
0
2 1π
∫
;
L∞ — prostir 2π-periodyçnyx, vymirnyx i sutt[vo obmeΩenyx funkcij, u qkomu
normu zadano formulog
f ∞ = ess sup ( )
t
f t ;
C — prostir 2π-periodyçnyx neperervnyx funkcij, normu v qkomu zadano takym
çynom:
f C = max ( )
t
f t .
Intehralamy Puassona sumovno] funkci] ϕ( )⋅ nazyvagt\ funkci] f x( ), wo
oznaçagt\sq za dopomohog rivnosti
f x( ) =
A0
2
+ 1
0
2
π
ϕ β
π
( – ) ( ),x t P t dtq∫ , A0 ∈R , (1)
u qkij P tq, ( )β — qdra Puassona z parametramy q ∈ (0, 1) i β ∈R, tobto funkci]
vyhlqdu
P tq, ( )β = q ktk
k
cos –
βπ
21
=
∞
∑ , q ∈ (0, 1), β ∈R.
MnoΩynu vsix funkcij, qki dopuskagt\ zobraΩennq u vyhlqdi (1) pry ϕ ∈
∈ �, de � — deqka pidmnoΩyna iz L1, poznaçatymemo çerez
L q
β�. V ramkax
dano] roboty rol\ � vidihravatymut\ mnoΩyny
Up
0 = ϕ ϕ ϕ∈ ≤ ⊥{ }L p p: ,1 1 .
Pry c\omu dlq zruçnosti poklademo L L Up
q q
pβ β, =df 0
.
KoΩnij funkci] f iz klasu
L q
β� postavymo u vidpovidnist\ tryhonometryç-
© A. S. SERDGK, 2008
976 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
NABLYÛENNQ INTEHRALIV PUASSONA ODNYM LINIJNYM METODOM … 977
nyj polinom U f xn – ( ; )1
∗
vyhlqdu
U f xn – ( ; )1
∗ =
A0
2
+ λk
n
k k
k
n
a k x b k x( )
–
cos sin+( ){
=
∑
1
1
+ νk
n
k ka k x b k x( ) sin – cos( )},
de ak = ak ( )ϕ , bk = bk ( )ϕ , k = 1, 2, … , — koefici[nty Fur’[ funkci] ϕ, a çys-
la λk
n( ) = λk
n( ) ( ; )q β i νk
n( ) = νk
n( ) ( ; )q β , k = 1, 2, … , n – 1, n ∈N , oznaçagt\sq za
dopomohog rivnostej
λk
n( ) = q q qk n k n k– – cos–2 2
2
+( ) βπ
,
νk
n( ) = q q qk n k n k– sin–2 2
2
+( )+ βπ
.
Polinom U f xn – ( ; )1
∗
moΩna rozhlqdaty qk linijnyj metod nablyΩennq, wo vy-
znaça[t\sq systemog çysel λ1
( )n{ , … , λn
n
–
( )
1, ν1
( )n , … , νn
n
–
( )
1}. Uperße cej metod
(u bil\ß zahal\nomu vypadku) bulo rozhlqnuto v roboti [1]; tam Ωe bulo dos-
lidΩeno deqki aproksymatyvni vlastyvosti vkazanoho metodu na zaprovadΩenyx
O.II.IStepancem [2, s.I33] klasax ( , )ψ β -dyferencijovnyx funkcij. Zokrema, v
[1] dovedeno, wo dlq deqkyx klasiv neskinçenno dyferencijovnyx funkcij da-
nyj metod [ najkrawym (v sensi syl\no] asymptotyky) linijnym metodom nably-
Ωen\ tryhonometryçnymy polinomamy v rivnomirnij metryci. Analohiçnyj re-
zul\tat ma[ misce i dlq nablyΩen\ u metryci prostoru L1.
U danij roboti vstanovymo asymptotyçni rivnosti dlq velyçyn
E L Up
q
n Cβ, –; 1
∗( ) = sup ( ) – ( ; )
,
–
f L
n C
p
q
f x U f x
∈
∗
β
1 ,
E L Uq
n Ls
β, –;1 1
∗( ) = sup ( ) – ( ; )
,
–
f L
n sq
f x U f x
∈
∗
β 1
1
pry dovil\nyx 1 ≤ p, s ≤ ∞.
