Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m)
We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space S (p,q)(σ m), m > 3, in terms of the best approximations and modules of continuity. We consider constructive characteristics of functional classes defined by majorants of mo...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3355 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509431888871424 |
|---|---|
| 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 | 2020-03-18T19:52:14Z |
| description | We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space S (p,q)(σ m), m > 3, in terms of the best approximations and modules of continuity.
We consider constructive characteristics of functional classes defined by majorants of modules of continuity of their elements. |
| first_indexed | 2026-03-24T02:41:00Z |
| format | Article |
| fulltext |
UDK 517.51
R. A. Lasuryq (Abxaz. un-t, Suxum)
PRQMÁE Y OBRATNÁE TEOREMÁ
PRYBLYÛENYQ FUNKCYJ, ZADANNÁX NA SFERE,
V PROSTRANSTVE S((((
p,
q
))))
(((( σσσσm
))))
We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space
S p q m( )( ), σ , m ≥ 3, in terms of the best approximations and modules of continuity. We consider
constructive characteristics of functional classes defined by majorants of modules of continuity of their
elements.
Dovedeno prqmi ta oberneni teoremy nablyΩennq funkcij, zadanyx na sferi, u prostori
S p q m( )( ), σ , m ≥ 3, u terminax najkrawyx nablyΩen\ i moduliv neperervnosti ta rozhlqnuto
konstruktyvni xarakterystyky funkcional\nyx klasiv, wo zadani maΩorantamy moduliv nepe-
rervnosti ]xnix elementiv.
1. Pust\ R
m
, m ≥ 3, — m-mernoe evklydovo prostranstvo, x = ( x1, … , xm ) —
eho vektor¥, σm
— edynyçnaq sfera v prostranstve R
m
s centrom v naçale ko-
ordynat, Lp ( σm
) , p ≥ 1, — prostranstvo funkcyj f ( x ) s normoj
|| f || p = || f || Lp
= f x d xp
p
m
( ) ( )
∫ σ
σ
1/
< ∞,
L ( σm
) = L1 ( σm
), L∞ ( σm
) ≡ M = { = ( ) < ∞}
∈
f f f xM
x m
: sup vrai
σ
.
Pust\, dalee,
S [ f ] = Y f xn
n
λ( )
=
∞
∑ ,
0
, λ =
m − 2
2
, (1)
— rqd Fur\e – Laplasa f ∈ L ( σm
),
Y f x f Y xn j
n
j
n
j
an
λ( ) = ( )( ) ( )
=
∑, ˆ
1
,
ˆ ,f f Yj
n
j
n( ) ( )= ( ), j = 1, … , an ,
— koπffycyent¥ Fur\e – Laplasa funkcyy f ( x ), { }( )( )Y xj
n
, j = 1, … , an , —
sferyçeskye harmonyky stepeny n, obrazugwye ortohonal\n¥j bazys v pro-
stranstve Hn sferyçeskyx harmonyk stepeny n razmernosty an = ( + − )2 2n m ×
× ( + − )
( − )
n m
n m
3
2
!
! !
.
Pry kaΩd¥x fyksyrovann¥x p ∈ ( 0, ∞ ], q ∈ ( 0, ∞ ) oboznaçym
S f L Y f xp q m m
k p
q
k
( )
=
∞
( ) = ∈ ( ) ( ) < ∞
∑, : ,σ σ λ
0
.
∏lement¥ f ( x ), g ( x ) ∈ S p q m( )( ), σ sçytaem toΩdestvenn¥my, esly f̂ j
n( ) = ĝ j
n( )
pry vsex j = 1, … , an , n = 0, 1, … .
Budem hovoryt\, çto S p q m( )( ), σ poroΩdaetsq prostranstvom L ( σm
) , syste-
moj funkcyj
© R. A. LASURYQ, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7 901
902 R. A. LASURYQ
{ … }( ) ( ) ( )
=
∞
Y Y Yn n
a
n
n
n1 2
0
, , ,∪ (2)
y çyslamy p, q.
Normu kaΩdoho πlementa f ∈ S p q m( )( ), σ opredelym posredstvom ravenstva
f x f x Y f xS S n p
q
n
q
p q p q m( ) = ( ) = ( )
( ) ( )( )
=
∞
∑, , ,
/
σ
λ
0
1
. (3)
MnoΩestvo S p q m( )( ), σ pry p ∈ [ 1, ∞ ], q ∈ [ 1, ∞ ) obrazuet lynejnoe normyro-
vannoe prostranstvo, soderΩawee systemu (2), s operacyqmy sloΩenyq y umno-
Ωenyq na çyslo, opredelenn¥my na vsem prostranstve L ( σm
) . Pry p = q = 2 v
sylu ravenstva Parsevalq v L2 ( σm
) ymeem
f Y f x fn
n
S2 2
2
0
1 2
2 2= ( )
=
=
∞
∑ ( )
λ ,
/
, .
