Structural properties of functions defined on a sphere on the basis of Φ-strong approximation
Structural properties of functions defined on a sphere are determined on the basis of the strong approximation of Fourier-Laplace series.
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509522616909824 |
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| 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. |
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| datestamp_date | 2020-03-18T19:54:30Z |
| description | Structural properties of functions defined on a sphere are determined on the basis of the strong approximation of Fourier-Laplace series. |
| first_indexed | 2026-03-24T02:42:27Z |
| format | Article |
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UDK 517.51
R.�A.�Lasuryq (Abxaz. un-t, Suxum)
STRUKTURNÁE SVOJSTVA FUNKCYJ, ZADANNÁX
NA SFERE, NA OSNOVE ΦΦΦΦ -SYL|NOJ APPROKSYMACYY
Structural properties of functions defined on a sphere are established on the basis of the strong
approximation of Fourier – Laplace series.
Vstanovlggt\sq strukturni vlastyvosti funkcij, zadanyx na sferi, na osnovi syl\no] aproksy-
maci] rqdiv Fur’[ – Laplasa.
1. V nastoqwee vremq suwestvuet dostatoçno mnoho rezul\tatov, kasagwyxsq
obratn¥x teorem v termynax syl\noj summyruemosty tryhonometryçeskyx rq-
dov Fur\e (sm., naprymer, [1 – 6]).
V dannoj rabote rassmatryvagtsq strukturn¥e svojstva funkcyy, zadannoj
na edynyçnoj dvumernoj sfere S
2, v termynax Φ -syl\noj summyruemosty rq-
dov Fur\e – Laplasa. Pry πtom formulyrovky rezul\tatov, poluçenn¥x v rabo-
te, v yzvestnoj stepeny analohyçn¥ sootvetstvugwym rezul\tatam yz [5, 6].
Pust\ funkcyq f ( x ) = f ( θ , ϕ ) neprer¥vna na edynyçnoj sfere s normoj
f f xC S
x S
( ) max ( )2
2
=
∈
.
Pod modulem neprer¥vnosty funkcyy f ( x ) ∈ C ( S
2) budem ponymat\ vely-
çynu, opredelqemug ravenstvom [7]
ω ( f ; δ ) = sup ( ) ( ; )
( )0
2
< ≤
−
γ δ
γf x S f x
C S
, (1)
hde
S f x f y dt y
x y
γ
γπ γ
( )
sin
( ) ( )
( , ) cos
=
≤
∫1
2
,
( x , y ) — skalqrnoe proyzvedenye vektorov v evklydovom prostranstve R
3, d t —
πlement plowady poverxnosty { y : y ∈ S
2 : ( x , y ) = cos γ , 0 < γ < π }. PoloΩym
∆γ f ( x ) = f ( x ) – Sγ f ( x ) . Modul\ neprer¥vnosty k -ho porqdka, k ∈ N , opredelq-
etsq ravenstvom
ωk ( f ; δ ) = sup ( )
( )0
2
< ≤γ δ
γ∆
k
C S
f x ,
ω1 ( f ; δ ) = ω ( f ; δ ) ,
hde
∆ ∆ ∆γ γ γ
k kf x f x( ) ( )= ( )−1 , ∆γ
0 f x( ) ≡ f ( x ) .
