Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines
Exact estimates for the errors of approximation of functions of two variables and their derivatives by interpolation bilinear splines are obtained on certain classes.
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| Date: | 2005 |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509699867148288 |
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| author | Vakarchuk, S. B. Myskin, K. Yu. Вакарчук, С. Б. Мыскин, К. Ю. Вакарчук, С. Б. Мыскин, К. Ю. |
| author_facet | Vakarchuk, S. B. Myskin, K. Yu. Вакарчук, С. Б. Мыскин, К. Ю. Вакарчук, С. Б. Мыскин, К. Ю. |
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| datestamp_date | 2020-03-18T19:59:22Z |
| description | Exact estimates for the errors of approximation of functions of two variables and their derivatives by interpolation bilinear splines are obtained on certain classes. |
| first_indexed | 2026-03-24T02:45:16Z |
| format | Article |
| fulltext |
UDK 517.5
S. B. Vakarçuk, K. G. M¥skyn (Akad. tamoΩen. sluΩb¥ Ukrayn¥, Dnepropetrovsk)
NEKOTORÁE VOPROSÁ ODNOVREMENNOJ
APPROKSYMACYY FUNKCYJ DVUX PEREMENNÁX
Y YX PROYZVODNÁX YNTERPOLQCYONNÁMY
BYLYNEJNÁMY SPLAJNAMY
Exact estimates of approximation errors for two variable functions and their derivatives by bilinear
splines are obtained on some classes.
Na deqkyx klasax oderΩano toçni znaçennq ocinok poxybok nablyΩennq funkcij dvox zminnyx
ta ]x poxidnyx interpolqcijnymy bilinijnymy splajnamy.
1. Pust\ CD — klass neprer¥vn¥x v oblasty D =df
[ 0, 1 ] × [ 0, 1 ] funkcyj
f ( x, y ) , a C r sD
r s, , ∈( )+Z — klass funkcyj f ( x, y ) ∈ CD
, u kotor¥x neprer¥vn¥
proyzvodn¥e f x yi j( , )( , ) =df
∂
∂ ∂
+i j
i j
f
x y
, hde i ≤ r y j ≤ s; f x y( , )( , )0 0 =df
f ( x, y ) . Pry
πtom polahaem CD
0 0, =df
CD
. Dlq proyzvol\noj funkcyy f ( x, y ) ∈ CD
zapyßem
normu || f ||C = max ( , ) : ( , )f x y x y D∈{ } y poln¥j modul\ neprer¥vnosty
ω ( f; t, τ ) = f A f A A x y D A x y D( ) ( ) : ( , ) , ( , ) ,1 2 1 1 1 2 2 2−{ ∈ ∈
x x t y y1 2 1 2− ≤ − ≤ }, τ , t, τ ≥ 0.
Çerez CD
r s, ω( ) r s C CD D, , ; ( ),= ( ) =( )0 1 0 0 ω ωdf
oboznaçym klass funkcyj f ( x, y ) ∈
∈ CD
r s,
, udovletvorqgwyx uslovyg ω τf tr s( , ); ,( ) ≤ ω ( t, τ ) , 0 ≤ t, τ ≤ 1, hde
ω ( t, τ ) — nekotor¥j modul\ neprer¥vnosty.
Pust\ Ω ( t ) , t ≥ 0, — proyzvol\n¥j modul\ neprer¥vnosty y ρ ( A 1 , A 2 ) —
rasstoqnye meΩdu toçkamy A x y1 1 1( , ) y A x y2 2 2( , ) , prynadleΩawymy D.
Oboznaçym çerez CD
r s
,
,
ρ Ω( ) r s C CD D, , ; ( ),
,
,= ( ) =( )0 1 0 0
ρ ρΩ Ωdf
klass funkcyj
f ( x, y ) ∈ CD
r s,
, kotor¥e udovletvorqgt uslovyg
ωρ f tr s( , ),( ) =df
ω f tr s( , ),( ) = sup ( ) ( ) : , ,( , ) ( , )f A f A A A Dr s r s
1 2 1 2−{ ∈
ρ( , )A A t1 2 ≤ } ≤ Ω ( t ), 0 ≤ t ≤ dρ
,
hde dρ = max ( , ): ,ρ A A A A D1 2 1 2 ∈{ } .
Dalee v kaçestve ρ rassmotrym evklydovo
© S. B. VAKARÇUK, K. G. MÁSKYN, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 147
148 S. B. VAKARÇUK, K. G. MÁSKYN
ρe A A( , )1 2 = ( ) ( )x x y y1 2
2
1 2
2− + −
y xπmmynhovo
ρH A A( , )1 2 = x x y y1 2 1 2− + −
rasstoqnyq. Pry πtom d
eρ = 2 , a d
Hρ = 2.
Zadadym v oblasty D setku uzlov δn m =df
δ δn
x
m
y× (n, m ≥ 2), hde δn
x : x i =
= i / n, i = 0, n; δm
y : yj = j / m, j = 0, m , y postavym v sootvetstvye kaΩdoj funk-
cyy f ( x, y ) ∈ CD funkcyg S1, 1 ( f; x , y ) ∈ CD , opredelennug sledugwym
obrazom:
1) na kaΩdom prqmouhol\nyke Dij =df
x xi i, +[ ]1 × y yj j, +[ ]1 , i = 0 1, n − ,
j = 0 1, m − , S1, 1 ( f; x, y ) qvlqetsq alhebrayçeskym mnohoçlenom pervoj stepeny
po x y po y;
2) S1, 1 ( f; x i , yj ) = f ( x i, yj ), i = 0, n , j = 0, m .