PredloΩenye 1. Pust\
Pn ( x ) = Y x Y xk
k
n
j
k
j
k
j
a
k
n k
( ) = ( )
=
( ) ( )
==
∑ ∑∑
0 10
α
— proyzvol\n¥j polynom po sferyçeskym harmonykam Yk ( x ),
S f x Y f xn k
k
n
λ λ( ) = ( )
=
∑, ,
0
— çastn¥e summ¥ rqda (1). Sredy vsex summ vyda Pn ( x ) pry dannom n = 0, 1,
2, … naymenee uklonqetsq ot f ∈ S p q m( )( ), σ , q ∈ ( 0, ∞ ), p ∈ ( 0, ∞ ], çastnaq
summa Fur\e – Laplasa S f xn
λ( ), :
inf ,, ,
Y
n S n S
k
p q p qf x P x f x S f x( ) − ( ) = ( ) − ( )( ) ( )
λ
, (4)
pryçem
f x S f x f Y f xn S
q
S
q
k p
q
k
n
p q p q( ) − ( ) = − ( )( ) ( )
=
∑λ λ, ,, ,
0
. (5)
Dokazatel\stvo. V sylu (3) y toho, çto dlq lgboho Yl ( x ) ∈ Hl
Y Y x
k l
Y x k lk l
l
λ( ) =
≠
( ) =
,
, ,
, ,
0
ymeem
f x P x Y f P xn S
q
k n p
q
k
p q( ) − ( ) = ( − )( )
=
∞
∑, ,λ
0
=
= Y f x Y P x Y f x Y P xk k n p
q
k
n
k k n p
q
k n
λ λ λ λ( ) − ( ) + ( ) − ( )
= = +
∞
∑ ∑, , , ,
0 1
=
= Y f x Y P x Y f xk k n p
q
k
n
k p
q
k n
λ λ λ( ) − ( ) + ( )
= = +
∞
∑ ∑, , ,
0 1
. (6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRQMÁE Y OBRATNÁE TEOREMÁ PRYBLYÛENYQ FUNKCYJ, … 903
Mynymum pravoj çasty (6) dostyhaetsq, kohda raznost\ Y f x Y P xk k n
λ λ( ) − ( ), ,
ravna nulg pry proyzvol\nom x ∈ σm
:
Y f x Y P xk k n
λ λ( ) − ( ), , = 0 ∀x ∈ σm
,
( − ) ( )( ) ( ) ( )
=
∑ f̂ Y xj
k
j
k
j
k
j
ak
α
1
= 0.
Otsgda v sylu lynejnoj nezavysymosty { }( )( )Y xj
k
, j = 1, … , ak , k = 0, 1, … ,
sledugt ravenstva f̂ j
k( ) = α j
k( )
, t. e.
S f xn
λ( ), = Pn ( x ),
pry πtom yz (6) v¥tekaet (5).
2. Prqm¥e teorem¥. Pust\ f ∈ S p q m( )( ), σ , q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ],
Su f ( x ) =
1
1 2σ λm
x y u
u
f y dt y−
( )=
( ) ( )∫sin
, cos
, 0 < u ≤ π,
— sferyçeskyj sdvyh s ßahom u ∈ ( 0, π ], ( x, y ) — skalqrnoe proyzvedenye vek-
torov x, y v R
m
, | σm
–
1
| — plowad\ ( m – 1 )-mernoj sfer¥. Yzvestno [1,
s.J236; 2, s. 254], çto
Y S f x
P u
P
Y f xn u
n
n
n
λ
λ
λ
λ( ) = ( )
( )
( ),
cos
,
1
, n = 0, 1, … ,
hde P tn
λ( ) — mnohoçlen¥ Hehenbauπra.
Tohda rqd Fur\e – Laplasa funkcyy Su f ( x ) prynymaet vyd
S [ Su f ] =
P u
P
Y f xn
nn
n
λ
λ
λ( )
( )
( )
=
∞
∑ cos
,
10
.
Otsgda, polahaq ∆u f ( x ) = Su f ( x ) – f ( x ), poluçaem
∆u S
n
n
q
n p
q
n
q
f x
P u
P
Y f xp q( ) = ( )
( )
− ( )
( )
=
∞
∑,
cos
,
/
λ
λ
λ
1
1
0
1
, (7)
q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ], u ∈ ( 0, π ].
Velyçynu
ω( ) = ( )( ) ( )
< ≤
f t f x
S
u t
u Sp q p q, sup, ,
0
∆ (8)
budem naz¥vat\ modulem neprer¥vnosty f v prostranstve S p q m( )( ), σ . Yz (7)
sleduet, çto
lim ,
t
f t
→
( )
0
ω = 0
y, krome toho, ω( ) ( )f t
S p q, , ne ub¥vaet na ( 0, π ].