Pust\, dalee,
Tn ( x ) = Tn ( θ , ϕ ) = Yk
k
n
( , )θ ϕ
=
∑
0
— polynom sferyçeskoj harmonyky, hde Y k ( ⋅ ) — sferyçeskaq harmonyka po-
rqdka k , tak çto
D Yk ( θ , ϕ ) = – k ( k + 1) Yk ( θ , ϕ ) ,
© R.8A.8LASURYQ, 2006
20 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
STRUKTURNÁE SVOJSTVA FUNKCYJ, ZADANNÁX NA SFERE, NA OSNOVE … 21
D = 1 1
2
2
2sin
sin
sinθ
∂
∂θ
θ ∂
∂θ θ
∂
∂ϕ{ } +
— operator Laplasa na sfere,
En ( f ) = E fn C S
( )
( )2 = inf ( )( ) ( )T
n C S n C S
n
f T f T f
−
− = −− −
∗
1
2 21 1
— velyçyna nayluçßeho pryblyΩenyq funkcyy f sferyçeskymy summamy po-
rqdka ne8v¥ße n – 1. PoloΩym
ρ σk kf x f x f x( / ) ( / )( ; ) ( ) ( ; )1 2
1
1 2= − − ,
σ ν ν
ν
n
n
n
n
f x
A
A S f x( / )
/ –
/ ( / )( ; ) ( ; )1 2
1 2
1 2 1 2
0
1= −
=
∑ ,
hde σn f x( / )( ; )1 2 — srednye Çezaro ( C , 1 / 2 ) rqda Fur\e – Laplasa (sm., napry-
mer, [7]),
f ( x ) = f ( θ , ϕ ) � 1
4
2 1
0 0
2
0π
θ ϕ γ θ θ ϕ
ππ
( ) ( , ) (cos )sinj f P d d
j
j+ ′ ′ ′ ′ ′
=
∞
∑ ∫∫ , (2)
Pj ( t ) — mnohoçlen¥ LeΩandra, S f xν
( / )( ; )1 2 — çastyçn¥e summ¥ rqda (2), ν = 0,
1, 2, … .
2. Ymeet mesto sledugwee utverΩdenye.
Teorema 1. Pust\ funkcyq Φ ( ⋅ ) neprer¥vnaq stroho vozrastagwaq
v¥puklaq vnyz na [ 0 , + ∞ ) y ravnaq nulg v nule, a funkcyq Ψ ( ⋅ ) — obratnaq
k nej. Esly dlq f ( x ) = f ( θ , ϕ ) , zadannoj na sfere S
2 , suwestvuet posledo-
vatel\nost\ ( Tn ( x ) ) , n = 0, 1, 2, … , sferyçeskyx summ takyx, çto
Φ T x f xk
k C S
( ) ( )
( )
−( )
=
∞
∑
0 2
= M < + ∞ , (3)
to dlq modulq neprer¥vnosty v¥polnqetsq neravenstvo
ω ( f ; h ) ≤ K h u
u
du
h
2
3
1 Ψ( )∫ , (4)
hde K — poloΩytel\naq velyçyna, ne,zavysqwaq ot h > 0.
Dokazatel\stvo. Prymenqq neravenstvo Yensena [8, s. 92], s uçetom (3) na-
xodym
M ≥ Φ T x f xk
k n
n
C S
( ) ( )
( )
−( )
=
−
∑
2 1
2
≥ n
n
T x f xk
k n
n
C S
Φ 1 2 1
2
( ) ( )
( )
−
=
−
∑ =
= n
n
T x f x n
n
T x f xk
k n
n
C S
k
k n
n
C S
Φ Φ1 12 1 2 1
2 2
( ) ( ) ( ) ( )
( ) ( )
−
≥ −
=
−
=
−
∑ ∑ ≥
≥ n Φ ( E2n ( f ) ), n ∈ N . (5)
Yz uslovyj teorem¥ sleduet, çto funkcyq Ψ ( ⋅ ) takΩe neprer¥vna, stroho
vozrastaet, ravna nulg v nule y v¥pukla vverx na [ 0 , + ∞ ) . Tohda, kak yzvest-
no, Ψ ( ⋅ ) qvlqetsq modulem neprer¥vnosty, dlq kotoroho spravedlyvo svojst-
vo poluaddytyvnosty:
Ψ ( t1 + t2 ) ≤ Ψ ( t1 ) + Ψ ( t2 ) ∀t1 , t2 > 0. (6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
22 R.8A.8LASURYQ
Sohlasno (6) yz (5) poluçaem
E2n ( f ) ≤ Ψ Ψ Ψ ΨM
n
M
n
M
n
K
n
≤
+
≤ +( )
=
[ ]
[ ]
1
1 1 1 . (7)
Podbyraq k ∈ N tak, çtob¥ 2k ≤ n < 2k
+1, na osnovanyy (7) ymeem
En ( f ) ≤ E fk2
( ) ≤ K K
nkΨ Ψ1
2
1
1 1−
≤
. (8)
Ar.8S.8DΩafarov¥m [9] ustanovleno neravenstvo
ω ( f ; 1 / n ) ≤ K n k E fk
k
n
−
=
∑2
1
( ) . (9)
Sopostavlqq (8) y (9), naxodym
ω ( f ; 1 / n ) ≤ K
n
k
k
K
n
x
x
dx
k
n
k
k
k
n
2
1
1
2
12
1 1 1Ψ Ψ
≤ +
= −=
∑ ∫∑ ( ) ≤
≤
K
n
x
x
dx
K
n
u
u
du
n
n
1
2
1
2
3
1
1
1 1( )
( )
/
+
≤∫ ∫Ψ Ψ
. (10)
Pust\ 0 < h ≤ 1. Tohda, v¥byraq n tak, çtob¥ 1
1n +
≤ h < 1
n
, n ∈ N , yz (10)
poluçaem (4):
ω ( f ; h ) ≤ ω ( f ; 1 / n ) ≤ K h
u
u
du
h
3
2
3
1 Ψ( )∫ .