Funkcyy S1, 1 ( f; x, y ) naz¥vagt ynterpolqcyonn¥my splajnamy pervoj ste-
peny dvux peremenn¥x ·1, s. 54D–D58‚ yly ynterpolqcyonn¥my bylynejn¥my
splajnamy ·2‚.
Na mnoΩestve toçek ( x, y ) ∈ Di j
, i = 0 1, n − , j = 0 1, m − , ymeet mesto pred-
stavlenye
S1, 1 ( f; x, y ) =
p k
i p j k p i k jf x y H x H y
= =
+ +∑ ∑ ( )
0
1
0
1
, ( ) ( ), , ,
hde
H xi0, ( ) =df
n x xi( )+ −1 ,
p
p iH
=
∑
0
1
, ≡ 1,
H yj0, ( ) =df
m y yj( )+ −1 ,
k
k jH
=
∑
0
1
, ≡ 1.
Pry fyksyrovannom znaçenyy odnoj yz peremenn¥x, naprymer x ( y ) , S1, 1 ( f;
x, y ) qvlqetsq splajnom pervoj stepeny otnosytel\no druhoj peremennoj y ( x ) .
Netrudno ubedyt\sq v tom, çto dlq lgboj funkcyy f ( x, y ) splajn S1, 1 ( f; x, y )
edynstven ·1‚.
Oboznaçym
F x yi j( , ) =df
p k
p k
i p j kf x y
= =
+
+ +∑ ∑ −
0
1
0
1
1( ) ( , ) ,
F x yj i( , ) =df
p k
p
k j i p j kH y f x y
= =
+
+ +∑ ∑ −
0
1
0
1
11( ) ( ) ( , ), ,
F x yi j( , ) =df
p k
k
p i i p j kH x f x y
= =
+
+ +∑ ∑ −
0
1
0
1
11( ) ( ) ( , ), ,
hde i = 0 1, n − , j = 0 1, m − .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEKOTORÁE VOPROSÁ ODNOVREMENNOJ APPROKSYMACYY FUNKCYJ … 149
Napomnym, çto dlq reßenyq sformulyrovannoj v ·3‚ zadaçy polyhonal\noj
ynterpolqcyy V. N. Malozemov doopredelyl pervug proyzvodnug lomanoj
L n ( f, x ) , 0 ≤ x ≤ 1, v ee verßynax x i = i / n , i = 0, n , sledugwym obrazom:
L f xi
( )( , )1 =df
n f x f xi i( ) ( )+ −[ ]1 , i = 0 1, n − , L f( )( , )1 1 =df
n f f xn( ) ( )1 1−[ ]− , polahaq
pry πtom L f x( )( , )1 = n f x f xi i( ) ( )+ −[ ]1 , esly x ∈ [ xi , xi+1 ) , i = 0 2, n − , y
L f x( )( , )1 = n f f xn( ) ( )1 1−[ ]− , esly x ∈ [ x n – 1, 1 ] .
Pry reßenyy πkstremal\n¥x zadaç, sformulyrovann¥x v teoremax 1 – 3 sle-
dugweho punkta, voznykagt analohyçn¥e problem¥, svqzann¥e s doopredeleny-
em smeßannoj proyzvodnoj S f x y1 1
1 1
,
( , )( ; , ) y çastn¥x proyzvodn¥x S f x yr s
1 1,
( , )( ; , )
( r, s = 0, 1; r + s = 1) funkcyy S1, 1 ( f; x, y ) na toçeçn¥x mnoΩestvax
A i =df
( , ) : ,x y x x yi= ≤ ≤{ }0 1 , i = 1 1, n − ,
B j =df
( , ) : ,x y y y xj= ≤ ≤{ }0 1 , j = 1 1, m − ,
hde proyzvodn¥e preterpevagt razr¥v¥ pervoho roda.
Yspol\zuq yzloΩenn¥e v ·3‚ soobraΩenyq, doopredelym S f x y1 1
1 1
,
( , )( ; , ) na
mnoΩestve D. V rezul\tate πtoho dlq smeßannoj proyzvodnoj poluçym sle-
dugwye v¥raΩenyq ·4‚:
1) esly ( x, y ) ∈ ′Dij =df
[ xi , xi+1 ) × [ yj , yj+1 ), i = 0 2, n − , j = 0 2, m − , to
S f x y1 1
1 1
,
( , )( ; , ) = n m F ( x i , yj ) ;
2) esly ( x , y ) ∈ ′ −Dn j1, =df
[ x n – 1, x n ] × [ y j , y j+1 ), j = 0 2, m − , to
S f x y1 1
1 1
,
( , )( ; , ) = n m F ( x n – 1, yj ) ;
3) esly ( x , y ) ∈ ′ −Di m, 1 =df
[ xi , x i+1 ) × [ ym – 1, ym ] , i = 0 2, n − , to
S f x y1 1
1 1
,
( , )( ; , ) = n m F ( x i, ym – 1 ) ;
4) esly ( x, y ) ∈ ′ − −Dn m1 1, =df
[ x n – 1, xn ] × [ ym – 1, ym ] , to S f x y1 1
1 1
,
( , )( ; , ) =
= n m F ( x n – 1, ym – 1 ) .
Yspol\zuq ukazann¥e v¥ße soobraΩenyq, posle doopredelenyq proyzvodnoj
S f x y1 1
1 0
,
( , )( ; , ) na mnoΩestve D poluçym v¥raΩenyq ·5‚:
a 1) esly ( x, y ) ∈ ′′Dij =df
[ xi , xi+1 ) × [ yj , yj+1 ] , i = 0 2, n − , j = 0 1, m − , to
S f x y1 1
1 0
,
( , )( ; , ) = n Fj ( x i, y ) ;
b 1) esly ( x, y ) ∈ ′′−Dn j1, =df
[ x n – 1, x n ] × [ yj , y j+1 ] , j = 0 1, m − , to
S f x y1 1
1 0
,
( , )( ; , ) = n Fj ( x n – 1, y ) .