Sohlasno predloΩenyg 1, esly
E f f x Pn S Y
n
x
S
p q
K
p q( ) = ( ) −( ) ( )−
( )
, ,inf 1 , (9)
to
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
904 R. A. LASURYQ
E f f x Y f xn
q
S S
q
k p
q
k
n
p q p q( ) = ( ) − ( )( ) ( )
=
−
∑, , ,λ
0
1
= Y f xk p
q
k n
λ( )
=
∞
∑ , , (10)
q ∈ ( 0, ∞ ), p ∈ ( 0, ∞ ].
Zdes\ m¥ rassmatryvaem sferyçeskye analohy, v opredelennom sm¥sle, prqm¥x
teorem pryblyΩenyq, yssledovann¥x v [3] (sm. takΩe [4]).
Ymeet mesto sledugwee utverΩdenye.
Teorema 1. Pust\ f ∈ S p q m( )( ), σ , q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ], y f � const. Toh-
da dlq lgboho τ > 0
E f C f nn S n Sp q p q( ) ≤ ( ) ( )( ) ( ), ,, /τ ω τ , (11)
hde
C
In
M n
q
( ) = ( ) − ( )
( )
∈ ( )
τ µ τ µ
τ µµ τ
inf
,
/0 1
(12)
y
I I
P t n
P
d tn n q
k n
k
k
q
( ) = ( ) = ( )
( )
− ( )
≥ ∫τ µ τ µ µ
λ
λ
τ
, , inf
cos /
, 1
1
0
, k ∈ N, (13)
M ( τ ) — mnoΩestvo funkcyj µ, ohranyçenn¥x, neub¥vagwyx, otlyçn¥x ot kon-
stant na [ 0, τ ]. Pry πtom suwestvuet funkcyq µ* ∈ M ( τ ), realyzugwaq v
(12) toçnug nyΩngg hran\. Neravenstvo (11) neuluçßaemo na mnoΩestve
S p q m( )( ), σ v tom sm¥sle, çto dlq lgboho n ∈ N
Kn ( τ ) = Cn ( τ ), (14)
hde
Kn ( τ ) = sup
, /
: , const
,
,
,E f
f n
f S f
n S
S
p q mp q
p q
( )
( )
∈ ( ) /≡
( )
( )
( )
ω τ
σ . (15)
Dokazatel\stvo provedem po sxeme, predloΩennoj v [3]. Pust\ f ∈
∈ S p q m( )( ), σ . Na osnovanyy (7) y (10) ymeem
∆u S
k
k
q
k p
q
k n
f x
P u
P
Y f xp q( ) ≥ ( )
( )
− ( )( )
=
∞
∑,
cos
,
λ
λ
λ
1
1 =
=
I
E fn
n
q
S p q
( )
( ) − ( )
( ) ( )
τ µ
µ τ µ
,
,
0
+ Y f x
P u
P
I
k p
q
k n
k
k
q
nλ
λ
λ
τ µ
µ τ µ
( ) ( )
( )
− − ( )
( ) − ( )
=
∞
∑ ,
cos ,
1
1
0
. (16)
Otsgda dlq lgboho t ∈ [ 0, τ ]
E f
I
f Y f xn
q
S
n
t n S
q
k p
q
k n
p q p q( ) ≤ ( ) − ( )
( )
− ( )
( ) ( )
=
∞
∑, ,,
,/
µ τ µ
τ µ
λ0 ∆ ×
×
P t n
P
Ik
k
q
n
λ
λ
τ µ
µ τ µ
( )
( )
− − ( )
( ) − ( )
cos / ,
1
1
0
. (17)
V sylu neravenstva [1, s. 205; 2, s. 255]
max cos
0
1
≤ ≤ π
( ) ≤ ( )
u
n nP u Pλ λ
naxodym
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRQMÁE Y OBRATNÁE TEOREMÁ PRYBLYÛENYQ FUNKCYJ, … 905
P u
P
P u
P
k
k
q
u
n
n
q
λ
λ
λ
λ
( )
( )
− ≤
( )
( )
+
≤ ≤ πcos
max cos
1
1
1
1
0
≤ 2
q
. (18)
Ysxodq yz (18), zaklgçaem, çto rqd v pravoj çasty (17) ravnomerno po t maΩo-
ryruetsq sxodqwymsq poloΩytel\n¥m çyslov¥m rqdom. Na osnovanyy πtoho,
yntehryruq neravenstvo po dµ ( t ) v predelax ot 0 do τ, poluçaem
E f
I
f x d tn
q
S
n
t n S
q
p q p q( ) ( ) − ( ) ≤ ( ) − ( )
( )
( ) ( )( ) ( )( ) ∫, ,, /µ τ µ µ τ µ
τ µ
µ
τ
0
0
0
∆ –
– Y f x
P t n
P
d t Ik S
q
k n
k
k
q
np q
λ
λ
λ
τ
µ τ µ( ) ( )
( )
− ( ) − ( )
( )
=
∞
∑ ∫,
cos /
,,
1
1
0
. (19)
Uçyt¥vaq opredelenye velyçyn¥ (13), yz (19) naxodym
E f
I
f x d tn
q
S
n
t n S
q
p q p q( ) ≤
( )
( ) ( )( ) ( )∫, ,, /
1
0
τ µ
µ
τ
∆ ≤
1
0
I
f t n d t
n
q
S p q( )
( ) ( )( )∫τ µ
ω µ
τ
,
, / , .