Teorema 1 dokazana.
Sledstvye 1. Pust\ Φ ( u ) = up, p ≥ 1. Esly dlq f ( ⋅ ) , zadannoj na S
2,
suwestvuet posledovatel\nost\ ( Tn ( x ) ) , n = 0, 1, 2, … , sferyçeskyx summ
takyx, çto
T x f xk
p
k C S
( ) ( )
( )
−
=
∞
∑
0 2
= M < + ∞ ,
to
ω ( f ; h ) ≤ K h p1/ , 0 < h ≤ 1, K ≡ cont > 0.
Teorema 2. Pust\ funkcyq Φ ( ⋅ ) neprer¥vnaq stroho vozrastagwaq v¥-
puklaq vverx na [ 0 , + ∞) y ravnaq nulg v nule, a Ψ ( ⋅ ) — obratnaq k nej funk-
cyq, pryçem
Ψ ( t1 + t2 ) ≤ A [ Ψ ( t1 ) + Ψ ( t2 ) ] ∀t1 , t2 > 0. (11)
Esly dlq funkcyy f ( x ) = f ( θ , ϕ ) , zadannoj na sfere S
2,
Φ ρk
k C S
f x( / )
( )
( ; )1 2
0 2
( )
=
∞
∑ = M < + ∞ ,
to dlq modulq neprer¥vnosty
ω ( f ; h ) ≤ K h
u u
u u
du
h
2
3
1 2 Ψ | |
| |
( )∫ ln
ln
/
, 0 < h ≤ 1. (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
STRUKTURNÁE SVOJSTVA FUNKCYJ, ZADANNÁX NA SFERE, NA OSNOVE … 23
Dokazatel\stvo. Otpravlqqs\ ot analoha neravenstva Lebeha [7]
ρk kf x K kE f( / )( ; ) ln ( )1 2 ≤ ∀x ∈ S
2, K ≥ 1,
y uçyt¥vaq, çto v sluçae v¥puklosty vverx funkcyq Φ( )u
u
ub¥vaet, naxodym
M ≥ Φ ρk
k n
n
C S
f x( / )
( )
( ; )1 2
2 1
2
( )
=
−
∑ =
Φ ρ
ρ
ρk
k
k
k n
n
C S
f x
f x
f x
( / )
( / )
( / )
( )
( ; )
( ; )
( ; )
1 2
1 2
1 2
2 1
2
( )
=
−
∑ ≥
≥
Φ K n E f
K n E f
f xn
n
k
k n
n
C S
ln ( )
ln ( )
( ; )( / )
( )
( )
=
−
∑ ρ 1 2
2 1
2
≥
≥
n n E f
K n E f n
f x f xn
n
k
k n
n
C S
Φ ln ( )
ln ( )
( ; ) ( )( / )
( )
( ) −
=
−
∑1 1 2
2 1
2
σ ≥
n n E f E f
K n E f
n n
n
Φ ln ( ) ( )
ln ( )
( ) 2 .