Posle provedennoho sootvetstvugwym obrazom doopredelenyq çastnoj
proyzvodnoj S f x y1 1
0 1
,
( , )( ; , ) na D ymeem sledugwye v¥raΩenyq dlq dannoj
funkcyy ·5‚:
a 2) S f x y1 1
0 1
,
( , )( ; , ) = m Fi ( x, yj ) , esly ( x, y ) ∈ ′′′Dij =df
[ xi , xi+1 ] × [ yj , yj+1 ) , i =
= 0 1, n − , j = 0 2, m − ;
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
150 S. B. VAKARÇUK, K. G. MÁSKYN
b 2) S f x y1 1
0 1
,
( , )( ; , ) = m Fi ( x, ym – 1 ) , esly ( x, y ) ∈ ′′′ −Di m, 1 =df
[ xi , xi+1 ] × [ ym – 1,
ym ] , i = 0 1, n − .
Otmetym, çto zapysann¥e s uçetom provedennoho doopredelenyq v¥raΩenyq
dlq S f x yr s
1 1,
( , )( ; , ) ( r, s = 0, 1; 1 ≤ r + s ≤ 2) sovpadagt s formulamy dlq soot-
vetstvugwyx proyzvodn¥x tam, hde poslednye suwestvugt.
2. V monohrafyy N. P. Kornejçuka ·6, s. 323‚ otmeçalos\, çto po sravnenyg
s odnomern¥m sluçaem yssledovanye voprosov pryblyΩenyq funkcyj dvux y
bolee peremenn¥x znaçytel\no usloΩnqetsq vvydu poqvlenyq pryncypyal\no
nov¥x obstoqtel\stv, svqzann¥x s mnohomernost\g. Poπtomu na sehodnqßnyj
den\ ymeetsq nemnoho toçn¥x rezul\tatov, poluçenn¥x v zadaçax ocenky
pohreßnosty pryblyΩenyq v mnohomernom sluçae, v tom çysle y v zadaçax
mnohomernoj splajn-ynterpolqcyy.
Dlq proyzvol\noj funkcyy f ( x, y ) ∈ CD
r s, , r, s = 0, 1, oboznaçym
�n m
r s f x y,
, ( ; , ) =df
f x y S f x yr s r s( , )
,
( , )( , ) ( ; , )− 1 1 , ( x, y ) ∈ D,
hde �n m f x y,
, ( ; , )0 0 = �n m f x y, ( ; , ). Dlq lgboho klassa � ⊂ CD
r s,
zapyßem
En m
r s
,
, ( )� =df
sup ( ) :,
,� �n m
r s
C
f f ∈{ }.
Pry πtom polahaem En m,
, ( )0 0 � =df
En m, ( )� .
Pust\ ω ( t, τ ) y Ω ( t ) — proyzvol\n¥e v¥pukl¥e vverx moduly neprer¥v-
nosty. V rabotax ·7, 8‚ pokazano, çto
E Cn m D, ( )ω( ) = ω 1
2
1
2n m
,
y
E Cn m D e, , ( )ρ Ω( ) = Ω 1
2
1 1
2 2n m
+
.
Dlq proyzvol\noho modulq neprer¥vnosty ω ( t, τ ) v stat\e ·4‚ pokazano, çto
E Cn m D,
, , ( )1 1 1 1 ω( ) = nm t dt d
n m
0
1
0
1/ /
( , )∫ ∫ ω τ τ .
V rabote ·9‚ dokazana spravedlyvost\ sootnoßenyj
E Cn m D,
, , ( )1 0 1 0 ω( ) = n t
m
dt
n
0
1
1
2
/
,∫
ω ,
esly ω ( t, τ ) — modul\ neprer¥vnosty, v¥pukl¥j vverx po peremennoj τ;
E Cn m D,
, , ( )0 1 0 1 ω( ) = m
n
d
m
0
1
1
2
/
,∫
ω τ τ,
esly ω ( t, τ ) — modul\ neprer¥vnosty, v¥pukl¥j vverx po peremennoj t;
dlq v¥pukloho vverx modulq neprer¥vnosty Ω ( t ) ymegt mesto ravenstva
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEKOTORÁE VOPROSÁ ODNOVREMENNOJ APPROKSYMACYY FUNKCYJ … 151
E Cn m D e,
,
,
, ( )1 0 1 0
ρ Ω( ) = n t
m
dt
n
0
1
2
2
1
4
/
∫ +
Ω ,
E Cn m D e,
,
,
, ( )0 1 0 1
ρ Ω( ) = m
n
d
m
0
1
2
21
4
/
∫ +
Ω τ τ ,
a dlq proyzvol\noho modulq neprer¥vnosty Ω ( t )
E Cn m D e,
,
,
, ( )1 1 1 1
ρ Ω( ) = nm t dt d
n m
0
1
0
1
2 2
/ /
∫ ∫ +( )Ω τ τ .