(20)
Otsgda poluçaem (11) y sootnoßenye
K
I
Cn
q
M n
n
q( ) ≤ ( ) − ( )
( )
= ( )
∈ ( )
τ µ τ µ
τ µ
τ
µ τ
inf
,
0
. (21)
PoloΩym
˜ ˜ cos /
: ,,W t
P t n
P
n q k
k
k
q
k k
k nk n
= ( ) = ( )
( )
− ≥ =
=
∞
=
∞
∑∑ω ρ ρ ρ
λ
λ 1
1 0 1 ,
˜ inf ˜, ˜ ˜ ,
,
In q
W C
n q
( ) =
∈ [ ]τ ω
ω τ0 .
Prynymaq vo vnymanye ravenstvo
ω
λ
λ
λq
S
u t
k
k
q
k p
q
k
f t
P u
P
Y f xp q( ) = ( )
( )
− ( )( )
< ≤ =
∞
∑, sup
cos
,,
0 0 1
1 , (22)
a takΩe sootnoßenyq (10) y (15), ymeem
K
P u
P
n
q kk n
u t
k
k
k
q
k n
k
( ) =
( )
( )
−
≥
=
∞
< ≤ =
∞
∑
∑
τ
ρ
ρ
ρ λ
λ
sup
sup
cos0
0 1
1
, (23)
hde sup rassmatryvaetsq po vsem posledovatel\nostqm çysel ρk takyx, çto
ρkk =
∞∑ 1
< ∞. Dalee, povtorqq rassuΩdenyq yz [3] (sm. takΩe [4, s. 392]), pryxo-
dym k ravenstvu v sootnoßenyy (21).
Teorema 1 dokazana.
Teorema 2. Dlq lgboj funkcyy f ( x ) ∈ S p q m( )( ), σ , q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ],
ymegt mesto ravenstva
E f
I q
f t n t dtn
q
S
n
q
Sp q p q( ) ≤
( )
( )( ) ( )∫, ,, / sin
1
0
ω
π
, n ∈ N, (24)
hde
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
906 R. A. LASURYQ
I q
P t n
P
t dtn
k n
k
k
q
( ) = ( )
( )
−
≥
π
∫inf
cos /
sin
λ
λ 1
1
0
. (25)
Pry πtom
C t t dt I q C t t dtq
n
q
2
2
0
1
2
0
( ) ≤ ( ) ≤ ( )∫ ∫λ λ
π π
sin sin , λ =
m − 2
2
, (26)
y neravenstvo
E f C
t t dt
f t n t dtn
q
q
q
S p q( ) ≤ ( ) ( )
∫
∫ ( )3 2
0
0
1λ ωπ
π
sin
, / sin, (27)
neuluçßaemo po porqdku na mnoΩestve S p q m( )( ), σ , Ci ( λ ), i = 1, 2, 3, — polo-
Ωytel\naq konstanta, zavysqwaq tol\ko ot λ.
Dokazatel\stvo. PoloΩym v neravenstve (20) τ = π, µ ( t ) = 1
1
1
1
− ( )
( )
P t
P
λ
λ
cos
.
Poskol\ku
P t t1 2
1
0 1
λ λ λ
( ) =
−
cos cos = 2λ cos t,
P1 1
1 2
2
2 2
2
λ λ
λ
λ λ
λ
( ) = ( + )
( )
= ( )
( )
Γ
Γ
Γ
Γ
= 2λ,
to µ ( t ) = 1 – cos t, hde Γ ( ⋅ ) — hamma-funkcyq. Otsgda pryxodym k (24).
Yzvestno [1, s. 242] (sm. takΩe [2, s. 255]) predstavlenye
P t
P
n t nt
o tn
n
λ
λ λ
λ
λ
( )
( )
− = −
( + )
−
+
+ ( )cos
1
1
2 2 1 2 1
2 2 2
3
, (28)
yz kotoroho, v çastnosty, sledugt neravenstva
P t
P
Cn tn
n
λ
λ
( )
( )
− ≥cos
1
1 2 2 , C = C ( λ ), (29)
P t
P
C n tn
n
λ
λ
( )
( )
− ≤cos
1
1 1
2 2 , C1 = C1 ( λ ), n → ∞, t → 0. (30)
Ysxodq yz (29), poluçaem
I q
P t n
P
t dtn
k n
k
k
q
( ) = ( )
( )
−
≥
π
∫inf
cos /
sin
λ
λ 1
1
0
≥ C
k
n
t t dt
k n
q
qinf sin
≥
∫
2
2
0
π
=
= C t t dtq2
0
sin
π
∫ , C = C ( λ ).
