Asymptotics of approximation of ψ-differentiable functions of many variables
We investigate approximative characteristics of classes of ψ-differentiable multivariable functions introduced by A. I. Stepanets. We give asymptotics of the approximation of functions from these classes.
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| Datum: | 2008 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509273120833536 |
|---|---|
| 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 | 2020-03-18T19:48:39Z |
| description | We investigate approximative characteristics of classes of ψ-differentiable multivariable functions introduced by A. I. Stepanets.
We give asymptotics of the approximation of functions from these classes. |
| first_indexed | 2026-03-24T02:38:29Z |
| format | Article |
| fulltext |
UDK 517.5
R. A. Lasuryq (Abxaz. un-t, Suxum)
ASYMPTOTYKA PRYBLYÛENYQ
ψψψψ-DYFFERENCYRUEMÁX FUNKCYJ
MNOHYX PEREMENNÁX
We investigate approximative characteristics of classes of ψ-differentiable multivariable functions
introduced by A. I. Stepanets. We give asymptotics of the approximation of functions from these
classes.
DoslidΩugt\sq aproksymatyvni xarakterystyky klasiv ψ-dyferencijovnyx funkcij bahat\ox
zminnyx, uvedenyx O. I. Stepancem. Navedeno asymptotyku nablyΩennq funkcij cyx klasiv.
1. Pust\ T
m = [ – π, π ] — m-mern¥j tor, Z
m
— celoçyslennaq reßetka v m-
mernom evklydovom prostranstve R
m
, x y = x1 y1 + … + xm ym , x = xx , C ( T
m
)
— prostranstvo neprer¥vn¥x na T
m
funkcyj f ( x ), 2π-peryodyçeskyx po kaΩ-
doj peremennoj s normoj || f ||C = max
x T m
f x
∈
( ) ,
S f c f e
k
k
ikx
m
[ ] = ( )
∈
∑
Z
(1)
— rqd Fur\e funkcyy f ∈ C ( T
m
),
c f f y e dyk
m iky
T m
( ) = ( π) ( )− −∫2 , k ∈ Z
m
,
— koπffycyent¥ Fur\e funkcyy f ( x ).
Pust\ ψ ( t ) — proyzvol\naq funkcyq, opredelennaq pry vsex dejstvytel\-
n¥x t > 0,
1
0ψ( )
df= 0, ψ ( | k | ) ≠ 0, k ∈ Z
m
\ { 0 }. Otpravlqqs\ ot rqda (1), pred-
poloΩym, çto rqd
k
k
ikx
m k
c f e
∈
∑ ( ) ( )
Z
1
ψ
qvlqetsq rqdom Fur\e funkcyy yz C ( T
m
) , kotoraq oboznaçaetsq f ψ ( x ) y na-
z¥vaetsq ψ-proyzvodnoj funkcyy f. MnoΩestvo funkcyj f ∈ C ( T
m
) , udov-
letvorqgwyx takym uslovyqm, oboznaçaetsq çerez Cψ
C ( T
m
).
Opredelenye klassov ψ-dyfferencyruem¥x funkcyj odnoj peremennoj
moΩno najty, naprymer, v rabote [1]. V sluçae ψ = | k | –
2r
, r ∈ N = { 1, 2, 3 },
hovorqt, çto f ∈ C ( T
m
) ymeet r-j obobwenn¥j laplasyan y Sr
— klass funk-
cyj, ymegwyx r-j obobwenn¥j laplasyan ( f
ψ = ∆̃r f , r ∈ N ). Pry ψ =
= ( | k1 | + … + | km | )
–
r
ymeem klass funkcyj, obladagwyx r-j obobwennoj pro-
yzvodnoj Ryssa ( f ψ = Dr
f ).