Otsgda poluçaem
E2n ( f ) ≤
K n E f
n n E f
n
n
1 ln ( )
ln ( )Φ( )
. (13)
S pomow\g yndukcyy pokaΩem v¥polnymost\ neravenstva
E fk2
( ) ≤ ln
ln
2
2
2
1k
k
k
C( )
−
Ψ , C ≡ cont > 0. (14)
Pry k = 1
M ≥ Φ ρ2
1 2
2
( / )
( )
( ; )f x
C S( ) ≥ Φ ( E2 ( f ) ) ≥ Φ ( E2 ( f ) ln 2),
t.8e.
E2 ( f ) ≤ ln
ln
2
2
2
1( )
− Ψ C ,
hde C = max
ln
,2
2
2 1
M AK
, postoqnnaq A udovletvorqet neravenstvu Ψ ( 2u ) ≤
≤ A Ψ ( u ) .
Vsledstvye (13)
E fk2 1+ ( ) ≤ K
E f
E f
k
k k
k
k
1
2
2
2
2 2
ln ( )
ln ( )Φ( ) ≤
K C
C
K C
C
k
k
k k k
k k
k
1
1
2
2
2 2 2
2 2
2
Ψ
Φ Ψ
Ψ
ln
ln
ln
ln
( )( ) =
⋅( )
−
−
≤
≤
2 2 2
2
2 2
2
1
1 1
1
1 1
1
A K C
C
Ck k
k
k k
k
Ψ Ψ⋅( )
≤
⋅( )− − +
+
− − +
+
ln
ln
ln
ln
.
Otsgda sleduet v¥polnenye (14) pry lgbom k ∈ N .
Podberem k ∈ N tak, çtob¥ 2k ≤ n < 2k
+1. Tohda yz (14), (11) sleduet
En ( f ) ≤ E fk2
( ) ≤
Ψ C k k
k
⋅( )−2 2
2
ln
ln
≤
A
n
C n
n
A n n
n
1 22
ln
ln ln
ln
/Ψ Ψ
≤
( )( )
. (15)
Sohlasno (9) y (15) ymeem
ω ( f ; 1 / n ) ≤ K n k E f K n
k k k
k
dxk
k
n
k
k
k
n
−
=
−
+
=
−
∑ ∫∑≤ ( )( )2
1
1
2
1
2
1
( )
ln
ln
/Ψ
≤
≤ K n
x x x
x
dx K n
u u
u u
du
k
k
k
n
k
k
k
n
2
2
1
2
1
2
2
3
1 1
1
2
1
−
+
=
−
−
+=
−( )( ) = ( )∫∑ ∫∑ | |
| |
Ψ Ψln
ln
ln
ln
/
/
/
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
24 R.8A.8LASURYQ
= K n
u u
u u
du
n
2
2
3
1
1 2
− | |
| |
( )∫ Ψ ln
ln/
/
.
Podbyraq n ∈ N tak, çtob¥ 1
1n +
≤ h < 1
n
, h > 0, naxodym (12):
ω ( f ; h ) ≤ ω ( f ; 1 / n ) ≤ K h
u u
u u
du
h
3
2
3
1 2 Ψ | |
| |
( )∫ ln
ln
/
.
Teorema 2 dokazana.
Sledstvye 2. Pust\ Φ ( u ) = up, p ∈ ( 0 , 1 / 2 ] . Esly dlq f ( x ) , zadannoj na
sfere S
2,
ρk
p
k C S
f x( / )
( )
( ; )1 2
0 2=
∞
∑ = M < + ∞ ,
to
ω ( f ; h ) ≤
K h p
K h
h
p h
2
2 2
0 1
2
1 1
2
0 1
2
, , ,
ln , , .
∈
= < ≤
Teorema 3. Pust\ funkcyq Φ ( ⋅ ) neprer¥vnaq stroho vozrastagwaq
v¥puklaq vverx na [ 0 , + ∞ ) y ravnaq nulg v nule, a Ψ ( ⋅ ) — obratnaq k nej
funkcyq, pryçem
Ψ ( t1 + t2 ) ≤ A [ Ψ ( t1 ) + Ψ ( t2 ) ] ∀t1 , t2 > 0.