V ·5‚ poluçen¥ sledugwye rezul\tat¥:
esly ω* ( t, τ ) = ω1 ( t ) + ω2 ( τ ) , hde ω i ( ⋅ ) , i = 1, 2, — proyzvol\n¥e moduly
neprer¥vnosty, to
E Cn m D,
, ,
*( )1 1 1 1 ω( ) = n t dt m d
n m
0
1
1
0
1
2
/ /
∫ ∫( ) + ( )ω ω τ τ ;
esly ˜ ( , )ω τt = ω1 ( t ) + ω2 ( τ ) , hde ω1 ( t ) — proyzvol\n¥j (v¥pukl¥j vverx),
a ω2 ( τ ) — v¥pukl¥j vverx (proyzvol\n¥j) moduly neprer¥vnosty, to
E Cn m D,
, , ( ˜ )1 0 1 0 ω( ) = n t dt
m
n
0
1
1 2
1
2
/
∫ ( ) +
ω ω
E C
n
m dn m D
m
,
, ,
/
( ˜ )0 1 0 1
1
0
1
2
1
2
ω ω ω τ τ( ) =
+ ( )
∫ .
Dannaq stat\q prodolΩaet ukazannug tematyku y osnovnoe ee soderΩanye
sostavlqgt sledugwye teorem¥.
Teorema 1. Pust\ Ω ( t ) — proyzvol\n¥j v¥pukl¥j vverx modul\ neprer¥v-
nosty. Tohda dlq lgb¥x natural\n¥x çysel n, m ≥ 2 spravedlyv¥ ravenstva
E Cn m D H, , ( )ρ Ω( ) = Ω 1
2
1
2n m
+
. (1)
Teorema 2. Esly uslovyq teorem¥ 1 v¥polnen¥, to
E Cn m D H,
,
,
, ( )1 0 1 0
ρ Ω( ) = n t dt
m
n m
1 2
1 1 2
/( )
/ /( )+
∫ ( )Ω , (2)
E Cn m D H,
,
,
, ( )0 1 0 1
ρ Ω( ) = m t dt
n
n m
1 2
1 2 1
/( )
/( ) /+
∫ ( )Ω , (3)
hde natural\n¥e çysla n, m ≥ 2.
Teorema 3. Pust\ Ω ( t ) — proyzvol\n¥j modul\ neprer¥vnosty. Tohda dlq
lgb¥x natural\n¥x çysel n, m ≥ 2 ymegt mesto ravenstva
E Cn m D H,
,
,
, ( )1 1 1 1
ρ Ω( ) = nm t dt d
n m
0
1
0
1/ /
∫ ∫ +( )Ω τ τ =
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
152 S. B. VAKARÇUK, K. G. MÁSKYN
= nm
t t dt
n
t dt
n m
t t dt m n
t t dt
m
t dt
n m
t t dt m
n
n
m
m
m n
m
m
n
n
n m
0
1
1
1
1
1 1
0
1
1
1
1
1 1
1 1 1
1 1 1
/
/
/
/
/ /
/
/
/
/
/ /
, ,
,
∫ ∫ ∫
∫ ∫ ∫
( ) + ( ) + + −
( ) <
( ) + ( ) + + −
( ) >
+
+
Ω Ω Ω
Ω Ω Ω nn
t t dt
n
t t dt m n
n
n
n
,
, .
/
/
/
0
1
1
2
2∫ ∫( ) + −
( ) =
Ω Ω
(4)
3. Dokazatel\stvo teorem¥ 1. Uçyt¥vaq yzloΩenn¥e v p. 1 svojstva
funkcyj H xp i, ( ) y H yk j, ( ), p , k = 0, 1, hde ( x , y ) ∈ Di j, i = 0 1, n − ,
j = 0 1, m − , dlq proyzvol\noj funkcyy f ( x, y ) ∈ CD zapys¥vaem
�n m f x y, ( ; , ) =
p k
p i k j i p j kH x H y f x y f x y
= =
+ +∑ ∑ −[ ]
0
1
0
1
, ,( ) ( ) ( , ) ( , ) .
Yspol\zuq opredelenye modulq neprer¥vnosty, otsgda ymeem
�n m f x y, ( ; , ) ≤
p k
p i k j i p j kH x H y f x x y y
H
= =
+ +∑ ∑ − + −( )
0
1
0
1
, ,( ) ( ) ;ωρ . (5)
Poskol\ku Ω ( t ) — v¥puklaq vverx funkcyq, ysxodq yz neravenstva (5), zapy-
s¥vaem
�n m f x y, ( ; , ) ≤
≤ Ω
p k
p i k j
p
i p
k
j kH x H y x x y y
= =
+ +∑ ∑ − −( ) + − −( )( )
0
1
0
1
1 1, ,( ) ( ) ( ) ( ) , (6)
hde ( x, y ) ∈ Di j, i = 0 1, n − , j = 0 1, m − . Yspol\zuq opredelenyq funkcyj
H xp i, ( ) y H yp j, ( ), p = 0, 1, yz (6) ymeem
�n m f x y, ( ; , ) ≤ Ω
p k
p i k j p i k jH x H y
n
H x
m
H y
= =
− −∑ ∑ +
0
1
0
1
1 1
1 1
, , , ,( ) ( ) ( ) ( ) =
= Ω 1 1
0
1
1
0
1
0
1
1
0
1
n
H x H x H y
m
H y H y H x
p
p i p i
k
k j
k
k j k j
p
p i
=
−
= =
−
=
∑ ∑ ∑ ∑+
, , , , , ,( ) ( ) ( ) ( ) ( ) ( ) =
= Ω 2 2
0 1 0 1n
H x H x
m
H y H yi i j j, , , ,( ) ( ) ( ) ( )+
,
hde ( x, y ) ∈ Di j, i = 0 1, n − , j = 0 1, m − . Poskol\ku
max ( ) ( ); ( ) ( )
( , )
, ; ,
, , , ,
x y D
i n j m
i i j j
ij
H x H x H y H y
∈
= − = −( )
{ }
0 1 0 1
0 1 0 1 = 1
4
, (7)
dlq lgboj toçky ( x, y ) ∈ Di j, i = 0 1, n − , j = 0 1, m − , poluçaem
�n m f x y, ( ; , ) ≤ Ω 1
2
1
2n m
+
y, sledovatel\no,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEKOTORÁE VOPROSÁ ODNOVREMENNOJ APPROKSYMACYY FUNKCYJ … 153
E Cn m D H, , ( )ρ Ω( ) ≤ Ω 1
2
1
2n m
+
. (8)
Dlq poluçenyq ocenky snyzu v kaçestve πkstremal\noj rassmotrym funkcyg
f 0( x, y ) = Ω min , min ,x x x x y y y yi i j j− −( ) + − −( )( )+ +1 1 ,
hde ( x, y ) ∈ Di j, i = 0 1, n − , j = 0 1, m − . Provodq standartn¥e rassuΩdenyq
(sm., naprymer, ·4, 5, 8D–D9‚), moΩno ubedyt\sq v prynadleΩnosty f 0( x, y ) klassu
CD H, ( )ρ Ω . Poskol\ku f 0( x i , yj ) = 0, i = 0, n , j = 0, m , to
E Cn m D H, , ( )ρ Ω( ) ≥
�n m C
f, ( )0 = f C0 ≥
≥ f
n m0
1
2
1
2
,
= Ω 1
2
1
2n m
+
. (9)
Sravnyvaq ocenky sverxu (8) y snyzu (9), poluçaem ravenstvo (1), çto y zaver-
ßaet dokazatel\stvo teorem¥ 1.