Pust\, dalee, metod summyrovanyq Un ( f; Λ ) , n ∈ N, rqda Fur\e (1) funkcyj
klassa Cψ
C ( T
m
) zadan posledovatel\nost\g funkcyj Λ = ( )( )λk
n
, k ∈ Z
m
, n ∈
© R. A. LASURYQ, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8 1051
1052 R. A. LASURYQ
∈ N, pryçem
S U f x c f en
k
k
n
k
ikx
m
[ ]( ) = ( )
∈
( )∑; ; Λ
Z
λ ,
a µ(
n
) = µk
n( )
, k ∈ Z
m
, n ∈ N, opredelqet posledovatel\nost\ mul\typlykato-
rov v prostranstve C ( T
m
) , t. e. rqd
k
k
n
k
ikx
m
c f e
∈
( )∑ ( )
Z
µ qvlqetsq rqdom Fur\e
nekotoroj funkcyy
˜ ; ;U f xn( )µ ∈ C ( T
m
) y
µ µ µk
n
M
n
C C f
n C
U f x( ) ( )
→ ≤
= = ( )sup ˜ ; ;
1
.
Esly
sup
n
k
n
C∈
( )
N
µ < ∞,
to posledovatel\nost\ µ(
n
)
naz¥vaetsq ravnomerno ohranyçennoj v C T m( ) .
Dostatoçn¥e uslovyq ravnomernoj ohranyçennosty mul\typlykatorov v C ( T
m
)
v sluçae m = 1 xoroßo yzvestn¥ (sm., naprymer, [2, s. 100]). V sluçae kratn¥x
rqdov Fur\e, kohda
µ µk
n k
n
( ) =
, k ∈ Z
m
, n ∈ N,
takye uslovyq soderΩatsq, naprymer, v [3, s. 39; 4]. Odno yz uslovyj zaklgça-
etsq v prynadleΩnosty funkcyy µ ( u ), u ∈ R
m
, alhebre B ( Rm
) -preobrazova-
nyj Fur\e koneçn¥x borelev¥x mer na R
m
:
µ ( x ) = e d uiux
m
− ( )∫ ν
R
s normoj
|| µ || B = inf var ν < ∞.
Pust\, dalee, ψi ( | k | ) ≠ 0,
1
0ψ i ( )
df= 0, i = 1, 2, k ∈ Z
m
\ { 0 }, — proyzvol\n¥e
funkcyy. Sleduq [1, s. 35] (sm. takΩe [2, s. 145]), budem hovoryt\, çto velyçyna
ψ1 c-predßestvuet velyçyne ψ2 , y zapys¥vat\ ψ1 ≤
c
ψ2 , esly yz vklgçenyq
f ∈ Cψ2
C ( T
m
) sleduet suwestvovanye f ψ1 ∈ C ( T
m
) . V πtom sluçae spravedlyvo
sootnoßenye
S f S f[ ]( ) = [ ]ψ ψ ψ ψ1 2 1 2/
.
Dejstvytel\no, rqd¥
k i
k
ikx
m k
c f e
∈
∑ ( ) ( )
Z
1
ψ
, i = 1, 2,
qvlqgtsq rqdamy Fur\e funkcyj f ψ1
y f ψ2
sootvetstvenno. Pry i = 1
S f c f e
k
k
ikx
m
[ ] = ( )
∈
∑ψ ψ1 1
Z
,
hde
c f
k
c fk k( ) = ( ) ( )ψ
ψ
1
1
1
, k ∈ Z
m
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ASYMPTOTYKA PRYBLYÛENYQ ψ-DYFFERENCYRUEMÁX FUNKCYJ … 1053
Otsgda
S f
k
k
c f e S f
k
k
ikx
m
[ ]( ) = ( )
( ) ( ) = [ ]
∈
∑ψ ψ ψ ψ ψψ
ψ
1 2 1 1 21
2
/
Z
ψ
ψ
1
2
0
0
0
( )
( )
=
df
.
Sledovatel\no,
f C C T mψ ψ ψ1 2 1∈ ( )/
∀f ∈ Cψ2
C ( T
m
).
