Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series
We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of ψ-integrals of multivariable functions
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509743798288384 |
|---|---|
| author | Bodraya, V. I. Novikov, O. A. Rukasov, V. I. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. |
| author_facet | Bodraya, V. I. Novikov, O. A. Rukasov, V. I. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. |
| author_sort | Bodraya, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:05Z |
| description | We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of ψ-integrals of multivariable functions
|
| first_indexed | 2026-03-24T02:45:57Z |
| format | Article |
| fulltext |
UDK 517.5
V. Y. Rukasov, O. A. Novykov, V. Y. Bodraq (Slavqn. ped. un-t)
PRYBLYÛENYE KLASSOV ψψ-YNTEHRALOV
PERYODYÇESKYX FUNKCYJ MNOHYX
PEREMENNÁX PRQMOUHOL|NÁMY
LYNEJNÁMY SREDNYMY YX RQDOV FUR|E
We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of
ψ -integrals of multivariable functions
OderΩano asymptotyçni rivnosti dlq vidxylen\ prqmokutnyx linijnyx serednix rqdiv Fur’[ na
klasax ψ -intehraliv funkcij bahat\ox zminnyx.
Klass¥ ψ -yntehralov peryodyçeskyx funkcyj odnoj peremennoj b¥ly vve-
den¥ v 1996 h. A. Y. Stepancom (sm. [1]) sledugwym obrazom.
Pust\ L — prostranstvo summyruem¥x 2π-peryodyçeskyx funkcyj, f ∈ L y
S [ f ] =
a
a kx b kx A f xk k
k
k
k
0
1 02
+ ( + ) ≡ ( )
=
∞
=
∞
∑ ∑cos sin ;
— rqd Fur\e funkcyy f. Pust\, dalee, ψ = ( ψ1 , ψ2 ) — para proyzvol\n¥x
fyksyrovann¥x system çysel ψ1 ( k ) y ψ2 ( k ), k = 0, 1, … , takyx, çto ψ1 ( 0 ) =
= 1, ψ2 ( 0 ) = 0, ψ( )k = ψ ψ1
2
2
2( ) + ( )k k ≠ 0, k ∈ N , y
˜ ;A f xk ( ) = ak sin k x –
– bk cos k x.
Esly rqd
ψ
ψ
ψ
ψ
1
2
2
2
1
( )
( )
( ) − ( )
( )
( )
=
∞
∑ k
k
A f x
k
k
A f xk k
k
; ˜ ;
qvlqetsq rqdom Fur\e nekotoroj summyruemoj funkcyy, to πtu funkcyg obo-
znaçagt f ψ
y naz¥vagt ψ -proyzvodnoj funkcyy f. Funkcyg f pry πtom na-
z¥vagt ψ -yntehralom funkcyy f ψ
.
Esly funkcyq f neprer¥vna y v¥polneno uslovye ess sup f xψ( ) ≤ 1, to
polahagt f ∈ C∞
ψ
.
Yssledovanyg approksymatyvn¥x svojstv klassov C∞
ψ
posvqwen obßyrn¥j
kruh rabot (sm., naprymer, [1 – 6]). V rabotax [2, 5] moΩno najty byblyohrafyg
po voprosam, prym¥kagwym k πtoj tematyke. V to Ωe vremq vopros¥ prybly-
Ωenyq klassov ψ -yntehralov peryodyçeskyx funkcyj mnohyx peremenn¥x
yzuçen¥ v men\ßej mere. Zdes\ sleduet otmetyt\ rabot¥ [7, 8], v kotor¥x
yzuçaetsq povedenye uklonenyj prqmouhol\n¥x summ Fur\e na πtyx klassax. V
dannoj stat\e poluçen¥ asymptotyçeskye formul¥ dlq verxnyx hranej uklo-
nenyj razlyçn¥x prqmouhol\n¥x lynejn¥x srednyx rqdov Fur\e, vzqt¥x po
klassam ψ -yntehralov funkcyj m peremenn¥x. V çastnosty, najden¥ asymp-
totyçeskye ravenstva, kotor¥e obespeçyvagt reßenye zadaçy Kolmohorova –
Nykol\skoho (sm. [2, s. 8]) na πtyx klassax dlq prqmouhol\n¥x summ Valle
Pussena.