Teorema 1. Nexaj 1 ≤ p ≤ ∞, q ∈ (0, 1), β ∈R i n ∈N . Todi pry n → ∞
vykonu[t\sq asymptotyçna rivnist\
E L Up
q
n Cβ, –; 1
∗( ) = q
t
M O
q
n q
n
p
p
p q p p
2
1
1
1
1 1
/
/ , ( )
cos
( )
( – )
′
+ ′ ′ +
π σ ,
de ′p =
p
p – 1
,
Mq p, ′ =
1
2
1
1 2
2
2
–
– cos
q
q t q p+ ′
, (2)
σ( )p =
1
2 1
, ,
, ,
p
p
= ∞
≤ < ∞
a velyçyna O(1) rivnomirno obmeΩena po n, q, p i β.
Dovedennq. Zhidno z lemogI2 z roboty [1, s. 302], dlq bud\-qko] funkci]
f ∈
L q
β�, � ⊂ L1 , ma[ misce intehral\ne zobraΩennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
978 A. S. SERDGK
f x( ) – U f xn – ( ; )1
∗ =
=
2
2
0
2
q
x t nt t dt
n
qπ
ϕ βππ
( – ) cos – ( )∫
P –
q
x t t dt
n
q n
2
0
2
π
ϕ
π
β( – ) ( ), ,∫ P , (3)
v qkomu
Pq t( ) = 1
2 1
+
=
∞
∑ q ktk
k
cos , q ∈ (0, 1),
Pq n t, , ( )β = q ktk
k n
cos +
=
∞
∑ βπ
2
, q ∈ (0, 1), β ∈R.
Vnaslidok rivnosti (3) ta invariantnosti mnoΩyny Up
0
vidnosno zsuvu arhu-
mentu dlq dovil\noho 1 ≤ p ≤ ∞ ma[mo
E L Up
q
n Cβ, –; 1
∗( ) =
2
20
0
2
q
t nt t dt
n
U
q
Cp
π
ϕ βπ
ϕ
π
sup ( ) cos – ( )
∈
∫ P + Rn , (4)
de
Rn ≤
q
x t t dt
n
q n
Cp
2
1 0
2
π
ϕ
ϕ
β
π
sup ( – ) ( ), ,
≤
∫ P . (5)
Vidomo (dyv., napryklad, [3, s. 137, 138]), wo qkwo ϕ ∈L p , 1 ≤ p ≤ ∞, i
K L p∈ ′ ,
1
p
+ 1
′p
= 1, to
ϕ
π
( – ) ( )x t K t dt
C0
2
∫ ≤ ϕ p pK ′ . (6)
Zastosovugçy nerivnist\ (6) pry K t( ) = Pq n t, , ( )β , iz (5) oderΩu[mo ocinku
Rn ≤
q n
q n p
2
π βP , , ( )⋅ ′ ′p =
p
p – 1
. (7)
U roboti [4, s. 1083, 1087, 1088] oderΩano rezul\taty, z qkyx pry 1 ≤ ′p ≤ ∞
vyplyvagt\ rivnosti
P
P
P P
R
R
q n p
q n p
h
q n q n p
t
t
t h t
, ,
, ,
, , , ,
( )
inf ( ) –
sup ( ) – ( )
β
λ β
β β
λ
′
∈ ′
∈
′
+
1
2
= q
t
Zn p
p q p
cos
( ) /
′
′ ′
2 1π
+ O
q
n q s p
( )
( – ) ( )
1
1 ′
, (8)
de
Z tq( ) = 1
1 2 2– cosq t q+
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
NABLYÛENNQ INTEHRALIV PUASSONA ODNYM LINIJNYM METODOM … 979
s p( ) =
1 1
2 1
, ,
, , ,
p
p
=
∈ ∞( ]
(9)
a velyçyna O( )1 rivnomirno obmeΩena vidnosno parametriv ′p , q i n.
Vraxovugçy (7) i (8), ma[mo
Rn = O
q
q
n
( )
–
1
1
3
, (10)
de O( )1 — velyçyna, rivnomirno obmeΩena za vsima rozhlqduvanymy paramet-
ramy.