Pust\ teper\ rqd
k
ikx
m
k
k
e
∈ { }
−∑ ( )
( )
Z \ 0
2
1
ψ
ψ
(2)
qvlqetsq rqdom Fur\e nekotoroj summyruemoj 2π-peryodyçeskoj po kaΩdoj
peremennoj funkcyy Dψ ( x ), Dψ ∈ L ( T
m
), ψ df=
ψ
ψ
2
1
. Tohda rqd
k
k
ikx
m k
c f e
∈
∑ ( ) ( )
Z
1
1ψ
∀f ∈ Cψ2
C ( T
m
) (3)
qvlqetsq rqdom Fur\e nekotoroj neprer¥vnoj funkcyy f ψ1
( x ).
V samom dele, poskol\ku f ψ2 ∈ C ( T
m
), Dψ ∈ L ( T
m
), funkcyq
I ( x ) = ( π) ( − ) ( )− ∫2 2m
T
f x t D t dt
m
ψ
ψ
qvlqetsq neprer¥vnoj. Rassmatryvaq dlq nee rqd Fur\e, ubeΩdaemsq, çto on
sovpadaet s rqdom (3).
Esly
ψ
ψ
2
1
k
k
k r( )
( ) = −
, r > 0, k ∈ Z
m
\ { 0 }, y rqd (2) qvlqetsq rqdom Fur\e
summyruemoj funkcyy Dψ , to ψ1 ≤
c
ψ2 , v çastnosty pry ψ1 ( | k | ) = | k | –
r1
,
ψ2 ( | k | ) = | k | –
r2
, k ∈ Z
m
\ { 0 }, r1 , r2 > 0, yz yzloΩennoho v¥ße sleduet, çto
ψ1 ≤
c
ψ2 .
V dal\nejßem nam ponadobytsq sledugwyj fakt. Esly µ(
n
)
, n ∈ N, — po-
sledovatel\nost\ mul\typlykatorov, ravnomerno ohranyçennaq v C T m( ) , y
∀ k ∈ Z
m µk
n( ) → 0, n → ∞, to
˜ ; ;U f x on C
( ) = ( )µ 1 , n → ∞. (4)
Dejstvytel\no, v πtom sluçae
˜ ; ;U f x K fn C C( ) ≤µ ∀ f ∈ C ( T
m
),
hde K — postoqnnaq, ne zavysqwaq ot n ∈ N. Otsgda dlq lgboho polynoma
T ( x ) poluçaem
˜ ; ; ˜ ; ; ˜ ; ;U f x U f T x U T xn C n C n C
( ) = ( − ) + ( )µ µ µ ≤
≤ K f T U T xC n C
− + ( )˜ ; ; µ .
Za sçet nadleΩaweho v¥bora T ( x ) pervoe slahaemoe moΩno sdelat\ skol\
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1054 R. A. LASURYQ
uhodno mal¥m. Uçyt¥vaq, çto pry lgbom dostatoçno bol\ßom n summa
˜ ; ;U T xn( )µ soderΩyt odynakovoe çyslo çlenov, v sylu uslovyq µk
n( ) → 0, n →
→ ∞, zaklgçaem, çto
˜ ; ;U T x on C
( ) = ( )µ 1 , n → ∞.
2. Ymeet mesto sledugwee utverΩdenye.
Teorema 1. Pust\ ψs ( t ), s = 1, … , r, r ∈ N, — systema proyzvol\n¥x fun-
kcyj, opredelenn¥x pry vsex t > 0,
1
0ψ s( )
df= 0, ψs ( | k | ) > 0, k ∈ Z
m
\ { 0 }; ϕs ( n ),
s = 1, … , r, r ∈ N, — proyzvol\naq systema posledovatel\nostej dejstvy-
tel\n¥x poloΩytel\n¥x çysel, Λ = ( )( ) ( )=λ λk
n
k
n˜
, k ∈ Z
m
, n ∈ N, — matryca
çysel takaq, çto
λ̃ ϕ
ψ
ϕ
ψk
n
s
r
s
s
s
r
r
a
n
k
o
n
k
( )
=
= ( )
( ) + ( )
( )
∑
0
, n → ∞, (5)
ψ ϕ0 0k n( ) = ( ) ≡ 1,
hde as , s = 0, 1, … , r, — nekotor¥e dejstvytel\n¥e çysla, a0 df= 1.