Klass¥ ( ψ, β ) -dyfferencyruem¥x funkcyj m peremenn¥x y ψ -ynteh-
ralov funkcyj m peremenn¥x vvodqtsq v rabote [7, s. 545, 546] (sm. takΩe [8,
s.?911, 912]). Pryvedem opredelenye klassov ψ -yntehralov 2π-peryodyçeskyx
funkcyj m peremenn¥x, sleduq [7], odnako, s nekotor¥my yzmenenyqmy, neob-
xodym¥my v dannom yzloΩenyy.
© V. Y. RUKASOV, O. A. NOVYKOV, V. Y. BODRAQ, 2005
564 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
PRYBLYÛENYE KLASSOV ψ -YNTEHRALOV PERYODYÇESKYX FUNKCYJ … 565
Pust\ Rm
— prostranstvo m-mern¥x vektorov
�
x = ( x1 , x2 , … , xm ) , T m =
= [− π π]
=
∏ ;
i
m
1
— m-mern¥j kub s rebrom 2π,
N
m = {
�
x ∈ R
m
| xi ∈ N, i = 1, 2, … , m },
Nm
* = {
�
x ∈ R
m
| xi ∈ N* = N ∪ { 0 }, i = 1, 2, … , m }.
Çerez E
m
oboznaçym mnoΩestvo toçek yz R
m
, koordynat¥ kotor¥x prynymagt
odno yz dvux znaçenyj: 0 yly 1. Çerez L ( T
m
) oboznaçym mnoΩestvo 2π-pery-
odyçeskyx po kaΩdoj peremennoj, summyruem¥x na kube peryodov T
m
funkcyj
f x( )
�
= ( x1 , x2 , … , xm ) .
Pust\ f ∈ L ( T
m
) . Sleduq [7, s. 546], kaΩdoj pare toçek
�
s ∈ E
m
,
�
k ∈ Nm
*
postavym v sootvetstvye koπffycyent Fur\e funkcyy f
a f f x k x
s
dx
k
s
m
T
i i
i
i
m
i
m
�
� �
( ) =
π
( ) − π
∫ ∏
=
1
21
cos . (1)
KaΩdomu vektoru
�
k ∈ Nm
* postavym v sootvetstvye harmonyku
A f x a f k x
s
k k
s
s E
i i
i
i
m
m
� �
�
�
�
( ) = ( ) − π
∈ =
∑ ∏; cos
21
. (2)
Sleduq [7, s. 545], rqd Fur\e funkcyy f x( )
�
opredelym sootnoßenyem
S [ f ] =
1
2q k
k N
k
m
A f x
( )
∈
∑ ( )�
�
�
�
*
; , (3)
hde q k( )
�
— kolyçestvo nulev¥x koordynat vektora
�
k .
Pust\ m = { 1, 2, … , m } y µ — proyzvol\noe podmnoΩestvo yz m . Oboz-
naçym çerez | µ | kolyçestvo πlementov mnoΩestva µ y çerez µ ( r ) — r-πle-
mentnoe podmnoΩestvo yz m ( | µ ( r ) | = r ).
Dlq kaΩdoho µ ⊂ m vvedem ponqtye harmonyky, soprqΩennoj s A f x
k
�
�
( );
po peremenn¥m xi , i ∈ µ, sledugwym sootnoßenyem:
A f x a f k x
s
k x
s
k k
s
s E
i i
i
i m
j j
j
jm
� �
�
�
�µ
µ µ
( ) = ( ) − π
−
( + )π
∈ ∈ ∈
∑ ∏ ∏; cos cos
\ 2
1
2
.