4. Dokazatel\stvo teorem¥ 2. Ne umen\ßaq obwnosty, pokaΩem
spravedlyvost\ ravenstva (2), poskol\ku vse rassuΩdenyq, svqzann¥e s
poluçenyem sootnoßenyq (3), ymegt analohyçn¥j xarakter. Pust\, naprymer,
toçka ( x, y ) prynadleΩyt mnoΩestvu ′′Dij , i = 0 2, n − , j = 0 1, m − . Sohlasno
vvedenn¥m v p. 1 oboznaçenyqm
S f x y1 1
1 0
,
( , )( ; , ) = nF x yj i( , ) =
= n H y f x y f x y f x y
k
k j i j k j k i j k
=
+ + + +∑ ± −[ ]
0
1
1, ( ) ( , ) ( , ) ( , ) =
= n H y x x f x x y x x f x x y
k
k j i i j k i i j k
=
+ + + +∑ − + −[ ]
0
1
1 1, ( ) ( ) ( , ; ) ( ) ( , ; ) =
=
p k
p i k j i p j kH x H y f x x y
= =
− + +∑ ∑
0
1
0
1
1, ,( ) ( ) ( , ; ) , (10)
hde f x x yi p j k( , ; )− + +1 , p , k = 0, 1, — çastn¥e razdelenn¥e raznosty pervoho
porqdka funkcyy f ( x, y ) po peremennoj x pry y = yj k+ . Yspol\zuq v (10)
yntehral\noe predstavlenye razdelenn¥x raznostej ·10‚, dlq proyzvol\noj
funkcyy f ( x, y ) ∈ CD
1 0,
zapys¥vaem
�n m f x y,
, ( ; , )1 0 = f x y S f x y( , )
,
( , )( , ) ( ; , )1 0
1 1
1 0− =
=
p k
p i k j i p j kH x H y f x y f x x x y d
= =
− + +∑ ∑ ∫ − + −( )[ ]
0
1
0
1
0
1
1 0 1 0
1, ,
( , ) ( , )( ) ( ) ( , ) ( );τ τ ,
hde ( x, y ) ∈ ′′Dij , i = 0 2, n − , j = 0 1, m − . Otsgda sleduet, çto
�n m f x y,
, ( ; , )1 0 ≤
≤
p k
p i k j i p j kH x H y f x x y y d
H
= =
− + +∑ ∑ ∫ − + −( )
0
1
0
1
0
1
1 0
1, ,
( , )( ) ( ) ;ω τ τρ . (11)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
154 S. B. VAKARÇUK, K. G. MÁSKYN
Yspol\zuq opredelenye klassa CD H,
, ( )ρ
1 0 Ω , yz (11) pry ( x, y ) ∈ ′′Dij , i = 0 2, n − ,
j = 0 1, m − , poluçaem
�n m f x y,
, ( ; , )1 0 ≤
≤
p k
p i k j p i k jH x H y
n
H x
m
H y d
= =
−∑ ∑ ∫ +
0
1
0
1
0
1
1
1
, , , ,( ) ( ) ( ) ( )Ω τ τ . (12)
Uçyt¥vaq v¥puklost\ vverx modulq neprer¥vnosty Ω ( t ) y sootnoßenye (7), yz
(12) ymeem
�n m f x y,
, ( ; , )1 0 ≤ n H y u
m
H y du
p k
k j
n H x
k j
p i
= =
−∑ ∑ ∫
−
+
0
1
0
1
0
1
1
1
,
( )
,( ) ( )
,
Ω ≤
≤ n u
m
H y H y du
p
n H x
j j
p i
=
∑ ∫
−
+
0
1
0
0 1
1
2
, ( )
, ,( ) ( )Ω ≤ Ψi x( ), (13)
hde
Ψi x( ) =df
n u
m
du
p
n H xp i
=
∑ ∫
−
+
0
1
0
1
1
2
, ( )
Ω .