Pust\, dalee,
µ µ λ ϕ
ψ
ψ
ϕ
( ) ( ) ( )
=
= = − ( )
( )
( )
( )∑n
k
n
k
n
s
r
s
s
s
r
r
a
n
k
k
n
df
0
, n ∈ N, k ∈ Z
m
, (6)
poroΩdaet posledovatel\nost\ mul\typlykatorov, ravnomerno ohranyçennug
v C ( T
m
). Tohda esly
ψs ≤
c
ψr ∀s = 1, … , r – 1,
to dlq kaΩdoj funkcyy f ∈ Cψr
C ( T
m
) ravnomerno po x ymeet mesto asymp-
totyçeskoe ravenstvo
U f x f x a n f x o nn s s
s
r
r
s( ) − ( ) = ( ) ( ) + ( )
=
∑ ( ); ; Λ ϕ ϕψ
1
, n → ∞. (7)
Polahaq, v çastnosty, ψs ( | k | ) = | k | –
2s
, k ∈ Z
m
\ { 0 }, ϕs ( n ) = n–
2s
, s = 1, … , r,
r ∈ N, n ∈ N, λk
n( ) = λ k
n
, v uslovyqx teorem¥ 1 poluçaem asymptotyku
pryblyΩenyq funkcyj klassa Sr
, najdennug v rabote [5]:
U f x f x
a f x
n
o
n
n
s
s
s
s
r
r( ) − ( ) = ( ) +
=
∑; ;
˜
Λ ∆
2
1
2
1
, n → ∞. (8)
Ravenstva vyda (8) v sluçae m = 1 vosxodqt k rabotam E. V. Voronovskoj [6],
S. N. Bernßtejna [7]. V m-mernom sluçae ( m > 1 ) asymptotyka pryblyΩenyq
konkretn¥my metodamy summyrovanyq rqdov yssledovalas\ v rabotax [8, 9]. V
çastnosty, v klasse Sr
dlq srednyx typa Abelq – Puassona
λ λ λ
α
k
n
k
nk
n
k
n
e( )
−
=
=
=˜ , α > 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ASYMPTOTYKA PRYBLYÛENYQ ψ-DYFFERENCYRUEMÁX FUNKCYJ … 1055
(v πtom sluçae (sm. [5]) po teoreme Lefstrema [10] µ ( u ) ∈ B ( R
m
) ), asymptoty-
ka pryblyΩenyq ustanovlena v rabote [8].
Dokazatel\stvo teorem¥ 1. V sylu uslovyj teorem¥ rqd
k
k
n
r
k
ikx
m k
c f e
∈
( )∑ ( ) ( )
Z
µ
ψ
1
∀f ∈ Cψr
C ( T
m
)
qvlqetsq rqdom Fur\e nekotoroj funkcyy yz C ( T
m
).
Dalee, s uçetom toho, çto a0 = 1, ϕ0 = ψ0 ≡ 1,
I x
k
c f en
k
k
n
r
k
ikx
m
( ) = ( ) ( )
∈
( )∑df
Z
µ
ψ
1
=
=
k
k
n
s
r
s
s
s r
k
ikx
m
a
n
k n
c f e
∈
( )
=
∑ ∑− ( )
( )
( )
( )
Z
λ ϕ
ψ ϕ0
1
=
=
1
ϕ
λ
r k
k
n
k
ikx
k
k
ikx
n
c f e c f e
m m( )
( ) − ( )
∈
( )
∈
∑ ∑
Z Z
–
–
s
r
s s
k s
k
ikxa n
k
c f e
m= ∈
∑ ∑( ) ( ) ( )
1
1ϕ
ψ
Z
=
= S
n
U f x f x a n f x
r
n s s
s
r
s
1
1ϕ
ϕ ψ
( )
( ) − ( ) − ( ) ( )
=
∑; ; Λ .