Ponqtye ψ -proyzvodn¥x funkcyj mnohyx peremenn¥x vvedem po analohyy s
odnomern¥m sluçaem (sm. [1]). Poskol\ku nalyçye çastn¥x proyzvodn¥x po
otdel\n¥m peremenn¥m ne vsehda vleçet za soboj nalyçye smeßann¥x çastn¥x
proyzvodn¥x po πtym Ωe peremenn¥m y πtyx Ωe porqdkov, budem yspol\zovat\
dva nabora system çysel: odyn dlq opredelenyq çastn¥x proyzvodn¥x po ot-
del\n¥m peremenn¥m, vtoroj dlq opredelenyq smeßann¥x çastn¥x proyzvod-
n¥x.
Pust\ ψ i = ( ψi1 ( k ), ψi2 ( k ) ), Ψi = ( Ψi1 ( k ), Ψi2 ( k ) ), i ∈ m , k = 0, 1, 2, … , —
nabor¥ system çysel takyx, çto dlq vsex i ∈ m , k ∈ N v¥polnqgtsq uslovyq
ψi1 ( 0 ) = 1, Ψi1 ( 0 ) = 1, ψi2 ( 0 ) = 0, Ψi2 ( 0 ) = 0,
(4)
ψ ψ ψi i ik k k2
1
2
2
2( ) = ( ) + ( ) ≠ 0, Ψ Ψ Ψi i ik k k2
1
2
2
2( ) = ( ) + ( ) ≠ 0.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
566 V. Y. RUKASOV, O. A. NOVYKOV, V. Y. BODRAQ
Esly dlq fyksyrovannoho r ∈ m suwestvuet funkcyq f xrψ ( )
�
∈ L ( T
m
)
takaq, çto
S f
k
k A f x k A f xr
r
m q k
r rk N
r r k r r k
r[ ] =
( )
( ) ( ) − ( ) ( )
( )
∈
{ }∑ ( )ψ
ψ
ψ ψ1
2 2 1 2�
�
� �
� �
; ; , (5)
hde Nr
m
— mnoΩestvo toçek
�
k ∈ Nm
* , u kotor¥x kr ≠ 0, to budem hovoryt\,
çto f xrψ ( )
�
qvlqetsq çastnoj ψr -proyzvodnoj funkcyy f x( )
�
po peremennoj
xr . Dlq funkcyy f xrψ ( )
�
budem ynohda yspol\zovat\ estestvennoe dlq çastn¥x
proyzvodn¥x oboznaçenye
∂ ( )
∂
ψ r f x
xr
�
.
Dlq fyksyrovannoho nabora µ ⊂ m , µ = ( r1 , r2 , … , r| µ | ), smeßannoj Ψµ -
proyzvodnoj po peremenn¥m xi , i ∈ µ, po analohyy s opredelenyem ob¥knoven-
noj smeßannoj çastnoj proyzvodnoj, budem naz¥vat\ funkcyg f
Ψµ
, rqd Fu-
r\e kotoroj qvlqetsq rezul\tatom posledovatel\noho prymenenyq formul¥ (5)
k rqdu Fur\e funkcyy f, no s yspol\zovanyem vmesto system çysel ψij ( k ) so-
otvetstvenno Ψij ( k ), i ∈ µ, j = 1, 2:
f x
f x
x x x
r r r
r r r
Ψ
Ψ Ψ Ψ
µ
µ µ
µ µ
( ) = ∂ ∂ … ∂ ( )
∂ ∂ … ∂
−
−
� �1 1
1 1
.
MnoΩestvo funkcyj f ∈ L ( T
m
) takyx, çto suwestvugt ψ i -proyzvodn¥e
f iψ
dlq lgb¥x i ∈ m y smeßann¥e Ψµ -proyzvodn¥e f
Ψµ
dlq vsex µ ⊂ m ,
budem oboznaçat\ Lmψ
, a podmnoΩestvo neprer¥vn¥x funkcyj yz Lmψ
—
Cmψ
. MnoΩestvo funkcyj yz Cmψ
, udovletvorqgwyx uslovyqm
ess sup f xiψ ( )
�
≤ 1,
ess sup f x
Ψµ ( )
�
≤ K < ∞, i ∈ m , µ ⊂ m ,
budem oboznaçat\ Cm
∞
ψ
.
