Summation of Fourier-Laplace Series in the Space $L(S^m)$
We establish estimates of the rate of convergence of a group of deviations on a sphere in the space $L(S^m),\quad m > 3$.
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| Datum: | 2005 |
|---|---|
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509736591425536 |
|---|---|
| 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 | 2020-03-18T20:00:05Z |
| description | We establish estimates of the rate of convergence of a group of deviations on a sphere in the space $L(S^m),\quad m > 3$. |
| first_indexed | 2026-03-24T02:45:51Z |
| format | Article |
| fulltext |
UDK 517.51
R. A. Lasuryq (Abxaz. un-t, Suxum)
SUMMYROVANYE RQDOV FUR|E – LAPLASA
V PROSTRANSTVE L Sm( )
We establish estimates of the rate of convergence of a group of deviations on a sphere in the space
L Sm( ), m ≥ 3.
Vstanovleno ocinky ßvydkosti zbiΩnosti hrupy vidxylen\ na sferi u prostori L Sm( ), m ≥ 3.
1. Pust\ R
m
— m-mernoe evklydovo prostranstvo, m ≥ 3, S
m
— edynyçnaq
sfera v R
m
s centrom v naçale koordynat, L( S
m
) — prostranstvo funkcyj
f ( x ) s normoj
f f x dS xL S
S
m
m
( ) ( ) ( )= < +∞∫ ,
S f n f y P dS y
n S
n
m
[ ] = + ( )+
=
∞
∑ ∫Γ( )( ) ( ) cos ( )λ λ
π
γλ2 1
0
, λ = −m 2
2
, (1)
— rqd Fur\e – Laplasa funkcyy f x L Sm( ) ∈ ( ) , x x xm= ( )1, ... , ,
Γ( )α α= − −
∞
∫ e t dtt 1
0
, α > 0,
— hamma-funkcyq ∏jlera,
σ γλ
λ
λ λ
n
n k
n
n k kf x
A
n k A P( ) ( )( ; ) ( ) (cos )= +
=
−∑1
0
— srednye Çezaro ( c, λ ) rqda (1), hde
Φn
n k
n
n k kA
n k A P( ) ( )(cos ) ( ) (cos )λ
λ
λ λγ γ= +
=
−∑1
0
,
cos γ = ( x, y ) — skalqrnoe proyzvedenye vektorov
x x xm= ( , , )1 … , y y ym= ( , , )1 … , x ∈ S
m
, y ∈ S
m
,
S f xn
( )( ; )λ
— çastyçnaq summa rqda (1), P tk
( )( )λ
— mnohoçlen¥ Hehenbauπra
(ul\trasferyçeskye mnohoçlen¥), kotor¥e opredelqgtsq yz razloΩenyq
1 2 2
0
− +( ) =
−
=
∞
∑th h P t h
n
n
nλ λ( )( ) .
Pust\, dalee,
ρ σλ λ
k kf x f x f x( ) ( )( ; ) ( ; )= ( ) − ,
D f v
n
v f xn L S
k n
n
k k
L S
m
m
( )
( )
( )
( )
( ; ; ) ( ) ( ; )λ λα α ρ=
=
−
∑1 2 1
, (2)
© R. A. LASURYQ, 2005
496 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
SUMMYROVANYE RQDOV FUR|E – LAPLASA … 497
G f v v f xn L S
k n
k k
L S
m
m
( )
( )
( )
( )
( ; ; ) ( ) ( ; )λ λα α ρ=
=
∞
∑ , (3)
hde α = (αk( v )) , k ∈ N, v ∈ E, — nekotoraq neotrycatel\naq posledovatel\-
nost\ funkcyj, zadann¥x na mnoΩestve E, soderΩawem po krajnej mere odnu
predel\nug toçku.
2. Pryvedem snaçala utverΩdenye, soderΩawee ocenku skorosty sxodymos-
ty velyçyn¥ (2).