Poskol\ku f ψs
( x ) ∈ C ( T
m
), s = 1, … , r – 1, to Un ( f; x; Λ ) ∈ C ( T
m
).
V sylu uslovyq (5) µk
n( ) → 0, n → ∞ , k ∈ Z
m
. Otsgda, prynymaq vo vnyma-
nye sootnoßenye (4), ymeem
I x on C( ) = ( )1 , n → ∞.
Teorema 1 dokazana.
V rabote [5] yz sootnoßenyj vyda (8) ustanavlyvaetsq hladkost\ funkcyy
f ∈ C ( T
m
) . Poπtomu zdes\ rassmatryvaetsq vopros ob obrawenyy teorem¥ 1.
Ymeet mesto sledugwaq teorema.
Teorema 2. Pust\ Λ = ( )( ) ( )=λ λk
n
k
n˜
, λ0
( )n = 1, k ∈ Z
m
, n ∈ N, — matryca
çysel, zadagwaq posledovatel\nost\ mul\typlykatorov, ravnomerno ohrany-
çennug v C ( T
m
) , y v¥polnen¥ uslovyq (5), (6). Esly pry nekotorom r ∈ N
dlq f ∈ C ( T
m
) ravnomerno po x ymeet mesto asymptotyçeskoe ravenstvo
U f x f x g x n o nn s s
s
r
r( ) − ( ) = ( ) ( ) + ( )
=
∑ ( ); ; Λ ϕ ϕ
1
, n → ∞, (9)
hde gs ( x ) ∈ C ( T
m
), s = 1, … , r, to f ∈ Cψr
C ( T
m
) y as f ψs
( x ) = gs ( x ).
Dokazatel\stvo. Kak y v rabote [5], vospol\zuemsq pryemom Favara [11].
V sylu (9) pry r = 1 ymeem
U f x f x
n
on
C
( ) − ( )
( )
= ( ); ; Λ
ϕ1
1 , n → ∞. (10)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1056 R. A. LASURYQ
Rassmotrym summ¥ Boxnera – Ryssa
σδ
δ
N
k N
k
ikxF x
k
N
c F e( ) = −
( )
≤
∑; 1
2
2 , n ∈ N,
dlq funkcyy
F x
U f x f x
n
n( ) = ( ) − ( )
( )
df ; ; Λ
ϕ1
.
Uçyt¥vaq, çto pry δ >
m − 1
2
srednye σδ
N x(⋅ ); rehulqrn¥ (sm., naprymer,
[3]) y yz uslovyq (5) pry r = 1 sleduet
( − ) ( )
( )
= + ( )
( )λ ψ
ϕ
k
n k
n
a o
1
11
1
1 , n → ∞, (11)
v sylu (10) naxodym
σ
ϕ
δ
N
n
C
U f x f x
n
( ) − ( )
( )
; ; Λ
1
=
=
k N
k
n
k
ikx
C
k
N
k
n k
c f e
≤
( )
∑ −
( − ) ( )
( ) ( ) ( )1
12
2
1
1 1
δ
λ ψ
ϕ ψ
=
=
k N
k
ikx
C
k
N k
c f e o
≤
∑ −
( ) ( ) = ( )1
1
1
2
2
1
δ
ψ
, n → ∞, δ >
m − 1
2
. (12)
Otsgda vsledstvye ravenstva Parsevalq poluçaem
K
k
N k
c f e dx
k N
k
ikx
T m
≥ −
( ) ( )
≤
∑∫ 1
1
2
2
1
2δ
ψ
=
=
k N
k
k
N k
c f
≤
∑ −
( ) ( )1
1
2
2
2
1
2
2
δ
ψ
, K ≡ const > 0.
