Evaluation of the remainder term for the Thiele interpolation continued fraction
We present an estimate of the remainder term for the Thiele interpolation continued fraction.
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2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509325749911552 |
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| author | Pahirya, M. M. Пагіря, М. М. |
| author_facet | Pahirya, M. M. Пагіря, М. М. |
| author_sort | Pahirya, M. M. |
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| datestamp_date | 2020-03-18T19:49:31Z |
| description | We present an estimate of the remainder term for the Thiele interpolation continued fraction. |
| first_indexed | 2026-03-24T02:39:19Z |
| format | Article |
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UDK 517.518: 519.652
M. M. Pahirq (UΩhorod. nac. un-t)
OCINKA ZALYÍKOVOHO ÇLENA
INTERPOLQCIJNOHO LANCGHOVOHO DROBU TILE
An estimate of a remainder term of the Thiele interpolational continued fraction is obtained.
Poluçena ocenka ostatoçnoho çlena dlq ynterpolqcyonnoj cepnoj droby Tyle.
Vstup. Zadaça interpolqci] funkcij odni[] dijsno] zminno] na promiΩku [ α, β ]
dobre doslidΩena u vypadku linijno] interpolqci], tobto koly interpolqcijna
funkciq vybyra[t\sq u vyhlqdi uzahal\nenoho mnohoçlena [1, 2] g ( x ; f ; c0 ,
c1 , … , cn ) = c0
ϕ0 ( x ) + … + cn
ϕn ( x ), de { ϕi ( x ) } — systema funkcij Çebyßova.
U vypadku nelinijno] interpolqci] funkcig g ( x ; f ; c0 , c1 , … , cn ) çasto vyby-
ragt\ u vyhlqdi aproksymaci] Pade [3] abo lancghovyx drobiv [4]. Vperße
interpolqcijnyj lancghovyj drib buv rozhlqnutyj v 1909 roci T. N. Tile [5].
Formula zalyßkovoho çlena dlq takoho drobu bula vstanovlena Í. {. Mike-
ladze [6]. Uzahal\nennq rezul\tatu Mikeladze dlq interpolqcijnoho lancgho-
voho drobu, koly çastynni çysel\nyky i znamennyky mnohoçleny, bulo otrymano
v roboti [7]. V danij roboti ob©runtovugt\sq novi ocinky zalyßkovoho çlena
dlq interpolqcijnoho lancghovoho drobu Tile (teorema 2 ta teorema 3).
Interpolqciq funkcij lancghovym drobom. Formula zalyßkovoho çle-
na. Nexaj funkciq f ( x ) ∈ C(
n
+
1
)( [ α, β ] ) i zadana svo]my znaçennqmy v toçkax
mnoΩyny Λ = { xi : xi ∈ [ α, β ], i = 0, 1, … , n, xi ≠ xj pry i ≠ j }. Nexaj yi = f ( xi ),
i = 0, 1, … , n. NablyΩennq funkci] ßuka[mo u vyhlqdi funkcional\noho lan-
cghovoho drobu
b x
a x
b x
a x
b x
a x
b x
b x
a x
b x
n
n
k
n
k
k
0
1
1
2
2
0
1
( ) + ( )
( ) + ( )
( ) + … + ( )
( )
= ( ) + ( )
( )=
K ,
de an ( x ), bn ( x ) ∈ C [ α, β ], ak ( x ) � 0. Skinçennomu funkcional\nomu lancgho-
vomu drobu postavymo u vidpovidnist\ vidnoßennq dvox uzahal\nenyx mnoho-
çleniv
P x
Q x
b x
a x
b x
n
n k
n
k
k
( )
( )
= ( ) + ( )
( )=
0
1
K . (1)
Iz klasu funkcional\nyx lancghovyx drobiv (1) vydilymo pidklas interpolqcij-
nyx lancghovyx drobiv (ILD), tobto takyx, wo zadovol\nqgt\ spivvidnoßennq
P x
Q x
b x
a x
b x
yn i
n i
i
k
n
k i
k i
i
( )
( )
= ( ) + ( )
( )
=
=
0
1
K , i = 0, 1, … , n. (2)
Qkwo funkciq f ( x ) ∈ C(
n
+
1
)( [ α, β ] ), kanoniçni çysel\nyk Pn ( x ) ta znamennyk
Qn ( x ) lancghovoho drobu (1) [ mnohoçlenamy, deg Pn ( x ) ≤ n, to zalyßkovyj
çlen interpolqcijnoho lancghovoho drobu zada[t\sq formulog [7]
r x f x
P x
Q x
x x
n Q x
d
dx
f x Q xn
n
n
kk
n
n
n
n n x( ) = ( ) − ( )
( )
=
( − )
( + ) ( )
( ) ( )= +
+ =
∏
[ ]0
1
11 ! ζ , (3)
de ξ ∈ ( α, β ). Provedemo podal\ßi doslidΩennq ci[] formuly.