Teorema 1. Pust\ α = (αk( v )) , k ∈ N, v ∈ E, — neotrycatel\naq posle-
dovatel\nost\ funkcyj takaq, çto pry kaΩdom fyksyrovannom v ∈ E çysla
αk( v ) ne vozrastagt. Tohda ∀ f ( x ) ∈ L( S
m
) , m ≥ 3, ∀ v ∈ E
D f v K v E fn n n L Sm
( )
( )
( ; ; ) ( ) ( )α α α≤ , (4)
hde E fn L Sm( )
( )
— nayluçßee pryblyΩenye funkcyy f ∈ L( S
m
) sferyçeskymy
harmonykamy porqdka n v metryke prostranstva L ( S
m
) , K = K( λ ) — vely-
çyna, ravnomerno ohranyçennaq po n ∈ N, v ∈ E y f ∈ L( S
m
) .
Otmetym nekotor¥e fakt¥, v¥tekagwye yz teorem¥ 1.
Rassmatryvaq pry l ∈ N
Vl
l
k l
l
kf x
l
f x2 1
2 1
1−
=
−
= ∑, ( )( ; ) ( ; )λ λσ
— srednye Valle Pussena Vn p
n f x−
, ( ; )λ
summ σ λ
k f x( )( ; ) , v kotor¥x n = 2l – 1,
p = l – 1, v sylu neravenstva (4) pry λ k( v ) ≡ 1, v ∈ E, ∀ f ∈ L( S
m
) , m ≥ 3,
ymeem
f x f x KE fl
l
L S l L Sm m( ) ( ; ) ( ),
( ) ( )
− ≤−
V
2 1 λ .
Sledugwyj fakt sformulyruem v vyde teorem¥.
Teorema 2. Pust\ α = (αk( v )) , k ∈ N, v ∈ E, udovletvorqet uslovyqm
teorem¥ 1. Tohda dlq lgboj f ( x ) ∈ L( S
m
), m ≥ 3, ∀ v ∈ E
G f v K n v E fn L S n n L Sm m
( )
( ) ( )
( ; ; ) ( ) ( )λ α α≤
+
+ αk k L S
k n
v E f m( ) ( )
( )
=
∞
∑
, K = K( λ ) , (5)
hde G f vn
( )( ; ; )λ α — velyçyna, opredelqemaq ravenstvom (3).
Dokazatel\stvo teorem¥ 2. V prynqt¥x oboznaçenyqx v sylu neravenstva
(4) ymeem
G f v v f xn
i k n
n
k k
L S
i
i
m
( ) ( )
( )
( ; ; ) ( ) ( ; )λ λα α ρ=
=
∞
=
−
∑ ∑
+
0 2
2 1
1
≤
≤
i k n
n
k k
L S
i
i
m
v f x
=
∞
=
−
∑ ∑
+
0 2
2 1
1
α ρ λ( ) ( ; )( )
( )
=
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
498 R. A. LASURYQ
=
i
i
i
k n
n
k k
L S
n
n
v f x
i
i
m=
∞
=
−
∑ ∑
+
0 2
2 1
2
1
2
1
α ρ λ( ) ( ; )( )
( )
≤
≤ K n v E f
i
i
n n L Si i m
=
∞
∑
0
2 2
2 α ( ) ( )
( )
=
= K n v E f n v E fn n L S
i
i
n n L Sm i i mα α( ) ( ) ( ) ( )
( ) ( )
+
=
∞
∑
1
2 2
2 ≤
≤ K n v E f v E fn n L S
i k n
n
k k L Sm
i
i
mα α( ) ( ) ( ) ( )
( ) ( )
+
=
∞
=
∑ ∑
−1 2
2
1
=
= K n v E f v E fn n L S
k n
k k L Sm mα α( ) ( ) ( ) ( )
( ) ( )
+
=
∞
∑ ,
K = K( λ ) .
Neravenstvo (5) ustanovleno.