V sluçae, kohda m = 2 y suwestvugt system¥ çysel ψi ( k ), Ψi k( ) y çysla
βi , βi
*
takye, çto ψ i1 ( k ) = ψi ( k ) cos
βiπ
2
, ψ i2 ( k ) = ψi ( k ) sin
βiπ
2
, Ψi k1( ) =
= Ψi k( ) cos
*βi π
2
, Ψi k2( ) = Ψi ( k ) sin
*βi π
2
, i = 1, 2, Cm
∞
ψ
qvlqetsq klassom
( ψ, β ) -dyfferencyruem¥x funkcyj dvux peremenn¥x, kotor¥e b¥ly vveden¥ v
rabote [9]. Esly, krome toho, dlq çysel r > 0, s > 0, r1 ≥ r, s1 ≥ s v¥polnen¥
uslovyq Ψ1 ( k ) = k–
r
, Ψ2 ( k ) = k–
s
, ψ1 ( k ) = k–
r1
, ψ2 ( k ) = k–
s1
, to klass¥ Cm
∞
ψ
sovpadagt s klassamy Wr s
r s
1 1,
,
. V rabote [10] (sm. takΩe [3, 11]) yzuçen¥ vopros¥
pryblyΩenyq klassov Wr s
r s
1 1,
,
prqmouhol\n¥my summamy Fur\e
S f x S f x A f xn n n
q k
k
k
n
k
n
�
�
�
�
( ) = ( ) = ( )− ( )
==
∑∑; ; ;,1 2
2
2
1
1
2
00
.
Tam Ωe (sm. [10, s. 604]) dlq verxnyx hranej uklonenyj prqmouhol\n¥x summ
Fur\e S f xn
�
�
( ); , vzqt¥x po klassam Wr s
r s
1 1,
,
,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
PRYBLYÛENYE KLASSOV ψ -YNTEHRALOV PERYODYÇESKYX FUNKCYJ … 567
�( ) = (⋅) − ( ⋅)
∈
W S f S fr s
r s
n
f W
n C
r s
r s1 1
1 1
,
, ; sup ;
,
,
� �
pry ni → ∞, i = 1, 2, poluçeno asymptotyçeskoe ravenstvo
�( ) =
π
+
π
+ ( ) + +
W S
n
n
n
n
O
n n
n n n nr s
r s
n r s r s r s1 1 1 1 1 1
4 4
1 1 11
2
1
2
2
2
1 2
1 2 1 2
,
, ;
ln ln ln ln� . (6)
Sleduq [3] (sm. takΩe [2, 9]), prqmouhol\n¥e lynejn¥e srednye rqdov Fur\e
opredelym sledugwym obrazom.
Pust\ Λ = ( Λ1 , Λ2 , … , Λm ) — fyksyrovann¥j nabor beskoneçn¥x treuhol\-
n¥x çyslov¥x matryc, Λi = { }( )λk
n
i
i
, i = 1, 2, … , m,
�
n ∈ N
m
,
�
k ∈ Nm
* , λ0
( )ni = 1 y
λk
n
i
i( ) = 0 dlq ki ≥ ni . Pust\, dalee,
λ λ�
�
k
n
k
n
i
m
i
i( ) ( )
=
= ∏
1
y Gn
� = [ − ]
=
∏ 0 1
1
; ni
i
m
—
prqmouhol\n¥j parallelepyped, sootvetstvugwyj vektoru
�
n ∈ Nm
. Ponqtno,
çto
λ �
�
k
n( ) = 0 dlq lgb¥x
�
k ∉ Gn
� .