Oçevydno, çto funkcyq Ψi x( ) neprer¥vna na otrezke [ xi , x i+1 ] . Provodq
ob¥çn¥m obrazom ee yssledovanye na πkstremum na ukazannom mnoΩestve, po-
luçaem
max ( ) :Ψi i ix x x x≤ ≤{ }+1 = Ψi i kx( )+ , k = 0, 1. (14)
Tohda yz (12)D–D(14) ymeem
�n m f x y,
, ( ; , )1 0 ≤ n t dt
m
n m
1 2
1 1 2
/( )
/ /( )+
∫ ( )Ω , (15)
hde ( x, y ) ∈ ′′Dij , i = 0 2, n − , j = 0 1, m − . Na osnovanyy podobn¥x rassuΩdenyj
moΩno ubedyt\sq v spravedlyvosty ocenok, analohyçn¥x (15), kohda ( x, y ) ∈
∈ ′′−Dn j1, , j = 0 1, m − . Yz yzloΩennoho v¥ße sleduet ocenka sverxu
E Cn m D H,
,
,
, ( )1 0 1 0
ρ Ω( ) =
=
sup ( ) : ( ),
,
,
,�n m C Df f C
H
1 0 1 0∈{ }ρ Ω ≤ n t dt
m
n m
1 2
1 1 2
/( )
/ /( )+
∫ ( )Ω . (16)
Dlq poluçenyq ocenky snyzu rassmotrym funkcyg γ ( x, y ) , kotoraq za-
daetsq na mnoΩestve 0 2,
n
× 0 1,
m
sledugwym obrazom:
γ ( x, y ) =df
Ω ( ) ( )− −
+ − −
1 1 1 1
2
i j
n
x
m
y ,
esly ( x, y ) ∈ i
n
i
n
j
m
j
m
, ,+
× +
1
2
1
2
, i, j = 0, 1. Pry πtom ymegt mesto ravenstva
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEKOTORÁE VOPROSÁ ODNOVREMENNOJ APPROKSYMACYY FUNKCYJ … 155
γ x
n
y+
2 , = γ x y
m
, +
1 = γ ( x, y ) ,
t. e. γ ( x, y ) qvlqetsq
2
n
-peryodyçeskoj po peremennoj x y
1
m
-peryodyçeskoj
po peremennoj y. Polahaq
γ* ( x, y ) =df
γ( , )
/( )
/ /( )
x y n t dt
m
n m
− ( )
+
∫
1 2
1 1 2
Ω ,
vvodym funkcyg
f1 ( x, y ) =df
0
x
t y dt∫ ( )γ
*
, , ( x, y ) ∈ D.
Putem standartn¥x rassuΩdenyj moΩno ubedyt\sq v prynadleΩnosty f1 ( x, y )
klassu CD H,
, ( )ρ
1 0 Ω . Posle nesloΩn¥x v¥çyslenyj poluçaem f1 ( xi , yj ) = 0, i =
= 0, n , j = 0, m . Znaçyt, S f x y1 1 1, ; ,( ) ≡ 0 dlq lgb¥x toçek ( x, y ) ∈ D. Tohda
E Cn m D H,
,
,
, ( )1 0 1 0
ρ Ω( ) ≥
�n m C
f,
, ( )1 0
1 = f
C1
1 0( , ) ≥
≥ f
n m1
1 0 1 1
2
( , ) ,
= n t dt
m
n m
1 2
1 1 2
/( )
/ /( )+
∫ ( )Ω . (17)
Trebuemoe ravenstvo (2) poluçym, sopostavyv ocenku sverxu (16) y ocenku sny-
zuD(17).
Teorema 2 dokazana.
5. Dokazatel\stvo teorem¥ 3. Ne umen\ßaq obwnosty, provedem ras-
suΩdenyq dlq mnoΩestv ′Dij , i = 0 2, n − , j = 0 2, m − , poskol\ku dlq mnoΩestv
′ −Dn j1, , j = 0 2, m − , ′ −Di m, 1, i = 0 2, n − , y ′ − −Dn m1 1, xod rassuΩdenyj analohy-
çen. Yspol\zovav opredelenye y svojstva razdelenn¥x raznostej funkcyy
f ( x, y ) ∈ CD
1 1, [10], dlq proyzvol\noj toçky ( x, y ) ∈ ′Dij zapyßem sledugwye
ravenstva ·4‚:
�n m f x y,
, ( ; , )1 1 = f x y S f x y( , )
,
( , )( , ) ; ,1 1
1 1
1 1− ( ) =
= f x y nm F x y f x yi j
p
i p
( , )( , ) ( , ) ( , )1 1
0
1
− ±
=
+∑ = f x y( , )( , )1 1 –
– n H y f x y y f x y y
k
k j
p
p
i p j k j k
= =
− + − + − +∑ ∑ − ±
0
1
0
1
1 1 11, ( ) ( ) ( ; , ) ( ; , ) =
= f x y H x H y f x x y y
p k
p i k j i p j k
( , )
, ,( , ) ( ) ( ) ( , ; , )1 1
0
1
0
1
1 1−
= =
− + − +∑ ∑ , (18)
hde f x y yi p j k( ; , )− + − +1 1 y f x y yj k( ; , )− +1 , k, p = 0, 1 — çastn¥e razdelenn¥e raz-
nosty pervoho porqdka po peremennoj y, f x x y yi p j k( , ; , )− + − +1 1 , k , p = 0, 1, —
smeßann¥e razdelenn¥e raznosty pervoho porqdka po peremennoj x y po pere-
mennoj y. Predstavlqq smeßann¥e razdelenn¥e raznosty v (18) v yntehral\noj
forme ·10‚, dlq proyzvol\noj funkcyy f ( x, y ) ∈ CD
1 1,
poluçaem
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
156 S. B. VAKARÇUK, K. G. MÁSKYN
�n m f x y,
, ( ; , )1 1 =
p k
p i k jH x H y
= =
∑ ∑
0
1
0
1
, ,( ) ( ) ×
×
0
1
0
1
1 1 1 1
1 1∫ ∫ − + − + −( )[ ]− + − +f x y f x t x x y y y dt di p j k
( , ) ( , )( , ) ( ), ( )τ τ.