Yz teorem¥ 2, v svog oçered\, sleduet rqd druhyx faktov. Polahaq v (5)
n = 1, v pryvedenn¥x v¥ße uslovyqx ymeem
α ρ αλ
k k
k L S
k k L S
k
v f x K v E f
m
m( ) ( ; ) ( ) ( )( )
( )
( )
=
∞
=
∞
∑ ∑≤
1 1
. (6)
Krome toho, esly pry kaΩdom fyksyrovannom v ∈ E
αk
k
v( ) =
=
∞
∑ 1
1
,
to v sylu (6)
f x U f xv
L Sm( ) ( ; )( ),
( )
− λ
≤
≤ K v E fk k L S
k
mα ( ) ( )
( )
=
∞
∑
1
, (7)
hde
U f x v f xv
k k
k
( ), ( )( ; ) ( ) ( ; )λ λα σ=
=
∞
∑
1
.
Yz neravenstva (7) poluçaem ocenky uklonenyj v metryke prostranstva
L( S
m
) dlq dostatoçno ßyrokoho spektra lynejn¥x srednyx summ σ λ
n f x( )( ; ) .
Naprymer, pry v = n ∈ N y
αk n
n k n
k n
( )
, ,
, ,
=
≤ ≤
>
−1 1
0
naxodym ocenku uklonenyq srednyx Fejera summ σ λ
n f x( )( ; ) :
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
SUMMYROVANYE RQDOV FUR|E – LAPLASA … 499
f x
n
f xk
k
n
L Sm
( ) ( ; )( )
( )
−
=
∑1
1
σ λ
≤
≤
K
n
E fk L S
k
n
m( )
( )
=
∑
1
, K = K( α ).
Polahaq αk( v ) = (1 – v )v
k
–
1
, 0 < v < 1, poluçaem ocenku uklonenyq srednyx
Abelq summ σ λ
n f x( )( ; ) :
f x v v f xk
k
k
L Sm
( ) ( ) ( ; )( )
( )
− − −
=
∞
∑1 1
1
σ λ
≤
≤ K v v E fk
k L S
k
m( ) ( )
( )
1 1
1
−
−
=
∞
∑ .
Esly Ωe
αk n
k n k n
k n
( )
ln( ) , ,
, ,
=
+( ) ≤
>
−1
0
1
to ymeem ocenku uklonenyq loharyfmyçeskyx srednyx summ σ λ
n f x( )( ; ) :
f x
n k
f x
k
n
k
L Sm
( )
ln( )
( ; )( )
( )
−
+ =
∑1
1
1
1
σ λ ≤
≤ K
n k
E fk L S
k
n
m
1
1
1
1ln( )
( )
( )+
=
∑ , K = K( λ ) ,
y t. d.
Dokazatel\stvo teorem¥ 1. Pust\ Yn( x ) — sferyçeskaq harmonyka
nayluçßeho pryblyΩenyq funkcyy f v metryke L( S
m
) . Tohda, polahaq
δn ( f; x ) = f ( x ) – Yn ( x ) ,
poluçaem
δn L S n L S
f x E fm m( ; ) ( )( ) ( )
=
y, tak kak
σ λ
k n nY x Y x( )( ; ) ( )= ∀ k ≥ n,
ρ δ σ δλ λ
k n k nf x f x x( ) ( )( ; ) ( ; ) ( ; )= − .
Otsgda
ρ δ λ
π
δ γλ
λ
λ
k n n
S
kf x f x f y ds y
m
( ) ( )( ; ) ( ; )
( )
( ; ) (cos ) ( )= − + ∫Γ Φ
2 1 =
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
500 R. A. LASURYQ
= δ λ
π
γ γ γλ
λ
π
n x
n
kf x f d( ; )
( )
( ; ) (cos )( ) ( )− + ∫Γ ∆ Φ
2 1
0
=
= δ λ
πλn k
i
i
f x U f x( ; )
( )
( ; )( )− +
=
∑Γ
2 1
1
3
, (8)
hde
∆x
n
n
x y
f f y dt y( )
( , ) cos
( ; ) ( ; ) ( )γ δ
γ
=
=
∫ , (9)
U f x f dk
i
x
n
k
ei
( ) ( ) ( )( ; ) ( ; ) (cos )= ∫ ∆ Φγ γ γλ
, i = 1, 2, 3, (10)
e
k1 0
2 1
=
+
,
( )
π
, e
k k2 2 1 2 1
=
+
−
+
π π π
( )
,
( )
,
e
k3 2 1
= −
+
π π π
( )
, .