Funkcyy f ∈ L ( T
m
), ymegwej rqd Fur\e (3), postavym v sootvetstvye semej-
stvo tryhonometryçeskyx polynomov
U f x A f xn
q k
k
n
k
k Gn
�
�
�
�
�
�
� �
�
( ) = ( )− ( ) ( )
∈
∑; ; ;Λ 2 λ . (7)
Pry
λ �
�
k
n( ) ≡ 1,
�
k ∈ Gn
� , mnohoçlen¥ U f xn
�
�
( ); ; Λ = S f xn
�
�
( ); qvlqgtsq prqmo-
uhol\n¥my çastyçn¥my summamy rqda Fur\e.
Esly velyçyn¥ λ �
�
k
n( )
,
�
k ∈ Gn
� ,
�
n ∈ N
m
, zadagtsq sootnoßenyqmy
λk
n
i i i
i i i
i
i i i i i i ii
i
k n p
k n p
p
n p k n p N p n i m
( ) =
≤ ≤ −
− − + − ≤ ≤ − ∈ ≤ ∈
1 0
1 1
, ,
, , , , ,
to mnohoçlen¥ U f xn
�
�
( ); ; Λ ≡ V f xn p
� �
�
, ;( ) budem naz¥vat\ prqmouhol\n¥my sum-
mamy Valle Pussena.
Velyçyn¥ δ �
�
n f x( ); ; Λ = f x( )
�
– U f xn
�
�
( ); ; Λ qvlqgtsq uklonenyqmy mnoho-
çlenov U f xn
�
�
( ); ; Λ ,
�
n ∈ N
m
, ot funkcyy f x( )
�
.
Cel\g dannoj rabot¥ qvlqetsq yssledovanye asymptotyçeskoho povedenyq
velyçyn
�( ) = ( )∞
∈ ∞
C U f xm
n
f C
n C
m
ψ
ψ
δ; sup ; ;� �
�
Λ
pry ni → ∞, i = 1, 2, … , m.
Najdem yntehral\n¥e predstavlenyq velyçyn δ �
�
n f x( ); ; Λ . Dlq πtoho vos-
pol\zuemsq pryemamy, predloΩenn¥my v [2, s. 52 – 56].
Çerez { λni
( v ) },
�
n = ( n1 , n2 , … , nm ) ∈ N
m
, oboznaçym semejstvo zadann¥x y
neprer¥vn¥x na [ 0; 1 ] funkcyj takyx, çto λk
n
i
i( ) = λn
i
i
i
k
n
,
�
k ∈ Gn
� .
Vvedem funkcyy
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
568 V. Y. RUKASOV, O. A. NOVYKOV, V. Y. BODRAQ
τij ( v ) =
( )− ( ) ( ) ≤ ≤
( ) ≤ = … =
1 1 1
1 1 2 1 2
λ ψ
ψ
n ij i
i
ij i
i
n
n
n i m j
v v v
v v
, ,
, , , , , , , ,
(8)
Tij ( v ) =
( )− ( ) ( ) ≤ ≤
( ) ≤ = … =
1 1 1
1 1 2 1 2
λn ij i
i
ij i
i
n
n
n i m j
v v v
v v
Ψ
Ψ
, ,
, , , , , , , ,
(9)
kotor¥e na 0 1;
ni
zadan¥ tak, çto τij ( v ), Tij ( v ) neprer¥vn¥ na poloΩytel\-
noj poluosy y v¥polneno uslovye τij ( 0 ) = 0; Tij ( 0 ) = 0, i = 1, 2, … , m, j = 1, 2.
Dlq i ∈ m , j = 1, 2 poloΩym
ˆ cosτ τij ijt t
j
d( ) =
π
( ) − ( − )π
∞
∫1 1
2
0
v v v,
ˆ cosT t T t
j
dij ij( ) =
π
( ) − ( − )π
∞
∫1 1
2
0
v v v ,
A t t d dti i i( ) =
π
( ) + ( )( )
∞
− ∞
∞
∫∫τ τ τ1
1 2
0
v v v v vcos sin ,
a t dtij ij
t ni
( ) = ( )
≥ π
∫τ τ̂
/ 2
.