Yspol\zuq opredelenye modulq neprer¥vnosty, zapys¥vaem
�n m f x y,
, ( ; , )1 1 ≤
≤
p k
p i k j i p j kH x H y f t x x y y dt d
H
= =
− + − +∑ ∑ ∫ ∫ − + −( )
0
1
0
1
0
1
0
1
1 1
1 1, ,
( , )( ) ( ) ;ω τ τρ , (19)
hde ( x, y ) ∈ ′Dij , i = 0 2, n − , j = 0 2, m − .
Oboznaçym
Ψij x y( , ) =df
nm u dud
p k
n H x m H yp i k j
= =
∑ ∑ ∫ ∫
− −
+( )
0
1
0
1
0 0
1 1
, ,( ) ( )
Ω v v. (20)
Dlq kaΩdoj funkcyy f ( x, y ) ∈ CD H,
, ( )ρ
1 1 Ω v sylu (19), (20) ymeem
�n m f x y,
, ( ; , )1 1 ≤ Ψij x y( , ), (21)
hde ( x, y ) ∈ ′Dij , i = 0 2, n − , j = 0 2, m − . Yssleduq funkcyg Ψij x y( , ) na
πkstremum na mnoΩestve ′Dij , hde ′Dij — zam¥kanye ′Dij , poluçaem
max ( , ) : ( , )Ψij ijx y x y D∈ ′{ } = Ψij i p j kx y( , )+ + =
=
nm u dud
n m
0
1
0
1/ /
∫ ∫ +( )Ω v v , p, k = 0, 1. (22)
Uçyt¥vaq (21), (22) y yzloΩennoe v naçale dannoho punkta otnosytel\no
analohyçnoho xaraktera rassuΩdenyj na mnoΩestvax ′ −Dn j1, , j = 0 2, m − ,
′ −Di m, 1, i = 0 2, n − , y ′ − −Dn m1 1, , zapys¥vaem ocenku sverxu
E Cn m D H,
,
,
, ( )1 1 1 1
ρ Ω( ) ≤
nm u dud
n m
0
1
0
1/ /
∫ ∫ +( )Ω v v . (23)
Prymenqq dlq v¥çyslenyq dvukratnoho yntehrala standartn¥e metod¥ matema-
tyçeskoho analyza, yz (23) poluçaem
E Cn m D H,
,
,
, ( )1 1 1 1
ρ Ω( ) ≤
≤ nm
0
1
1
1
1
1 1
0
1
1
1
1
1 1
1 1 1
1 1 1
/
/
/
/
/ /
/
/
/
/
/ /
, ,
,
n
n
m
m
m n
m
m
n
n
n m
t t dt
n
t dt
n m
t t dt m n
t t dt
m
t dt
n m
t t dt m n
∫ ∫ ∫
∫ ∫ ∫
( ) + ( ) + + −
( ) <
( ) + ( ) + + −
( ) >
+
+
Ω Ω Ω
Ω Ω Ω ,,
, .
/
/
/
0
1
1
2
2
n
n
n
t t dt
n
t t dt m n∫ ∫( ) + −
( ) =
Ω Ω
(24)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEKOTORÁE VOPROSÁ ODNOVREMENNOJ APPROKSYMACYY FUNKCYJ … 157
Dlq poluçenyq ocenky snyzu rassmotrym funkcyg
ϕ ( x, y ) =df
Ω ( ) ( )− −
+ − −
1 1 1 1i j
n
x
m
y ,
hde ( x, y ) ∈ i
n
i
n
j
m
j
m
, ,+
× +
1 1
, i, j = 0, 1, y
ϕ x
n
y+
2 , = ϕ x y
m
, +
2 = ϕ ( x, y ) .
Polahaq
ϕ* ( x, y ) =df
ϕ( , )
/ /
x y nm u dud
n m
− +( )∫ ∫
0
1
0
1
Ω v v, ( x, y ) ∈ D,
vvodym funkcyg
f2 ( x, y ) =df
0 0
x y
t dt d∫ ∫ ( )ϕ τ τ
*
, , ( x, y ) ∈ D.
Netrudno ubedyt\sq v prynadleΩnosty f2 ( x, y ) klassu CD H,
, ( )ρ
1 1 Ω . Uçyt¥vaq,
çto f2 ( x i, yj ) = 0, i = 0, n , j = 0, n , a znaçyt, S f x y1 1 2, ; ,( ) ≡ 0, zapys¥vaem
E Cn m D H,
,
,
, ( )1 1 1 1
ρ Ω( ) ≥
�n m C
f,
, ( )1 1
2 ≥ f
n m2
1 1 1 1( , ) ,
=
nm u dud
n m
0
1
0
1/ /
∫ ∫ +( )Ω v v .
Sravnyvaq poluçennug ocenku snyzu s ocenkoj sverxu (24), poluçaem trebuemoe
sootnoßenye (4).
Teorema 3 dokazana.
1. Zav\qlov G. S., Kvasov B. Y., Myroßnyçenko V. L. Metod¥ splajn-funkcyj. – M.: Nauka,
1980. –D350 s.