Pust\, dalee, pry nekotorom δ ∈ ( 0 , 1 ) suwestvuet
Ψ u u
u u
dur
| |
| |
( )
−∫ ln
ln2 1
0
δ
.
Esly dlq f ( x ) = f ( θ , ϕ ) , zadannoj na S
2,
Φ ρk
k C S
f x( / )
( )
( ; )1 2
0 2
( )
=
∞
∑ = M < + ∞ ,
to
ωk
rD f h−( )1 ; ≤ K r k h
u u
u u
du
u u
u u
duk
k r
h
r
h
( , )
ln
ln
ln
ln( )
/
2
2 1
1 2
2 1
0
Ψ Ψ| |
| |
| |
| |
( ) + ( )
+ − −∫ ∫ , 0 < h ≤ 1
2
,
(16)
hde D fr−1 — rezul\tat prymenenyq r – 1 ( r ≥ 2) raz operatora Laplasa D,
E D fn
r−( )1 ≤ K r
n n n
n
u u
u u
du
r
r
n
( )
ln
ln
ln
ln
( ) /( )
/2 1
2 1
0
1 1−
−
+( )( ) + ( )
| |
| |∫
Ψ Ψ
. (17)
Dokazatel\stvo. Ar.8S.8DΩafarov¥m [9] ustanovleno neravenstvo
ωk
rD f n; /1( ) ≤ K r n E f E fk k r
n
r
n
( ) ( ) ( )( )− + −
=
−
= +
∞
∑ ∑+
2 2 1
1
2 1
1
ν νν
ν
ν
ν
(18)
pry uslovyy koneçnosty pravoj çasty.
Na osnovanyy (15) y (18) poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
STRUKTURNÁE SVOJSTVA FUNKCYJ, ZADANNÁX NA SFERE, NA OSNOVE … 25
ωk
rD f n−( )1 1; / ≤
≤ K r k
n
E f E fk
k r
n
k
n
( , ) ( ) ( )( ) ( )1
2
2 3
1
2 1 1
1
ν νν
ν
ν
ν
+ −
=
− −
= +
∞
∑ ∑+
≤
≤ K r k
n ek
k r
n
r
n
1 2
2 3
1
2 3
1
1( , )
ln
ln( )
ln
ln
( ) / /ν ν
ν
ν ν
ν
ν ν
ν ν
+ −
=
−
= +
∞( )( ) + ( )( )
∑ ∑Ψ Ψ
≤
≤ K r k
n
x x x
x
dx
x x x
x
dxk
k rn r
n
2 2
2 31
2
1 2 31
1
1( , )
ln
ln
ln
ln
/ /( )Ψ Ψ( )( ) + ( )( )
+ −+
=
− −+
= +
∞
∫∑ ∫∑
ν
ν
ν ν
ν
ν
=
= K r k
n
u u
u u u
du
u u
u u
duk k r
n
r
n
2 2 2 3 2
1 1
1
2
1
2 1
1 1
1
1
1( , )
ln
ln
ln
ln( )
/
/
/
/Ψ Ψ| |
| |
| |
| |
( ) + ( )
+ −
+=
−
−
+= +
∞
∫∑ ∫∑
ν
ν
ν ν
ν
ν
=
= K r k
n
u u
u u
du
u u
u u
duk k r
n
r
n
2 2 2 1
1
1 2
2 1
0
1 1
1( , )
ln
ln
ln
ln( )
/
/ /( )Ψ Ψ| |
| |
| |
| |
( ) + ( )
+ − −
+
∫ ∫ .
Podbyraq n tak, çtob¥ 1
1n +
≤ h < 1
n
, naxodym
ωk
rD f h−( )1 ; ≤ ωk
rD f n−( )1 1; / ≤
≤ K r k h
u u
u u
du
u u
u u
duk
k r
h
r
h
( , )
ln
ln
ln
ln( )
/
2
2 1
1 2
2 1
0
Ψ Ψ| |
| |
| |
| |
( ) + ( )
+ − −∫ ∫ , 0 < h ≤ 1
2
,
y neravenstvo (16) ustanovleno.