Yspol\zuq rassuΩdenyq rabot¥ [12, s. 259 – 262], nesloΩno poluçyt\ sledu-
gwee utverΩdenye.
Teorema 1. Pust\ funkcyy τ i j ( v ), T ij ( v ) opredelen¥ sootnoßenyqmy (8),
(9) y ymegt summyruem¥e na R preobrazovanyq Fur\e
ˆ
–
τij t dt( )
∞
∞
∫ < ∞, ˆ
–
T t dtij( )
∞
∞
∫ < ∞, i = 1, 2, … , m, j = 1, 2.
Tohda dlq lgboj funkcyy f ∈ Cm
∞
ψ
v kaΩdoj toçke
�
x ∈ T
m
v¥polnqetsq ra-
venstvo
δ τ τψ�
� � �
n
k
m
k
k k kf x f x t
n
e t t d dtk( ) =
π
−
( ) + ( )
− ∞
∞
=
∞
∫∑ ∫( ); ; cos sinΛ 1
1
1 2
0
v v v v v +
+ (− )
π
−
+
= ( )⊂ ∈ ( )
∑ ∑ ∫ ∑1 11
2
r
r
m
r
r m R
j
j
j
j r
f x
t
n
e
rµ µ
µΨ � �
×
×
( )( ) + ( )
∞
∈ ( )
∫∏ T t T t d dt
r
ν ν ν ν ν ν ν ν ν ν
ν µ
1 2
0
v v v v vcos cos .
V kaçestve sledstvyq yz teorem¥ 1 moΩno poluçyt\ sledugwee utverΩde-
nye, kotoroe qvlqetsq m-mern¥m analohom yzvestnoj lemm¥ Telqkovskoho [13].
Teorema 2. Pust\ funkcyy τ i j ( x ), Tij ( x ), i = 1, 2, … , m, j = 1, 2, zadan¥
sootnoßenyqmy (8), (9) y v¥polnen¥ uslovyq
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
PRYBLYÛENYE KLASSOV ψ -YNTEHRALOV PERYODYÇESKYX FUNKCYJ … 569
ˆ
–
τij t dt( )
∞
∞
∫ < ∞, ˆ
–
T t dtij( )
∞
∞
∫ < ∞, i = 1, 2, … , m, j = 1, 2.
Tohda pry ni → ∞, i = 1, 2, … , m, spravedlyvo ravenstvo
�( ) = ( ) + ( ) ( ) + ( )
∞
= ∈ = ∈ ( )( )⊂=
∑ ∑ ∏∑∑C U A O a A Tm
n i
i
m
ij
i m j
j
j ii mi
m
ψ
µµ
τ τ;
; ,
�
1 1 2 2
1 . (10)
Sleduq [1], vvedem mnoΩestva �, F. Budem polahat\, çto funkcyy ψij ( v ),
Ψij ( v ), i = 1, 2, … , m, j = 1, 2, qvlqgtsq funkcyqmy neprer¥vnoho arhumenta
v ≥ 0, sovpadagwymy pry v ∈ N s πlementamy odnoymenn¥x system çysel
ψij ( k ), Ψij ( k ), kotor¥e yspol\zovalys\ v¥ße dlq opredelenyq ψ i - y Ψµ -pro-
yzvodn¥x.
Çerez � oboznaçym mnoΩestvo funkcyj ψ ( x ), neprer¥vn¥x pry x ≥ 0,
monotonno ub¥vagwyx, v¥pukl¥x vnyz pry x ≥ 1 y udovletvorqgwyx uslovyg
lim
x→∞
ψ ( x ) = 0.
Funkcyy ψ ( x ) postavym v sootvetstvye funkcyg η ( x ) = η ( ψ, x ), svqzan-
nug pry x ≥ 1 s ψ ( x ) sootnoßenyem ψ ( η ( x ) ) = 1
2
ψ ( x ).
Çerez F oboznaçym mnoΩestvo funkcyj ψ ∈ �, dlq kotor¥x ′( )η t =
= η′ ( ψ, t ) ≤ const, t ≥ 1, η′ ( t ) =df
η′ ( t + 0 ).