Prynymaq vo vnymanye (8) y opredelenye (2), naxodym
D f vn
( )( ; ; )λ α =
=
1
2
2 1
1
1
3
n
v f x U f xk
k n
n
n k
i
i L Sm
α δ λ
πλ( ) ( ; ) ( ) ( ; )( )
( )=
−
+
=
∑ ∑−
Γ
≤
≤
1 2 1
n
v f xk n
k n
n
L Sm
α δ( ) ( ; )
( )=
−
∑ +
+
Γ( )
( ) ( ; )( )
( )
λ
π
αλ2
1
1
1
3 2 1
+
= =
−
∑ ∑
i
k
k n
n
k
i
L S
n
v U f x
m
≤
≤ α λ
π
αλn n L S
i
n
iv E f I f vm( ) ( )
( )
( ; ; )
( )
( )+ +
=
∑Γ
2 1
1
3
, (11)
hde
I f v
n
v U f xn
i
k n
n
k k
i
L Sm
( ) ( )
( )
( ; ; ) ( ) ( ; )α α=
=
−
∑1 2 1
.
Ocenym v otdel\nosty kaΩdoe slahaemoe v pravoj çasty (11).
Pust\
S f x
S h
f y dt yh m
x y h
( ; )
sin
( ) ( )
( , ) cos
= −
=
∫1
1 2λ
— sferyçeskyj sdvyh dlq funkcyy f ( x ) , x ∈ S
m
, m ≥ 3, s ßahom h > 0.
Yzvestno (sm., naprymer, [1, s. 7]), çto
S f x fh L S L Sm m( ; ) ( ) ( )≤ . (12)
Tohda v sylu neravenstva [2]
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
SUMMYROVANYE RQDOV FUR|E – LAPLASA … 501
Φk Ck( )(cos )λ λγ ≤ +2 1, 0 ≤ γ ≤ π, C > 0,
s uçetom (9), (10), (12) y uslovyq teorem¥ poluçaem
I f vn
( )( ; ; )1 α =
=
1 2 1
1
n
v f dk
k n
n
x
n
k
e L Sm
α γ γ γλ( ) ( ; ) (cos )( ) ( )
( )=
−
∑ ∫ ∆ Φ ≤
≤ K v
n
f k dn x
n
ek n
n
L Sm
α γ γλ( ) ( ; ) ( )
( )
1 2 1
2 1
1
∆ +
=
−
∫∑ ≤
≤
K v
n
k f dn
k n
n
x
n
e L Sm
α γ γλ( )
( ; )( )
( )
2 1
2 1
1
+
=
−
∑ ∫ ∆ =
=
K v
n
k f dn
k n
n
x
n
L S
e
m
α γ γλ( )
( ; )( )
( )
2 1
2 1
1
+
=
−
∑ ∫ ∆ =
=
K v
n
k S x dn
k n
n
n L S
e
m
1 2 1
2 1
2
1
α δ γ γλ
γ
λ( )
( ; ) sin
( )
+
=
−
∑ ∫ ≤
≤
K v
n
k f x dn
k n
n
n L S
e
m
1 2 1
2 1
2
1
α δ γ γλ λ( )
( ; ) sin( )
+
=
−
∑ ∫ ≤
≤ K v E fn n L Sm2α ( ) ( )
( )
. (13)
Analohyçn¥my rassuΩdenyqmy poluçaem
I f vn
( )( ; ; )3 α ≤
≤
K v
n
k f dn
x
n
k
ek n
n
L Sm
α γ γ γλ λ( )
; (cos )( ) ( )
( )
2 1
2 1
1
+
=
−
( )∫∑ ∆ Φ ≤
≤
K v
n
k f dn
x
n
L S
ek n
n
m
α γ γλ( )
( ; )( )
( )
2 1
2 1
3
+
=
−
∫∑ ∆ =
=
K v
n
k S x dn
n L S
ek n
n
m
1 2 1 2
2 1
3
α δ γ γλ
γ
λ( )
( ; ) sin
( )
+
=
−
∫∑ ≤
≤
K v
n
E f k dn
n L S
ek n
n
m
1 2 1 2
2 1
3
α γ γλ λ( )
( ) sin
( )
+
=
−
∫∑ =
=
K v
n