Analohyçno, prymenqq (30), pryxodym k pervomu neravenstvu v (26), a znaçyt,
s uçetom (24) poluçaem (27).
PokaΩem, çto na vsem mnoΩestve S p q m( )( ), σ neravenstvo (27) po porqdku
uluçßyt\ nel\zq. Pust\
Y x Y xn j
n
j
n
j
an
*( ) = ( )( ) ( )
=
∑ α
1
— proyzvol\naq sferyçeskaq harmonyka stepeny n.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRQMÁE Y OBRATNÁE TEOREMÁ PRYBLYÛENYQ FUNKCYJ, … 907
Prynymaq vo vnymanye (30) y (22), naxodym
ωq
n
u t n
u n S
q
Y t n Y x p q( ) = ( )
< ≤
( )
*
/
*, / sup ,
0
∆ = sup
cos
,
/
*
0 1
1
< ≤
( )
( )
− ( )
u t n
n
n
q
n n p
qP u
P
Y Y x
λ
λ
λ ≤
≤ C u n Y Y x C t Y Y x
u t n
q
n n p
q q
n n p
q
1
0
2 2
1
2( ) ( ) ( ) = ( )
< ≤
λ λ λsup , ,
/
* *
,
otkuda
ω
π
λ
π
q
n S n n p
q qY t n t dt C Y Y x t t dtp q( ) ≤ ( )( )∫ ∫* *, / sin , sin,
0
1
2
0
= C t t dt E Yq
n
q
n S p q1
2
0
sin *
,( ) ( )∫
π
y
E Y
C
t t dt
Y t n t dtn
q
n S q
q
n Sp q p q( ) ≥ ( )( ) ( )
∫
∫* *
, ,
sin
, / sin2
2
0
0
π
π
ω , C2 = C2 ( λ ).
Teorema 2 dokazana.
Vsledstvye toho çto
t t dt t dtq
q
2
0
2
2
2
sin sin
/
π π
∫ ∫≥ π
π
≥ 1,
pryxodym k sledugwemu utverΩdenyg.
Sledstvye 1. Dlq lgboj funkcyy f ( x ) ∈ S p q m( )( ), σ , q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ],
ymegt mesto neravenstva
E f C f
nn S S
p q
p q
( ) ≤ π
( )
( )
,
,
,ω , n ∈ N, (31)
hde C = C ( λ ) — poloΩytel\naq konstanta, zavysqwaq tol\ko ot λ.
Pust\ S p m( ∞)( ), σ — mnoΩestvo funkcyj f ∈ L ( σm
) s koneçnoj normoj
f x Y f xS
k
k p
p( ) = ( )( ∞)
∈
, sup ,
N0
λ
, p ∈ ( 0, ∞ ],
hde N0 = { 0 } ∪ N,
ω( ) = ( )( ∞) ( ∞)
< ≤
f t f x
S
u t
u Sp p, sup, ,
0
∆ .
Funkcyq ω( ) ( ∞)f t
S p, , ne ub¥vaet pry t > 0 y ω( ) ( ∞)f t
S p, ,
t→ →
0
0.
Teorema 3. Dlq lgboj funkcyy f ( x ) ∈ S p m( ∞)( ), σ , p ∈ ( 0, ∞ ], v¥polnqetsq
neravenstvo
E f C f
nn S p( ) ≤ π
( ∞), ,ω , n ∈ N, C = C ( λ ),
pryçem dannoe neravenstvo na mnoΩestve S p m( ∞)( ), σ po porqdku uluçßyt\
nel\zq.
Dokazatel\stvo. Dlq lgboj f ∈ S p m( ∞)( ), σ
E f f x S f x Y f xn S n S k n
k p
p p( ) = ( ) − ( ) = ( )( ∞) ( ∞)−
≥
, ,, max ,1
λ λ
.
Tohda s uçetom (29) poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
908 R. A. LASURYQ
ω f
n
f x
S u n
u Sp
p, sup
,
,
/
π
= ( )
( ∞)
( ∞)
< ≤ π0
∆ =
= sup max
cos
,
/0 0 1
1
< ≤ π ∈
( )
( )
− ( )
u n k
k
k
k p
P u
P
Y f x
N
λ
λ
λ ≥
≥ C Y f x k u C Y f x
k n
k p u n k n
k p
max , sup max ,
/≥ < ≤ π ≥
( ) ≥ ( )λ λ
0
2 2 = CE fn S p( ) ( ∞), ,
C = C ( λ ).
Otsgda pryxodym k trebuemomu neravenstvu.