S pomow\g teorem¥ 2 yzuçenye velyçyn �( )∞C Um
n
ψ ; � svodytsq k v¥çysle-
nyg odnomern¥x nesobstvenn¥x yntehralov A ( τ ). V çastnosty, spravedlyvo
sledugwee utverΩdenye.
Teorema 3. Pust\ Ψ i j ∈ F, ψ ij ∈ F, i = 1, 2, … , m, j = 1, 2, y ( ( )η ψ i in, –
– ni)−1 = O ( 1 ), ( )( ) − −η Ψi i in n, 1 = O ( 1 ). Pust\, dalee, suwestvugt çysla r i
1
( )
,
r i
2
( )
takye, çto η ψ( − )( )
i i
in r, 1 = ni
, η( − )( )Ψi i
in r, 2 = ni
, i = 1, 2, … , m. Polo-
Ωym r i
0
( ) = min ,{ }( ) ( )r ri i
1 2 y pi ∈ [ ]( )1 0, r i
, i = 1, 2, … , m. Tohda pry ni → ∞ , i =
= 1, 2, … , m, spravedlyvo ravenstvo
�( ) =
π
( )∞
( )
=
∑C V n
r
p
m
n p i i
i
ii
m
ψ ψ; ln,
� � 4
2
1
1
+
+ O n n
r
pi i
i
m
j j
j
jjr mr
m
( ) ( ) + ( )
=
( )
∈( )⊂=
∑ ∏∑∑1
1
2
2
ψ
µµ
Ψ ln . (11)
Dokazatel\stvo. Yz rezul\tatov rabot¥ [6] sleduet, çto pry v¥polnenyy
uslovyj dannoj teorem¥ ymegt mesto asymptotyçeskye ravenstva
A n
r
p
O ni i i
i
i
i i( ) =
π
( ) + ( ) ( )
( )
τ ψ ψ4 12
1ln ,
A T O n
r
pi i i
i
i
( ) = ( ) ( )
( )
1 2Ψ ln , a O ni i i( ) = ( ) ( )τ ψ1 .
Obæedynqq πty ravenstva y sootnoßenye (10), neposredstvenno ubeΩdaemsq v
spravedlyvosty formul¥ (11).
Teorema dokazana.
Zametym, çto v sluçae, kohda m = 2, dlq prqmouhol\n¥x summ Fur\e ( pi =
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
570 V. Y. RUKASOV, O. A. NOVYKOV, V. Y. BODRAQ
= 1, i = 1, 2, … , m ) na klassax Wr s
r s
1 1,
,
sootnoßenye (11) prynymaet vyd raven-
stva (6), poluçennoho v rabote [10].
1. Stepanec A. Y. Skorost\ sxodymosty rqdov Fur\e na klassax ψ -yntehralov // Ukr. mat.
Ωurn. – 1997. – 49, # 8. – S. 1069 – 1113.
2. Stepanec A. Y. Klassyfykacyq y pryblyΩenye peryodyçeskyx funkcyj. – Kyev: Nauk.
dumka, 1987. – 268 s.
3. Stepanec A. Y. Ravnomern¥e pryblyΩenyq tryhonometryçeskymy polynomamy. – Kyev:
Nauk. dumka, 1981. – 340 s.
4. Stepanec A. Y. Klassyfykacyq peryodyçeskyx funkcyj y skorost\ sxodymosty yx rqdov
Fur\e // Yzv. AN SSSR. Ser. mat. – 1986. – 50, # 1. – S. 101 – 136.
5. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj: V 2 ç. – Kyev: Yn-t matematyky NAN Ukra-
yn¥, 2002.
6. Rukasov V. Y., Novykov O. A., Çajçenko S. O. PryblyΩenye klassov C∞
ψ
summamy Valle
Pussena // Teoriq nablyΩennq funkcij ta ]] zastosuvannq: Zb. nauk. pr. – Ky]v: In-t mate-
matyky NAN Ukra]ny, 2000. – S. 396 – 406.