E f k dn
n L S
k
k n
n
m
1 2 1
0
2 12 1
2α γ γλ
π
λ( )
( ) sin
( )
/ ( )
+
+
=
−
∫∑ ≤
≤ K v E fn n L Sm2α ( ) ( )
( )
. (14)
Dalee vospol\zuemsq asymptotyçeskym v¥raΩenyem qdra Φk
( )(cos )λ γ [2]:
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
502 R. A. LASURYQ
Φk
( )(cos )λ γ =
=
λ
λ γ λπ
γ γ
η γ
γ γλ λ λ λ λ4
3 1
2
2
1
1 21 1 1
sin
(sin ) sin( / )
( )
(sin ) sin( / )
k
k
k
+ +
−
( )
+
+ ( )+ + + =
= Φ Φk k,
( )
,
( )(cos ) (cos )1 2
λ λγ γ+ ,
π γ π
2 2 1
− <
+
k
k( )
,
hde η γk K( ) < .
V πtyx oboznaçenyqx ymeem
I f vn
( )( ; ; )2 α ≤
≤
1 2 1
1n
v f d
k n
n
k x
n
k
e L Sk m=
−
∑ ∫α γ γ γλ( ) ( ; ) (cos )( )
,
( )
( )
∆ Φ +
+
1 2 1
2
2
n
v f d
k n
n
k x
n
k
e L Sm=
−
∑ ∫α γ γ γλ( ) ( ; ) (cos )( )
,
( )
( )
∆ Φ =
= I f v I f vn n,
( )
,
( )( ; ; ) ( ; ; )1
2
2
2α α+ . (15)
Ocenym velyçynu I f vn,
( )( ; ; )1
2 α . S πtoj cel\g zametym, çto
I f vn,
( )( ; ; )1
2 α ≤
≤
1 2 1
1 2
1 2
1n
v f d
k n
n
k
n
n
x
n
k
L Sm=
− −
∑ ∫α γ γ γ
π
λ( ) ( ; ) (cos )
/
/
( )
,
( )
( )
∆ Φ +
+
1 2 1
1 2
2 1
1n
v f d
k n
n
k
n
k
x
n
k
L Sm=
− +
∑ ∫α γ γ γ
π
λ( ) ( ; ) (cos )
/
/ ( )
( )
,
( )
( )
∆ Φ +
+
1 2 1
2 1
1 2
1n
v f d
k n
n
k
k
n
x
n
k
L Sm=
−
− +
−
∑ ∫α γ γ γ
π π
π
λ( ) ( ; ) (cos )
/ ( )
/
( )
,
( )
( )
∆ Φ =
= A f v B f v C f vn n n( ; ; ) ( ; ; ) ( ; ; )α α α+ + . (16)
Dalee,
A f vn( ; ; )α ≤
≤ 1
3 1
2
2
2 1
1 2
1 2
1n
v f
k
d
k n
n
k
n
n
x
n
L Sm
=
− −
+∑ ∫
+ −
( ) ( )
α γ
γ λ γ λπ
γ γ
γ
π
λ λ( ) ( ; )
sin cos
sin sin //
/
( )
( )
∆ +
+ 1
3 1
2
2
2 1
1 2
1 2
1n
v f
k
d
k n
n
k
n
n
x
n
L Sm
=
− −
+∑ ∫
+ −
( ) ( )
α γ
γ λ γ λπ
γ γ
γ
π
λ λ( ) ( ; )
cos sin
sin sin //
/
( )
( )
∆ =
= A f v A f vn n, ,( ; ; ) ( ; ; )1 2α α+ . (17)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
SUMMYROVANYE RQDOV FUR|E – LAPLASA … 503
Prynymaq vo vnymanye uslovye teorem¥ y prymenqq preobrazovanye Abelq,
naxodym
A f vn, ( ; ; )1 α ≤
≤
1
3 1
2
21 2
1 2
1
2 1
n
f
v k d
n
n x
n
k
k n
n
L Sm
/
/ ( )
( )
( ; )cos
sin sin( / )
( )sin
π
λ λ
γ λ γ λπ
γ γ
α γ γ
−
+
=
−