Pust\ teper\ f ∈ S p m( ∞)( ), σ , dlq kotoroj Y f xk p
λ( ), = 0 pry vsex k < n. V
sylu ocenky (18)
ω
λ
λ
λf
n
P u
P
Y f x
S u n k n
k
k
k pp
, sup max
cos
,
,
/
π
= ( )
( )
− ( )
( ∞) < ≤ π ≥0 1
1 =
= max , sup
cos
/
,
k n
k p u n
k
k
n S
Y f x
P u
P
E f p
≥ < ≤ π
( ) ( )
( )
− ≤ ( ) ( ∞)
λ
λ
λ
0 1
1 2 .
Takym obrazom,
E f f
nn S S
p
p
( ) ≥ π
( ∞)
( ∞)
,
,
,
1
2
ω ,
y teorema 3 dokazana.
3. Obratn¥e teorem¥. Osnovn¥m v πtom napravlenyy qvlqetsq sledugwee
utverΩdenye.
Teorema 4. Dlq lgboj funkcyy f ( x ) ∈ S p q m( )( ), σ , q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ],
v¥polnqetsq neravenstvo
ω f
n
C
n
k k E f
S
q q
k
q
S
k
n q
p q
p q,
,
,
/
π
≤ − ( − ) ( )
( )
( )( )
=
∑2
2 2
1
1
1 , n ∈ N, (32)
hde C = C ( λ ) — poloΩytel\naq konstanta, zavysqwaq tol\ko ot λ. Pry
πtom neravenstvo (32) na mnoΩestve S p q m( )( ), σ po porqdku uluçßyt\ nel\zq.
Dokazatel\stvo. Pust\ f ( x ) ∈ S p q m( )( ), σ , u ∈ ( 0, π / n ], n ∈ N. Tohda na
osnovanyy (7) ymeem
∆u S
q k
k
q
k
n
k p
q
f x
P u
P
Y f xp q( ) = ( )
( )
− ( )( )
=
−
∑,
cos
,
λ
λ
λ
1
1
0
1
+
+
P u
P
Y f xk
k
q
k n
k p
qλ
λ
λ( )
( )
− ( )
=
∞
∑ cos
,
1
1 = I1 + I2
. (33)
V sylu neravenstva (18) y (10)
I2 ≤
P u
P
Y f xk
k
q
k n
k p
qλ
λ
λ( )
( )
+
( )
=
∞
∑ cos
,
1
1 ≤
≤ 2 2q
k p
q
k n
q
n
q
S
Y f x E f p q
λ( ) = ( )
=
∞
∑ ( ), , .
(34)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRQMÁE Y OBRATNÁE TEOREMÁ PRYBLYÛENYQ FUNKCYJ, … 909
Uçyt¥vaq (30), naxodym
I1 =
P u
P
Y f xk
k
q
k n
n
k p
qλ
λ
λ( )
( )
− ( )
=
−
∑ cos
,
1
1
1
≤ C k u Y f xq q q
k p
q
k
n
2 2
1
1
λ( )
=
−
∑ , ≤
≤ C
n
k Y f xq
q
q
q
k p
q
k
nπ ( )
=
−
∑
2
2
2
1
1
λ , , C = C ( λ ). (35)
Sohlasno (34), (35) yz (33) poluçaem
∆u S
q q
n
q
S
q q
q
q
k p
q
k
n
f x E f
C
n
k Y f xp q p q( ) ≤ ( ) + π ( )( ) ( )
=
−
∑, , ,2
2
2
2
1
1
λ
. (36)
Zatem prymenqem ravenstvo (sm., naprymer, [4, s. 401])
α α α α αk k
k m
M
m k
k m
k k
k m
M
i
i k
M k
k M
c c c c
= =
∞
−
= + =
∞
= +
∞
∑ ∑ ∑ ∑ ∑= + ( − ) −1
1 1
(37)
pry uslovyy, çto
ck
k =
∞
∑
1
< ∞,
hde ( αk ), k ∈ N, — proyzvol\naq posledovatel\nost\ çysel m, M ∈ N, m ≤ M.
Polahaq v (37) αk = k2q
, ck = Y f xk p
qλ( ), , m = 1, M = n – 1, ymeem
k Y f x Y f xq
k p
q
k
n
k p
q
k
2
1
1
1
λ λ( ) = ( )
=
−
=
∞
∑ ∑, , +
+ ( )− ( − ) ( ) − ( − ) ( )
=
∞
=
−
=
∞
∑∑ ∑k k Y f x n Y f xq q
i p
q
i kk
n
q
k p
q
k n
2 2
2
1
21 1λ λ, , =
= ( )− ( − ) ( ) − ( − ) ( )( ) ( )
=
−
∑ k k E f n E fq q
k
q
S
k
n
q
n
q
Sp q p q
2 2
1
1
21 1, , . (38)
Sledovatel\no, yz (36) s uçetom (38) poluçaem
∆u S
q q q
q
q q
k
q
S
k
n
q
n
q
S
f x
C
n
k k E f n E fp q p q p q( ) ≤ π − ( − ) ( ) − ( − ) ( )
( ) ( ) ( )( )
=
−
∑, , ,
2
2
2 2
1
1
21 1 +
+ 2 11
2
2
2 2
1
q
n
q
S
q q
q
q q
k
q
S
k
n
E f
C
n
k k E fp q p q( ) ≤ π − ( − ) ( )( ) ( )( )
=
∑, , . (39)
Yz (39) sleduet (32).