Dlq ocinky perßoho dodanka u pravij çastyni rivnosti (4) skorysta[mos\
spivvidnoßennqm dvo]stosti (dyv., napryklad, [5, s. 27]):
inf ( ) –
λ
λ
∈ ′
R
x t p = sup ( ) ( )
y Up
x t y t dt
∈
∫
0
0
2π
, 1
p
+ 1
′p
= 1, x L p∈ ′ , 1 ≤ ′p ≤ ∞.
(11)
Pokladagçy v (11) x t( ) = cos nt
–
βπ
2
Pq t( ), y t( ) = ϕ( )t , otrymu[mo
sup ( ) cos – ( )
ϕ
π
ϕ βπ
∈
∫
U
q
p
t nt t dt
0 2
0
2
P =
inf cos – ( ) –
λ
βπ λ
∈ ′
R
nt tq
p2
P . (12)
Wob znajty toçnu asymptotyçnu ocinku velyçyny inf cos –
λ
βπ
∈
R
nt
2
Pq t( ) –
– λ
′p
, skorysta[mos\ nastupnym tverdΩennqm z roboty [4, s. 1083].
Lema 1. Nexaj 1 ≤ s ≤ ∞ i 2π-periodyçni funkci] g t( ) t a h t( ) magt\
obmeΩenu variacig, qkwo s = 1, abo naleΩat\ klasu Hel\dera K H1
, qkwo
1 < s ≤ ∞. Todi dlq funkci] ϕ( )t = g t( ) cos(nt + α) + h t( ) sin(nt + α), α ∈R,
n ∈N , vykonugt\sq asymptotyçni rivnosti
ϕ s = ( ) cos– /2 1π s
s st r + O Mn( ) –1 1, (13)
inf –
c sc
∈R
ϕ = ( ) cos– /2 1π s
s st r + O Mn( ) –1 1, (14)
sup ( ) – ( )
h
st h t
∈
+
R
ϕ ϕ = ( ) cos– /2 1π s
s st r + O Mn( ) –1 1, (15)
v qkyx
r t( ) = g t h t2 2( ) ( )+ ,
M = Ms =
V V
V
– –
– –
–
( ) ( ) ,
( ) ,
,
π
π
π
π
π
π
g h s
K s r r s
K s
s
s s
+ =
+ < < ∞
= ∞
pry
pry
pry
1
11 1
a velyçyny O( )1 rivnomirno obmeΩeni vidnosno usix rozhlqduvanyx parametriv.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
980 A. S. SERDGK
Zapysugçy funkcig cos nt( –
βπ
2
Pq t( ) u vyhlqdi
cos – ( )nt tq
βπ
2
P =
cos ( ) cos
βπ
2
Pq t nt
+
sin ( ) sin
βπ
2
Pq t nt
(16)
i zastosovugçy lemuI1, pokladagçy v ]] umovax s = ′p ,
g t( ) =
cos ( )
βπ
2
Pq t , h t( ) =
sin ( )
βπ
2
Pq t ,
iz rivnosti (14) oderΩu[mo
inf cos – ( ) –
λ
βπ λ
∈
′R
nt q t
p2
P =
cos
( )
( )/
t p
q t
pp
′
′′2 1π
P + O
n p
( )1 γ ′ , (17)
de
γ ′p =
V
V
–
–
–
( ) ,
( ) ( ) ,
( ) .