∫ ∑
+ −
( ) ( )
∆
≤
≤
K
n
f v
d
n
n
x
n
n
L Sm1 2
1 2
12/
/ ( )
( )
( ; )
sin sin( / )
( )
π
λ λ
γ
γ γ
α
γ
γ
−
+∫ ( ) ( )
∆
. (18)
Na osnovanyy neravenstva (12) yz (18) poluçaem
A f vn, ( ; ; )1 α ≤
≤
K v
n
S x dn
n
n
n L Sm
1
1 2
1 2 2
12
α δ γ
γ γ
γ
γ
π γ
λ
λ λ
( ) ( ; ) sin
sin sin( / )/
/
( )
−
+∫ ( ) ( )
≤
≤ K v E fn n L Sm2α ( ) ( )
( )
. (19)
Analohyçno ocenyvaetsq velyçyna A f vn, ( ; ; )2 α :
A f v K v E fn n n L Sm, ( )
( ; ; ) ( ) ( )2 α α≤ . (20)
Sledovatel\no, v sylu (19) y (20) yz (17) poluçaem
A f v K v E fn n n L Sm( ; ; ) ( ) ( )
( )
α α≤ . (21)
Dalee,
B f vn( ; ; )α ≤
≤
K v
n
f
dn
k n
n
n
k
x
n
L Smα γ
γ γ
γ
π
λ λ
( ) ( ; )
sin sin( / )/
/ ( ) ( )
( )
=
− +
+∑ ∫ ( ) ( )
2 1
1 2
2 1
12
∆
≤
≤
K v
n
E f dn
n L S
k n
n
n
k
m
1
2 1
1 2
2 1
12
α γ
γ
γ
π λ
λ
( )
( )
(sin )
sin( / )( )
/
/ ( )
=
− +
+∑ ∫ ( )
≤
≤ K v E fn n L Sm2α ( ) ( )
( )
. (22)
S uçetom (12) naxodym
C f vn( ; ; )α ≤
≤
K v
n
E f dn
n L S
k n
n
k
n
m
α γ
γ
γ
π π
π λ
λ
( )
( )
(sin )
sin( / )( )
/ ( )
/
=
−
− +
−
+∑ ∫ ( )
2 1
2 1
1 2
12
≤
≤
K v
n
E f dn
n L S
k n
n
k
n
m
α γ γ
π π
π
λ( )
( ) (sin )
( )
/ ( )
/
=
−
− +
−
∑ ∫
2 1
2 1
1 2
≤
≤ K v E fn n L Sm1α ( ) ( )
( )
. (23)
Takym obrazom, vsledstvye (21) – (23) yz (16) poluçaem
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
504 R. A. LASURYQ
I f v K v E fn n n L Sm,
( )
( )
( ; ; ) ( ) ( )1
2 α α≤ . (24)
Sohlasno opredelenyg velyçyn Φk,
( ) (cos )2
λ γ
I f vn,
( ) ( ; ; )2
2 α ≤
≤
K v
n k
f
dn
k n
n
x
n
e L Sm
α γ γ
γ γλ λ
( )
( ; )
sin sin( / )
( )
( )=
−
+ +∑ ∫+ ( ) ( )
2 1
1 1
1
1 2
2
∆ ≤
≤
K v
n k
S x
dn
k n
n n L S
e
m
1
2 1 2
1 1
1
1 2
2
α δ γ
γ γ
γ
γ
λ
λ λ
( ) ( ; ) sin
sin sin( / )
( )
=
−
+ +∑ ∫+ ( ) ( )
≤
≤
K v
n
E f
k
dn
n L S
k n
n
e
m
2
2 1
21
1
2
α γ γ( )
( )
( )
=
−
−∑ ∫+
≤
≤ K v E fn n L Sm3α ( ) ( )
( )
. (25)
Sohlasno (24), (25) yz (15) poluçaem
I f v K v E fn n n L Sm
( )
( )
( ; ; ) ( ) ( )2 α α≤ . (26)
Obæedynqq sootnoßenyq (13), (14), (26) y (11), pryxodym k utverΩdenyg teore-
m¥ 1.