PokaΩem, çto (32) po porqdku uluçßyt\ nel\zq. Dlq lgboj funkcyy f ∈
∈ S p q m( )( ), σ
ω
λ
λ
λq
S
n S
q k
k
q
k
k p
q
f
n
f x
P n
P
Y f x
p q p q,
cos /
,
, ,/
π
≥ ( ) = ( π )
( )
− ( )
( ) ( )π
=
∞
∑∆
1
1
0
≥
≥
P n
P
Y f xk
k
q
k
n
k p
qλ
λ
λ( π )
( )
− ( )
=
∑ cos /
,
1
1
1
≥
C
n
k Y f x
q q
q
q
k p
q
k
n
2
2
2
2
1
1π ( )
=
−
∑ λ , , C2 = C2 ( λ ).
(40)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
910 R. A. LASURYQ
Prymenqq ravenstvo (37) pry αk = k2q
, ck = Y f xk p
qλ( ), , m = 1, M = n, yz (40)
naxodym
ωq
S
q q
q
q q
k
q
S
k
n
q
n
q
S
f
n
C
n
k k E f n E f
p q
p q p q,
,
, ,
π
≥ π − ( − ) ( ) − ( )
( )
( ) ( )( )
=
+∑2
2
2
2 2
1
2
11 . (41)
Pust\ f = Pn — polynom po sferyçeskym harmonykam stepeny ne v¥ße n.
Tohda E Pn n S p q+ ( ) ( )1 , = 0.
Sledovatel\no, yz (41) v¥tekaet neravenstvo
ω P
n
C
n
k k E Pn
S
q q
k
q
n S
k
n q
p q
p q,
,
,
/
π
≥ π − ( − ) ( )
( )
( )( )
=
∑2
2
2
2 2
1
1
1 , C2 = C2 ( λ ).
Teorema dokazana.
Yz neravenstva (32) poluçaem ocenku
ω f
n
C q
n
k E f
S
q
q
k
q
S
k
n q
p q
p q,
,
,
/ /
π
≤ ( ) ( )
( )
( )
−
=
∑2 1
2
2 1
1
1
, C = C ( λ ). (42)
Otsgda, v çastnosty, v¥tekaet sledugwee utverΩdenye.
Sledstvye 2. Pust\ f ( x ) ∈ S p q m( )( ), σ , q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ], y pry neko-
torom α > 0
E f O
n
n S p q( ) =
( ),
1
α , n ∈ N.
Tohda
ω
α
α
α
α
( ) =
( ) < <
( ) =
( ) >
( )f t
O t
O t t
O t
S
q
p q,
, ,
ln , ,
, .
,
/
0 2
2
2
2 1
2
4. Pust\ funkcyq ϕ ( t ) opredelena na [ 0, π ] y v¥polnen¥ sledugwye us-
lovyq:
1) ϕ ( t ) monotonno vozrastaet y neprer¥vna na [ 0, π ];
2) ϕ ( 0 ) = 0.
V πtom sluçae budem hovoryt\, çto ϕ prynadleΩyt klassu Φ.
Oboznaçym çerez H
S
m
p q( ) ( ),
ω σ , ω ∈ Φ, klass vsex funkcyj f ∈ S p q m( )( ), σ ,
q ∈ [ 1, ∞ ), p ∈ ( 0, ∞ ], udovletvorqgwyx uslovyg
ω ω( ) ≤ ( )( )f t C t
S p q, , , t ∈ ( 0, π ], (43)
hde C = C ( f ) — poloΩytel\naq postoqnnaq, voobwe hovorq, zavysqwaq ot f.
Funkcyq ϕ ∈ Φ udovletvorqet uslovyg Bary ( Br ), r ≥ 1, esly
k
k
O n
n
r
k
n
r−
=
π
= π
∑ 1
1
ϕ ϕ .
Teorema 5. Pust\ ω ( t ) ∈ Φ takova, çto ωq
( t ), q ∈ [ 1, ∞ ), udovletvorq-
et uslovyg ( B2q ). Tohda dlq toho çtob¥ funkcyq f ∈ S p q m( )( ), σ , p ∈ ( 0, ∞ ],
prynadleΩala klassu H
S
m
p q( ) ( ),
ω σ , neobxodymo y dostatoçno, çtob¥
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRQMÁE Y OBRATNÁE TEOREMÁ PRYBLYÛENYQ FUNKCYJ, … 911
E f O
nn S p q( ) = π
( ), ω . (44)
Dokazatel\stvo. Pust\ f ∈ H
S
m
p q( ) ( ),
ω σ . Tohda yz (43) y (31) sleduet (44).