7. Stepanec A. Y., Paçulya N. L. Kratn¥e summ¥ Fur\e na mnoΩestvax ( ψ, β ) -dyfferency-
ruem¥x funkcyj // Ukr. mat. Ωurn. – 1991. – 43, # 4. – S. 545 – 555.
8. Lasuryq R. A. Kratn¥e summ¥ Fur\e na mnoΩestvax ψ -dyfferencyruem¥x funkcyj //
Tam Ωe. – 2003. – 55, # 7. – S. 911 – 918.
9. Zaderej P. V. Yntehral\n¥e predstavlenyq uklonenyj lynejn¥x srednyx rqdov Fur\e na
klassax dyfferencyruem¥x peryodyçeskyx funkcyj dvux peremenn¥x // Nekotor¥e vop-
ros¥ teoryy approksymacyy funkcyj: Sb. nauçn. tr. / Otv. red. V. K. Dzqd¥k. – Kyev: Yn-t
matematyky AN USSR, 1985. – S. 16 – 28.
10. Stepanec A. Y. PryblyΩenye nekotor¥x klassov dyfferencyruem¥x peryodyçeskyx
funkcyj dvux peremenn¥x summamy Fur\e // Ukr. mat. Ωurn. – 1973. – 25, # 5. – S. 599 –
609.
11. Stepanec A. Y. Yssledovanye po πkstremal\n¥m zadaçam teoryy summyrovanyq rqdov Fu-
r\e: Dys. … d-ra fyz.-mat. nauk. – Kyev, 1974. – 305 s.
12. Rukasov V. Y., Novykov O. A., Bodraq V. Y. PryblyΩenye klassov ψ -yntehralov peryody-
çeskyx funkcyj dvux peremenn¥x lynejn¥my metodamy // Problemy teori] nablyΩennq
funkcij ta sumiΩni pytannq: Zb. prac\ In-tu matematyky NAN Ukra]ny / Vidp. red. O. I. Ste-
panec\. – Ky]v: In-t matematyky NAN Ukra]ny, 2004. – 1, # 1. – S. 250 – 269.
13. Telqkovskyj S. A. O normax tryhonometryçeskyx polynomov y pryblyΩenyy dyfferency-
ruem¥x funkcyj lynejn¥my srednymy yx rqdov Fur\e. I // Tr. Mat. yn-ta AN SSSR. – 1961.
– 62. – S. 61 – 97.
Poluçeno 16.04.2003,
posle dorabotky — 17.01.2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
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| id | umjimathkievua-article-3622 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:57Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2e/676b8dc229a71298db610f18c3cfd12e.pdf |
| spelling | umjimathkievua-article-36222020-03-18T20:00:05Z Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series Приближение классов у ψ-интегралов периодических функций многих переменных прямоугольными линейными средними их рядов Фурье Bodraya, V. I. Novikov, O. A. Rukasov, V. I. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of &psi;-integrals of multivariable functions Одержано асимптотичні рівності для відхилень прямокутних лінійних середніх рядів Фур'є на класах &psi;-інтегралів функцій багатьох змінних. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3622 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 564–570 Український математичний журнал; Том 57 № 4 (2005); 564–570 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3622/3975 https://umj.imath.kiev.ua/index.php/umj/article/view/3622/3976 Copyright (c) 2005 Bodraya V. I.; Novikov O. A.; Rukasov V. I. |
| spellingShingle | Bodraya, V. I. Novikov, O. A. Rukasov, V. I. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. Бодрая, В. И. Новиков, О. А. Рукасов, В. И. Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series |
| title | Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series |
| title_alt | Приближение классов у ψ-интегралов периодических функций многих переменных прямоугольными линейными средними их рядов Фурье |
| title_full | Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series |
| title_fullStr | Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series |
| title_full_unstemmed | Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series |
| title_short | Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series |
| title_sort | approximation of classes of ψ-integrals of periodic functions of many variables by rectangular linear means of their fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3622 |
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