Otmetym, çto analohyçn¥e ocenky v sluçae tryhonometryçeskyx rqdov
Fur\e ustanovlen¥ v rabotax [3, 4].
1. Topuryq S. B. Rqd¥ Fur\e – Laplasa na sfere. – Tbylysy: Yzd-vo Tbylys. un-ta, 1987. –
356 s.
2. Kogbetliantz E. Recherches sur la summabilitié des seris ultraspheriques par la methode des
moyennes arithmetiques // J. Math. Pures Appl. – 1924. – 9, # 3. – P. 107 – 187.
3. Stepanec A. Y., Paçulya N. L. Skorost\ sxodymosty hrupp¥ uklonenyj v prostranstve Lβ
ψ
// Vopros¥ summyrovanyq prost¥x y kratn¥x rqdov Fur\e. – Kyev, 1987. – S. 3 – 8. –
(Preprynt / AN USSR. Yn-t matematyky; 87.40).
4. Paçulya N. L. O syl\noj summyruemosty rqdov Fur\e // Tam Ωe. – S. 44 – 50.
Poluçeno 17.09.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
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| id | umjimathkievua-article-3616 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:51Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/61/64279d92fc6edac169bcf790c0178c61.pdf |
| spelling | umjimathkievua-article-36162020-03-18T20:00:05Z Summation of Fourier-Laplace Series in the Space $L(S^m)$ Суммирование рядов Фурье - Лапласа в пространстве $L(S^m)$ Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. We establish estimates of the rate of convergence of a group of deviations on a sphere in the space $L(S^m),\quad m > 3$. Встановлено оцінки швидкості збіжності групи відхилень на сфері у просторі $L(S^m),\quad m > 3$. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3616 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 496–504 Український математичний журнал; Том 57 № 4 (2005); 496–504 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3616/3963 https://umj.imath.kiev.ua/index.php/umj/article/view/3616/3964 Copyright (c) 2005 Lasuriya R. A. |
| spellingShingle | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. Summation of Fourier-Laplace Series in the Space $L(S^m)$ |
| title | Summation of Fourier-Laplace Series in the Space $L(S^m)$ |
| title_alt | Суммирование рядов Фурье - Лапласа в пространстве $L(S^m)$ |
| title_full | Summation of Fourier-Laplace Series in the Space $L(S^m)$ |
| title_fullStr | Summation of Fourier-Laplace Series in the Space $L(S^m)$ |
| title_full_unstemmed | Summation of Fourier-Laplace Series in the Space $L(S^m)$ |
| title_short | Summation of Fourier-Laplace Series in the Space $L(S^m)$ |
| title_sort | summation of fourier-laplace series in the space $l(s^m)$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3616 |
| work_keys_str_mv | AT lasuriyara summationoffourierlaplaceseriesinthespacelsm AT lasuriâra summationoffourierlaplaceseriesinthespacelsm AT lasuriâra summationoffourierlaplaceseriesinthespacelsm AT lasuriyara summirovanierâdovfurʹelaplasavprostranstvelsm AT lasuriâra summirovanierâdovfurʹelaplasavprostranstvelsm AT lasuriâra summirovanierâdovfurʹelaplasavprostranstvelsm |