S druhoj storon¥, v sylu (42)
ω f
n
C
n
k E f
S
q
k
q
S
k
n q
p q
p q,
,
,
/
π
≤ ( )
( )
( )
−
=
∑1
2
2 1
1
1
, f ∈ S p q m( )( ), σ ,
y s uçetom (43) naxodym
ω ωf
n
O
n
k
nS
q q
k
n q
p q
,
,
/
π
= π
( )
−
=
∑1
2
2 1
1
1
.
Vsledstvye toho çto ωq
( ⋅ ) udovletvorqet uslovyg ( B2q ), poluçaem
ω ωf
n
O
nS p q
,
,
π
= π
( )
,
y f ∈ H
S
m
p q( ) ( ),
ω σ , tak kak ω( + ) ( )t t
S p q1 2 , ≤ ω( ) ( )t
S p q1 , + ω( ) ( )t
S p q2 , .
Funkcyq ω ( t ) = tα udovletvorqet uslovyqm teorem¥ 5 pry α ∈ ( 0, 2 ).
Poπtomu esly H
S
m
p q( ) ( ),
α σ — klass H
S
m
p q( ) ( ),
ω σ pry ω ( t ) = tα, poluçaem takoe
utverΩdenye.
Sledstvye 3. Pust\ α ∈ ( 0, 2 ). Dlq toho çtob¥ funkcyq f ∈ S p q m( )( ), σ
prynadleΩala klassu H
S
m
p q( ) ( ),
α σ , p ∈ ( 0, ∞ ], q ∈ [ 1, ∞ ), neobxodymo y dosta-
toçno, çtob¥
E f O
n
n S p q( ) =
( ),
1
α .
1. Berens H., Butzer P. L., Pawelke S. Limitierungsverfahren vor Reihen mehrdimensionaler
Kugelfunktionen und deren Saturationsverhalten // Publs. Res. Inst. Math. Sci. A. – 1968. – 4, # 2.
– P. 201 – 268.
2. Topuryq S. B. Rqd¥ Fur\e – Laplasa na sfere. – Tbylysy: Yzd-vo Tbyl. un-ta, 1987. –
356Js.
3. Stepanec A. Y., Serdgk A. S. Prqm¥ y obratn¥e teorem¥ pryblyΩenyq funkcyj v prost-
ranstve S
p
// Ukr. mat. Ωurn. – 2002. – 54, # 1. – S. 106 – 124.
4. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj // Matematyka ta ]] zastosuvannq: Pr. In-tu
matematyky NAN Ukra]ny. – 2002. – 40, ç. 2. – 467 s.
Poluçeno 03.03.2004,
posle dorabotky — 04.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
|
| id | umjimathkievua-article-3355 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:41:00Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/90/0ab00a7082a08087bfa4623995ed5890.pdf |
| spelling | umjimathkievua-article-33552020-03-18T19:52:14Z Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) Прямые и обратные теоремы приближения функций, заданных на сфере, в пространстве S (p,q)(σ m) Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space S (p,q)(&sigma; m), m > 3, in terms of the best approximations and modules of continuity. We consider constructive characteristics of functional classes defined by majorants of modules of continuity of their elements. Доведено прямі та обернені теореми наближення функцій, заданих на сфері, у просторі S (p,q)(&sigma; m), m > 3, у термінах найкращих наближень і модулів неперервності та розглянуто конструктивні характеристики функціональних класів, що задані мажорантами модулів неперервності їхніх елементів. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3355 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 901-911 Український математичний журнал; Том 59 № 7 (2007); 901-911 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3355/3454 https://umj.imath.kiev.ua/index.php/umj/article/view/3355/3455 Copyright (c) 2007 Lasuriya R. A. |
| spellingShingle | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) |
| title | Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) |
| title_alt | Прямые и обратные теоремы приближения функций, заданных на сфере, в пространстве S (p,q)(σ m) |
| title_full | Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) |
| title_fullStr | Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) |
| title_full_unstemmed | Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) |
| title_short | Direct and inverse theorems on approximation of functions defined on a sphere in the space S (p,q)(σ m) |
| title_sort | direct and inverse theorems on approximation of functions defined on a sphere in the space s (p,q)(σ m) |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3355 |
| work_keys_str_mv | AT lasuriyara directandinversetheoremsonapproximationoffunctionsdefinedonasphereinthespacespqsm AT lasuriâra directandinversetheoremsonapproximationoffunctionsdefinedonasphereinthespacespqsm AT lasuriâra directandinversetheoremsonapproximationoffunctionsdefinedonasphereinthespacespqsm AT lasuriyara prâmyeiobratnyeteoremypribliženiâfunkcijzadannyhnasferevprostranstvespqsm AT lasuriâra prâmyeiobratnyeteoremypribliženiâfunkcijzadannyhnasferevprostranstvespqsm AT lasuriâra prâmyeiobratnyeteoremypribliženiâfunkcijzadannyhnasferevprostranstvespqsm |