π
π
π
π
P
P P P
P
q
q C q p
p
q
p
q C
p
p
p
p
pry
pry
pry
′ =
⋅ +
′
⋅ ( ) < ′ < ∞
⋅ ′ = ∞
′
′
′
′ ′
1
1 1
1
(18)
Oskil\ky, qk nevaΩko perekonatys\,
V
–
( )
π
π
Pq = 2 0P Pq q( ) – ( )π( ) =
4
1 2
q
q–
,
′ ⋅Pq C
( ) ≤ kqk
k =
∞
∑
1
=
q
q( – )1 2 ,
V
–π
π
Pq
p′( ) = 2 1 2– ( ) ( )
–
q p t h t dtq
p
q( ) ′ ∫ ′
π
π
P ≤
2 1 2– ( ) ( )q p hq C q p
p( ) ′ ⋅ ⋅ ′
′
P ,
de
h t
q t
q t q
q( )
sin
– cos
=
+
df
1 2 2 ,
to na pidstavi formul (17) i (18) ta oçevydnyx nerivnostej
hq C
( )⋅ ≤
q
q1 –
, Pq p
( )⋅ ′ ≤
( ) ( )
( – )
/2 1
2 1
1π ′ +p q
q
otrymu[mo rivnist\
inf cos – ( ) –
λ
βπ λ
∈ ′
R
nt tq
p2
P =
cos
( )
( )
/
t
tp
p q p
′
′ ′2 1π
P + O
q
n q s p
( )
( – ) ( )
1
1 ′ , (19)
u qkij s p( )′ oznaça[t\sq formulog (9), a velyçyna O( )1 rivnomirno obmeΩena
vidnosno usix rozhlqduvanyx parametriv.
Ob’[dnugçy formuly (4), (10), (12) i (19), ma[mo
E L Up
q
n Cβ, –; 1
∗( ) =
q
t
t O
q
n q
n
p
p
p q p s p
2
1
1
1
1 1
/
/ ( )
cos
( ) ( )
( – )
′
+ ′ ′ ′+
π
P .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
NABLYÛENNQ INTEHRALIV PUASSONA ODNYM LINIJNYM METODOM … 981
TeoremuI1 dovedeno.
Teorema 2. Nexaj 1 ≤ p ≤ ∞, q ∈ (0, 1), β ∈R i n ∈N . Todi pry n → ∞
vykonu[t\sq asymptotyçna rivnist\
E L Uq
n L p
β, –;1 1
∗( ) = q
t
M O
q
n q
n
p
p
p q p s p
2
1
1
1 1
1 1
– /
/ , ( )
cos
( )
( – )π + +
,
de velyçyny Mq p, ta s p( ) vyznaçagt\sq za dopomohog formul (2) i (9) vid-
povidno, a velyçyna O( )1 rivnomirno obmeΩena po n, q, p i β.
Dovedennq. Vyxodqçy iz zobraΩennq (3), dlq dovil\noho 1 ≤ p ≤ ∞ moΩemo
zapysaty rivnist\
E L Uq
n L p
β, –;1 1
∗( ) =
2
2
1
0
0
2
q
x t nt t dt
n
U
q
p
π
ϕ βπ
ϕ
π
sup ( – ) cos – ( )
∈
∫ P + R̃n , (20)
de
R̃n ≤
q
x t t dt
n
U
q n
p
2
0
2
1
0π
ϕ
ϕ
β
π
sup ( – ) ( ), ,
∈
∫ P . (21)
Vykorystovugçy tverdΩennq 1.5.5 iz [5, s. 43], zhidno z qkym pry ϕ ∈L1,
K L p∈ , 1 ≤ p ≤ ∞,
ϕ
π
( – ) ( )x t K t dt
p0
2
∫ ≤ K p ϕ 1,
iz (21) oderΩu[mo nerivnist\
R̃n ≤
q n
q n p
2
π βP , , ( )⋅ ,
qka razom iz formulamy (8) dozvolq[ zapysaty ocinku
R̃n = O
q
q
n
( )
–
1
1
3
, (22)
de O( )1 — velyçyna, rivnomirno obmeΩena za vsima rozhlqduvanymy paramet-
ramy.
Dlq ostatoçnoho dovedennq teoremy neobxidno znajty asymptotyçno toçnu
ocinku perßoho dodanka u pravij çastyni rivnosti (20).