Dlq ustanovlenyq neravenstva (17) vnov\ vospol\zuemsq rezul\tatom
Ar.8S.8DΩafarova [9]:
E D fn
r( ) ≤ K r h E f E fr
n
r
n
( ) ( ) ( )2 2 1
1
+
−
= +
∞
∑ ν ν
ν
.
Otsgda s uçetom (15) ymeem
E D fn
r–1( ) ≤ K r h E f E fr
n
r
n
( ) ( ) ( )( ) ( )2 1 2 1 1
1
− − −
= +
∞
+
∑ ν ν
ν
≤
≤ K r
n n n
n
u u
u u
du
r
r
n
1
2 1
2 1
0
1 1
( )
ln
ln
ln
ln
( ) /( )
/−
−
+( )( ) + ( )
| |
| |∫
Ψ Ψ
.
1. Freud G. Über die Sättigungsklasse der starken Approximation durch Teilsummen der Fouriershen
Reihe // Acta math. Acad. sci. hung. – 1969. – 20. – P. 275 – 279.
2. Leindler L. Strong approximation of Fourier series and structural properties of functions // Ibid. –
1979. – 33, # 1 – 2. – P. 105 – 125.
3. Leindler L., Nikišin E. Note on strong approximation by Fourier series // Ibid. – 1973. – 24. –
P. 223 – 227.
4. Szabados J. On a problem of Leindler concerning strong approximation by Fourier series // Anal.
math. – 1976. – 2. – P. 155 – 161.
5. Totik V. On the modules continuity connection with a problem of Szabados concerning strong
approximation // Ibid. – 1978. – 4. – P. 145 – 152.
6. Totik V. On structural properties of functions arising from strong approximation of Fourier series //
Acta sci. math. – 1979. – 41. – P. 227 – 251.
7. Topuryq,S.,B. Rqd¥ Fur\e – Laplasa na sfere. – Tbylysy: Tbylys. un-t, 1987. – 3568s.
8. Xardy,H., Lyttl\vud,D., Polya,H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. – 4568s.
9. DΩafarov ,Ar.,S. O porqdke nayluçßyx pryblyΩenyj neprer¥vn¥x na edynyçnoj sfere
funkcyj posredstvom koneçn¥x sferyçeskyx summ // Yssled. po sovr. probl. konstr. teo-
ryy funkcyj. – Baku: Yzd-vo AN8AzSSR, 1965. – S.846 – 52.
Poluçeno 17.09.82003,
posle dorabotky — 14.03.82005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
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| id | umjimathkievua-article-3431 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:42:27Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-34312020-03-18T19:54:30Z Structural properties of functions defined on a sphere on the basis of Φ-strong approximation Структурные свойства функций, заданных на сфере, на основе Φ-сильной аппроксимации Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. Structural properties of functions defined on a sphere are determined on the basis of the strong approximation of Fourier-Laplace series. Встановлюються структурні властивості функцій, заданих на сфері, на основі сильної апроксимації рядів Фур'є - Лапласа. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3431 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 20–25 Український математичний журнал; Том 58 № 1 (2006); 20–25 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3431/3602 https://umj.imath.kiev.ua/index.php/umj/article/view/3431/3603 Copyright (c) 2006 Lasuriya R. A. |
| spellingShingle | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. Structural properties of functions defined on a sphere on the basis of Φ-strong approximation |
| title | Structural properties of functions defined on a sphere on the basis of Φ-strong approximation |
| title_alt | Структурные свойства функций, заданных на сфере, на основе Φ-сильной аппроксимации |
| title_full | Structural properties of functions defined on a sphere on the basis of Φ-strong approximation |
| title_fullStr | Structural properties of functions defined on a sphere on the basis of Φ-strong approximation |
| title_full_unstemmed | Structural properties of functions defined on a sphere on the basis of Φ-strong approximation |
| title_short | Structural properties of functions defined on a sphere on the basis of Φ-strong approximation |
| title_sort | structural properties of functions defined on a sphere on the basis of φ-strong approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3431 |
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