Znaçyt,
k N
k
k
N k
c f
≤
∑ −
( ) ( )
0
1
1
2
2
2
1
2
2
δ
ψ
≤ K ∀N0
, 0 ≤ N0 ≤ N.
Perexodq k predelu pry N → ∞, naxodym
k
k
m k
c f
∈
∑ ( ) ( )
Z
1
1
2
2
ψ
< ∞.
Sledovatel\no, suwestvuet f ψ1
( x ) s summyruem¥m kvadratom, f ψ1 ∈ L2 ( T
m
), y
S f
k
c f e
k
k
ikx
m
[ ] = ( ) ( )
∈
∑ψ
ψ
1
1
1Z
.
Poskol\ku σδ
N f( ) → f, N → ∞ , ∀f ∈ L2 ( T
m
) poçty vsgdu, vsledstvye (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ASYMPTOTYKA PRYBLYÛENYQ ψ-DYFFERENCYRUEMÁX FUNKCYJ … 1057
ubeΩdaemsq v ohranyçennosty poçty vsgdu funkcyy f ψ1
( x ).
Na osnovanyy sootnoßenyq (9) ymeem
σ
ϕ
δ
N
n
C
U f x f x
n
g x
( ) − ( )
( )
− ( )
; ; Λ
1
1 =
=
k N
k
n
k
ikxk
N n
c f e
≤
( )
∑ −
−
( )
( )1
12
2
1
δ
λ
ϕ
–
k N
k
ikx
C
k
N
c g e
≤
∑ −
( )1
2
2 1
δ
=
=
k N
k
n
k
ikx
C
k
N
k
n k
c g e o
≤
( )
∑ −
( − ) ( )
( ) ( ) − ( )
= ( )1
1
1
2
2
1
1 1
1
δ
λ ψ
ϕ ψ
, n → ∞.
Perexodq k predelu pry n → ∞, s uçetom (11) poluçaem
k N
k k
ikx
C
k
N
a
k
c f c g e
≤
∑ −
( ) ( ) − ( )
1
2
2
1
1
1
δ
ψ
= 0.
Prynymaq vo vnymanye, çto
c f
k
c fk k( ) = ( ) ( )ψ
ψ
1
1
1
,
zaklgçaem, çto a c fk1
1( )ψ = c gk ( )1 , y poπtomu a f1
1ψ
πkvyvalentna g1
.
PredpoloΩym teper\, çto teorema spravedlyva dlq lgboho r ≤ r0
. Dokaza-
tel\stvo ee pry r = r0 + 1 poluçaetsq putem prymenenyq pred¥duwyx rassuΩ-
denyj k funkcyy
U f x f x a f x nn s
s
r
s
s( ) − ( − ( ) ( )
=
∑; ; Λ ψ ϕ
0
.
1. Stepanec A. Y. Klassyfykacyq y pryblyΩenye peryodyçeskyx funkcyj. – Kyev: Nauk.
dumka, 1987. – 268 s.
2. Stepanec A. Y. Metod¥ teoryy pryblyΩenyj. Ç. I. – Kyev: Yn-t matematyky NAN Ukray-
n¥, 2002. – 40. – 426 s.
3. Stejn Y., Vejs H. Vvedenye v harmonyçeskyj analyz na evklydov¥x prostranstvax. – M.:
Myr, 1974. – 331 s.
4. Tryhub R. M. Absolgtnaq sxodymost\ yntehralov Fur\e, summyruemost\ rqdov Fur\e y
pryblyΩenye polynomamy funkcyj na tore // Yzv. AN SSSR. Ser. mat. – 1980. – 44, # 6. –
S.Q1378 – 1409.
5. Kuznecova O. Y. Asymptotyçeskoe pryblyΩenye hladkyx funkcyj // Teoryq otobraΩenyj
y pryblyΩenye funkcyj. – Kyev: Nauk. dumka, 1989. – S. 75 – 81.