Zhidno z lemogI1 iz roboty [6, s. 1398], qkwo K t( ) ∈ L p , 1 ≤ p ≤ ∞, to dlq
velyçyny
E( )K p = sup ( – ) ( )
–ϕ π
π
π
ϕ
∈
∫
U
p
x t K t dt
1
0
1
vykonu[t\sq spivvidnoßennq
1
2π
sup ( ) – ( )
h
pK K h
∈
⋅ ⋅ +
R
≤ E( )K p ≤ 1
π
K p . (23)
Vykorystavßy nerivnosti (23) pry K t( ) = cos nt
–
βπ
2
Pq t( ) , otryma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
982 A. S. SERDGK
1
2 2 2π
βπ βπ
sup cos ( ) – ( ) – cos ( ) – ( )
h
q q
p
n n h h
∈
⋅
⋅ ⋅ +
⋅ +
R
P P ≤
≤
sup ( – ) cos – ( )
ϕ
π
ϕ βπ
∈
∫
U
q
p
x t nt t dt
1
0 2
0
2
P ≤
≤
cos ( ) – ( )n q
p
⋅
⋅βπ
2
P , 1 ≤ p ≤ ∞. (24)
Na osnovi zobraΩennq (16) i lemyI1 pokladagçy
v ]] umovax s = p , ϕ( )t =
= cos nt
–
βπ
2
Pq t( ) iz lancgΩka nerivnostej (24) i asymptotyçno] rivnos-
tiI(19) bezposeredn\o oderΩu[mo
sup ( – ) cos – ( )
ϕ
π
ϕ βπ
∈
∫
U
q
p
x t nt t dt
1
0 2
0
2
P =
=
cos
( )
( )/
t
tp
p q p2 1π
P +I O
q
n q s p( )
( – ) ( )1
1
. (25)
Ob’[dnugçy formuly (20), (22) i (25), ma[mo
E L Uq
n L p
β, –;1 1
∗( ) =
q
t
t O
q
n q
n
p
p
p q p s p
2
1
1
1
1 1
/
/ ( )
cos
( ) ( )
( – )
′
+ +
π
P .
TeoremuI2 dovedeno.
1. Serdgk A. S. Pro odyn linijnyj metod nablyΩennq periodyçnyx funkcij // Problemy
teori] nablyΩennq funkcij ta sumiΩni pytannq: Zb. prac\ In-tu matematyky NAN Ukra]ny. –
2004. – 1, # 1. – S. 295 – 336.
2. Stepanec A. Y. Klassyfykacyq y pryblyΩenye peryodyçeskyx funkcyj. – Kyev: Nauk.
dumka, 1987. – 268 s.
3. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: V 2 ç. // Praci In-tu matematyky NAN Ukra]-
ny. – Kyev: Yn-t matematyky NAN Ukrayn¥, 2002. – 40, ç. 1. – 427 s.
4. Serdgk A. S. NablyΩennq klasiv analityçnyx funkcij sumamy Fur’[ v rivnomirnij metryci
// Ukr. mat. Ωurn. – 2005. – 57, # 8. – S. 1079 – 1096.
5. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyq. – M.: Nauka, 1987. – 423 s.
6. Serdgk A. S. NablyΩennq klasiv analityçnyx funkcij sumamy Fur’[ v metryci prostoru
L p // Ukr. mat. Ωurn. – 2005. – 57, # 10. – S. 1395 – 1408.
OderΩano 09.02.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
|
| id | umjimathkievua-article-3213 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:18Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/78/dbf9dc82f655c2fc87dc3b1aa8df2d78.pdf |
| spelling | umjimathkievua-article-32132020-03-18T19:48:23Z Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics Наближення інтегралів Пуассона одним лінійним методом наближення в рівномірній та інтегральних метриках Serdyuk, A. S. Сердюк, А. С. We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces C and Lp . Найдены асимптотические равенства для точных верхних граней приближений классов интегралов Пуассона периодических функций некоторым линейным методом приближения специального вида в метриках пространств C и Lp . Institute of Mathematics, NAS of Ukraine 2008-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3213 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 7 (2008); 976–982 Український математичний журнал; Том 60 № 7 (2008); 976–982 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3213/3172 https://umj.imath.kiev.ua/index.php/umj/article/view/3213/3173 Copyright (c) 2008 Serdyuk A. S. |
| spellingShingle | Serdyuk, A. S. Сердюк, А. С. Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics |
| title | Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics |
| title_alt | Наближення інтегралів Пуассона одним лінійним методом наближення в рівномірній та інтегральних метриках |
| title_full | Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics |
| title_fullStr | Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics |
| title_full_unstemmed | Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics |
| title_short | Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics |
| title_sort | approximation of poisson integrals by one linear approximation method in uniform and integral metrics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3213 |
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