Kanoniçnyj znamennyk Qn ( x ) lancghovoho drobu (1) vyznaça[t\sq çerez ele-
© M. M. PAHIRQ, 2008
1548 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
OCINKA ZALYÍKOVOHO ÇLENA INTERPOLQCIJNOHO … 1549
menty c\oho drobu ai ( x ), bi ( x ) za dopomohog formuly Ojlera – Mindin©a [8, 9]
Q x B x X x X x X x X x X xn
n
i
i
n
i
i
n
j
j i
n
i
i
n
i
i i
n
( ) = ( )
+ ( ) + ( ) ( ) + ( ) ( )[ ]
=
−
=
−
= +
−
=
−
= +
−
∑ ∑ ∑ ∑ ∑1
1
1
1
3
2
1
1
5
2
3
1
1
1
2
2 1
×
× X x X x X x X xi
i i
n
i
i
n l
i
i i
n l
i
i i
n
l
l l
3
3 2
1
1
2
2 1 12
1
1
1 2
2
3 2
2
1
( ) + … + ( ) ( ) … ( )
= +
−
=
+ −
= +
+ −
= +
−
∑ ∑ ∑ ∑
−
, (4)
de l = [ n / 2 ], [ ⋅ ] — cila çastyna çysla, X x
a x
b x b xi
i
i i
( ) =
( )
( ) ( )
+
+
1
1
, B xn
1
[ ]( ) =
= b xii
n ( )=∏ 1
.
MoΩna pokazaty, wo kil\kist\ dodankiv v odynarnij sumi formuly (4) doriv-
ng[ n – 1, v podvijnij sumi — ( n – 3 ) ( n – 2 ) / 2!, v potrijnij sumi — ( n – 5 ) ( n –
– 4 ) ( n – 3 ) / 3!, … , v k-j sumi —
n k i
ki
k − +
=∏ 2
1 !
.
Kanoniçnyj znamennyk Qn ( x ) moΩe buty podanyj u vyhlqdi [9]
Q x B x R xn
n
k
n
k
l
( ) = ( ) ( )[ ] [ ]
=
∑1 1
0
, , (5)
de
R x X x X x X xk s
n
i
i s
n k
i
i i
n k
i
i i
n
k
k k
,
[ ]
=
+ −
= +
+ −
= +
−
( ) = ( ) ( ) … ( )∑ ∑ ∑
−
1
1
2
2 1 1
1 2
2
3 2
2
1
, k = 0, 1, … , l, (6)
R xt
n
0,
[ ]( ) = 1.