6. Voronovskaq E. V. Opredelenye asymptotyçeskoho vyda pryblyΩenyq funkcyj polynoma-
my S. N. Bernßtejna // Dokl. AN SSSR. Ser. A. – 1932. – # 4. – S. 79 – 85.
7. Bernßtejn S. N. Dobavlenye k stat\e E. V. Voronovskoj „Opredelenye asymptotyçeskoho
vyda pryblyΩenyq funkcyj polynomamy S. N. Bernßtejna”. – Sobr. soç. – M.: Yzd-vo AN
SSSR, 1954. – T. 2. – S. 155 – 158.
8. Holubov B. Y. Ob asymptotyke kratn¥x synhulqrn¥x yntehralov dlq dyfferencyruem¥x
funkcyj // Mat. zametky. – 1981. – 30, # 5. – S. 749 – 762.
9. D\qçkov A. M. Asymptotyka synhulqrn¥x yntehralov y dyfferencyal\n¥e svojstva fun-
kcyj. – M., 1986. – 52 s. – Dep. v VYNYTY, # 7383-V86.
10. Löfström J. Some theoreme on interpolation spaces with application to approximation in Lp //
Math. Ann. – 1967. – 172, # 3. – P. 176 – 196.
11. Favard F. Sur la saturation des procedes sommation // J. math. pures et appl. – 1957. – 36, # 4. –
P. 359 – 372.
Poluçeno 13.06.05
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
|
| id | umjimathkievua-article-3223 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:29Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9a/d78316bba24288df695c2392ae4d289a.pdf |
| spelling | umjimathkievua-article-32232020-03-18T19:48:39Z Asymptotics of approximation of ψ-differentiable functions of many variables Асимптотика приближения ψ-дифференцируемых функций многих переменных Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. We investigate approximative characteristics of classes of ψ-differentiable multivariable functions introduced by A. I. Stepanets. We give asymptotics of the approximation of functions from these classes. Досліджуються апроксимативні характеристики класів &psi;-диференційовних функцій багатьох змінних, уведених O. I. Степанцем. Наведено асимптотику наближення функцій цих класів. Institute of Mathematics, NAS of Ukraine 2008-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3223 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 8 (2008); 1051–1057 Український математичний журнал; Том 60 № 8 (2008); 1051–1057 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3223/3192 https://umj.imath.kiev.ua/index.php/umj/article/view/3223/3193 Copyright (c) 2008 Lasuriya R. A. |
| spellingShingle | Lasuriya, R. A. Ласурия, Р. А. Ласурия, Р. А. Asymptotics of approximation of ψ-differentiable functions of many variables |
| title | Asymptotics of approximation of ψ-differentiable functions of many variables |
| title_alt | Асимптотика приближения ψ-дифференцируемых функций многих переменных |
| title_full | Asymptotics of approximation of ψ-differentiable functions of many variables |
| title_fullStr | Asymptotics of approximation of ψ-differentiable functions of many variables |
| title_full_unstemmed | Asymptotics of approximation of ψ-differentiable functions of many variables |
| title_short | Asymptotics of approximation of ψ-differentiable functions of many variables |
| title_sort | asymptotics of approximation of ψ-differentiable functions of many variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3223 |
| work_keys_str_mv | AT lasuriyara asymptoticsofapproximationofpsdifferentiablefunctionsofmanyvariables AT lasuriâra asymptoticsofapproximationofpsdifferentiablefunctionsofmanyvariables AT lasuriâra asymptoticsofapproximationofpsdifferentiablefunctionsofmanyvariables AT lasuriyara asimptotikapribliženiâpsdifferenciruemyhfunkcijmnogihperemennyh AT lasuriâra asimptotikapribliženiâpsdifferenciruemyhfunkcijmnogihperemennyh AT lasuriâra asimptotikapribliženiâpsdifferenciruemyhfunkcijmnogihperemennyh |