2. Íumylov B. M. O lokal\noj approksymacyy bylynejn¥my splajnamy // Metod¥ splajn-
funkcyj (v¥çyslytel\n¥e system¥). – 1979. – V¥p. 81. – S. 42D–D47.
3. Malozemov V. N. K polyhonal\noj ynterpolqcyy // Mat. zametky. –D1967. – 1, # 5. – S.
537D–D540.
4. Vakarçuk S. B. K ynterpolqcyy bylynejn¥my splajnamy // Tam Ωe. –D1990. – 47, # 5. – S.
26D–D30.
5. Íabozov M. Í. Toçn¥e ocenky odnovremennoho pryblyΩenyq funkcyj dvux peremenn¥x y
yx proyzvodn¥x bylynejn¥my splajnamy // Tam Ωe. –D1996. – 59, # 1. – S. 142D–D152.
6. Kornejçuk N. P. Splajn¥ v teoryy pryblyΩenyj. – M.: Nauka, 1984. –D352 s.
7. Storçaj V. F. PryblyΩenye neprer¥vn¥x funkcyj dvux peremenn¥x splajn-funkcyqmy v
metryke C // Yssledovanyq po sovremenn¥m problemam summyrovanyq y pryblyΩenyq
funkcyj y yx pryloΩenyqm. – Dnepropetrovsk: Dnepropetr. un-t, 1972. – S. 66D–D68.
8. Storçaj V. F. PryblyΩenye neprer¥vn¥x funkcyj dvux peremenn¥x mnohohrann¥my
funkcyqmy y splajn-funkcyqmy v ravnomernoj metryke // Tam Ωe. – 1975. – S. 82D–D89.
9. Íabozov M. Í. O pohreßnosty ynterpolqcyy bylynejn¥my splajnamy// Ukr. mat. Ωurn. –
1994. – 46, # 11. – S. 1554D–D1560.
10. Mykeladze Í. E. Çyslenn¥e metod¥ matematyçeskoho analyza. – M.: Hostexyzdat, 1953.
–D528 s.
Poluçeno 17.06.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
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| id | umjimathkievua-article-3583 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:16Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f8/f219e0e0646c022bd2142c96f29307f8.pdf |
| spelling | umjimathkievua-article-35832020-03-18T19:59:22Z Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines Некоторые вопросы одновременной аппроксимации функций двух переменных и их производных интерполяционными билинейными сплайнами Vakarchuk, S. B. Myskin, K. Yu. Вакарчук, С. Б. Мыскин, К. Ю. Вакарчук, С. Б. Мыскин, К. Ю. Exact estimates for the errors of approximation of functions of two variables and their derivatives by interpolation bilinear splines are obtained on certain classes. На деяких класах одержано точні значення оцінок похибок наближення функцій двох змінних та їх похідних інтерполяційними білінійними сплайнами. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3583 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 147–157 Український математичний журнал; Том 57 № 2 (2005); 147–157 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3583/3898 https://umj.imath.kiev.ua/index.php/umj/article/view/3583/3899 Copyright (c) 2005 Vakarchuk S. B.; Myskin K. Yu. |
| spellingShingle | Vakarchuk, S. B. Myskin, K. Yu. Вакарчук, С. Б. Мыскин, К. Ю. Вакарчук, С. Б. Мыскин, К. Ю. Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines |
| title | Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines |
| title_alt | Некоторые вопросы одновременной аппроксимации функций двух переменных и их производных интерполяционными билинейными сплайнами |
| title_full | Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines |
| title_fullStr | Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines |
| title_full_unstemmed | Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines |
| title_short | Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines |
| title_sort | some problems of simultaneous approximation of functions of two variables and their derivatives by interpolation bilinear splines |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3583 |
| work_keys_str_mv | AT vakarchuksb someproblemsofsimultaneousapproximationoffunctionsoftwovariablesandtheirderivativesbyinterpolationbilinearsplines AT myskinkyu someproblemsofsimultaneousapproximationoffunctionsoftwovariablesandtheirderivativesbyinterpolationbilinearsplines AT vakarčuksb someproblemsofsimultaneousapproximationoffunctionsoftwovariablesandtheirderivativesbyinterpolationbilinearsplines AT myskinkû someproblemsofsimultaneousapproximationoffunctionsoftwovariablesandtheirderivativesbyinterpolationbilinearsplines AT vakarčuksb someproblemsofsimultaneousapproximationoffunctionsoftwovariablesandtheirderivativesbyinterpolationbilinearsplines AT myskinkû someproblemsofsimultaneousapproximationoffunctionsoftwovariablesandtheirderivativesbyinterpolationbilinearsplines AT vakarchuksb nekotoryevoprosyodnovremennojapproksimaciifunkcijdvuhperemennyhiihproizvodnyhinterpolâcionnymibilinejnymisplajnami AT myskinkyu nekotoryevoprosyodnovremennojapproksimaciifunkcijdvuhperemennyhiihproizvodnyhinterpolâcionnymibilinejnymisplajnami AT vakarčuksb nekotoryevoprosyodnovremennojapproksimaciifunkcijdvuhperemennyhiihproizvodnyhinterpolâcionnymibilinejnymisplajnami AT myskinkû nekotoryevoprosyodnovremennojapproksimaciifunkcijdvuhperemennyhiihproizvodnyhinterpolâcionnymibilinejnymisplajnami AT vakarčuksb nekotoryevoprosyodnovremennojapproksimaciifunkcijdvuhperemennyhiihproizvodnyhinterpolâcionnymibilinejnymisplajnami AT myskinkû nekotoryevoprosyodnovremennojapproksimaciifunkcijdvuhperemennyhiihproizvodnyhinterpolâcionnymibilinejnymisplajnami |