V svog çerhu, R xk s
n
,
[ ]( ) zadovol\nq[ rekurentne spivvidnoßennq
R x X x R xk s
n
i k i
n
i s
n k
, ,
[ ]
− +
[ ]
=
+ −
( ) = ( ) ( )∑ 1 2
1 2
. (7)
Zhidno iz formulog Lejbnica poxidno] m-ho porqdku dobutku dvox funkcij, z
(5) ma[mo, wo
( ) ( ) ( )( ) = ( ) ( )( ) [ ] ( − )
=
[ ] ( )
=
∑ ∑Q x C B x R xn
m
m
j n m j
j
m
k
n j
k
l
1
0
1
0
, ,
a todi
d
dx
f x Q x f x Q x C f x
n
n n
n
n n
m n m
m
n+
+
( + )
+
( + − )
=
+
[ ]( ) ( ) = ( ) ( ) + ( )∑
1
1
1
1
1
1
1
×
× C B x R xm
j n
j
m
m j
k
n
k
l
j( ) ( )[ ]
=
( − ) [ ]
=
( )∑ ∑( ) ( )1
0
1
0
, . (8)
Krim toho, iz (7) vyplyva[ nastupna rekurentna formula:
( ) ( )[ ] ( )
=
+ −
( )
− +
[ ]
=
( − )( ) = ( ) ( )∑ ∑R x C X x R xk
n m
j
n k
m
i
j
i
k j
n
i
m
m i
, ,1
1
1 2
1 2
0
. (9)
Ocinka zalyßkovoho çlena dlq ILD Tile. Rozhlqnemo ILD Tile [5, 7, 10]
P x
Q x
b
x x
b
n
n k
n k
k
( )
( )
= +
−
=
−
0
1
1K . (10)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1550 M. M. PAHIRQ
Koefici[nty bk , k = 0, 1, … , n, ILD (10) vyznaçagt\sq z umovy (2) nastupnym
çynom: abo obçyslg[t\sq poslidovnist\ obernenyx podilenyx riznyc\ Φk [ x0 ,
x1 , … , xk ], k = 0, 1, … , n, za formulog [4, 11]
Φk [ x0 , … , xk ] =
x x
x x x x x x
k k
k k k k k k
−
[ … ] − [ … ]
−
− − − − −
1
1 0 2 1 0 2 1Φ Φ, , , , , ,
, Φk [ x ] = f ( x ),
a todi bk = Φk [ x0 , … , xk ], k = 0, 1, … , n; abo za dopomohog rekurentnoho spivvid-
noßennq [9, 12]
b0 = y0
, bk =
x x
b
x x
b
x x
y b
k k
k
k
k
k
−
− + … + −
− + −
−
−
−
1
1
1
1
0
0
, k = 1, 2, … , n.
Teorema 1 ([4]). ILD Tile (10) [ drobovo-racional\nog funkci[g. Stepeni
mnohoçleniv kanoniçnoho çysel\nyka Pn ( x ) ta kanoniçnoho znamennyka Qn ( x )
zadovol\nqgt\ nerivnosti deg Pn ( x ) ≤ [ ( n + 1 ) / 2 ], deg Qn ( x ) ≤ [ n / 2 ].
Teorema 2. Nexaj funkciq f ( x ) ∈ C (
n
+
1
)( [ α, β ] ). Za znaçennqmy funkci]
f ( x ) v toçkax mnoΩyny Λ pobudovanyj ILD Tile (10). Todi dlq zalyßkovoho
çlena ILD Tile ma[ misce nerivnist\
f x
P x
Q x
x x
n Q x
f bn
n
kk
n
n
n
n n
n( ) − ( )
( )
≤
−
( + ) ( )
( )
( + + ) − ( − + )
+
=
+ +
+
∏ 0
1 1
11
1 1 4 1 1 4
2 1 4!
* * ρ ρ
ρ
+
+ C
b i
n m i jn
m
m
l
m
i
i
l m
j
m i
+
= =
−
=
+
∑ ∑ ∏( − ( + ) + )
1
1
2
0 1
1
2
* !
ρ
, (11)
de
d = β – α, b b
i n
i* min=
≤ ≤1
, ρ =
d
b*
2 ,
b b
i n
i
* max=
≤ ≤1
, f f x
i l x
n i* max max= ( )
≤ ≤ ≤ ≤
( + − )
0
1
α β
.
Dovedennq. U vypadku ILD Tile Xj ( x ) = ( x – xj ) / ( bj bj + 1 ). Oskil\ky Yj =
= ′ ( )X xj = 1 / ( bj bj + 1 ) dlq j = 1, 2, … , n – 1, i X xj
k( )( ) ≡ 0, koly k = 2, 3, … , n –
– 1, to formula (9) nabuva[ vyhlqdu
( ) ( ) ( )[ ] ( )
=
+ −
− +
[ ] ( )
− +
[ ] ( − )( ) = ( ) ( ) + ( )( )∑R x X x R x mY R xk
n m
j
n k
j k j
n m
j k j
n m
, , ,1
1
1 2
1 2 1 2
1
. (12)
Z (6) u vypadku ILD Tile ma[mo, wo
( )[ ] ( )( )R xk
n m
,1 = 0 pry k < m. (13)
Krim toho, zhidno iz teoremog 1 u c\omu vypadku deg Qn ( x ) ≤ l i B n
1
[ ]
ne zale-
Ωyt\ vid x, tomu formula (8) perepyßet\sq tak:
d
dx
f x Q x f x Q x
n
n n
n
n
+
+
( + )[ ]( ) ( ) = ( ) ( )
1
1
1 + B C f x R xn
n
m n m
m
l
k
n
k
l
m
1 1
1
1
1
0
[ ]
+
( + − )
=
[ ]
=
( )∑ ∑( ) ( )( ), .
(14)
Znajdemo poxidni ( )[ ] ( )( )R xk
n m
,1 pry k = m, m + 1, … , l. Koly k = m , z (12) z ura-
xuvannqm (13) otrymu[mo, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
OCINKA ZALYÍKOVOHO ÇLENA INTERPOLQCIJNOHO … 1551
( ) ( )[ ] ( )
=
+ −
− +
[ ] ( − )
=
+ −
= +
+ −
( ) = ( ) = ( − )∑ ∑ ∑R x mY R x mY m Ym
n m
i
n m
i m i
n m
i
n m
i
i i
n m
i, ,1
1
1 2
1 2
1
1
1 2
2
3 2
1
1
1
2 1
2
1 ×
× ( )− +
[ ] ( − )
=
+ −
= +
+ −
= +
−
( ) = … = …∑ ∑ ∑
−
R x m Y Y Ym i
n m
i
n m
i
i i
n m
i
i i
n
i
m m
m2 2
2
1
1 2
2
3 2
2
1
2
1
1
2 1
2
1
, ! .
Poznaçymo çerez
[ ] [ ]
=
+ −
= +
+ −
= +
−
= …∑ ∑ ∑
−
0
1
1 2
2
3 2
2
1
1
1
2 1
2
1
M Y Y Ym t
n
j
n m
j
j j
n m
j
j j
n
j
m m
m, , m = 1, 2, … , l, (15)
[ ] [ ] [ ]= =0
0 0 1M Rt
n
t
n
, , .
Lehko baçyty, wo ma[ misce spivvidnoßennq
[ ] [ ]
=
+ − [ ]
− +
[ ]= ∑0 1 2 0
1 2M Y Mm t
n
jj t
n m
m j
n
, , ,
a todi ( )[ ] ( ) [ ] [ ]( ) =R x m Mm
n m
m
n
, ,!1
0
1. Pry k = m + 1 z (12) ma[mo
( ) ( ) ( )+
[ ] ( )
=
− −
+
[ ] ( )
+
[ ] ( − )( ) = ( ) ( ) + ( )( )∑R x X x R x mY R xm
n m
i
n m
i m i
n m
i m i
n m
11
1
1 2
2 2
1
, , , =
=
i
n m
i m i
n
i m i
n mX x m M mY R x
=
− −
[ ]
+
[ ]
+
[ ] ( − )∑ ( ) + ( )( )( )
1
1 2
0
2 2
1! , , = …
… =
i
n m
i m i
n
i
i i
n m
i m i
nX x m M mY X x m M
1
1 1 1
2 1
2 2
1
1 2
0
2
2
1 2
0
1 21
=
− −
[ ]
+
[ ]
= +
+ −
[ ]
− +
[ ]∑ ∑
( ) +
( )( − )! !, , +
+ ( − )
( )( − ) + ( − )
…
= +
+ −
[ ]
− +
[ ]
= +
+ −
∑ ∑m Y X x m M m Yi
i i
n m
i m i
n
i
i i
n m
1 2 2
2
3 2
3 3 3
4 32
3 2
0
2 2
2
5 2
! !, +
+ 2
1
1 2
3
0
1 2 1 2Y X x M Y R xi
i i
n
i i
n
i i
n
m
m m
m m m m−
−= +
−
[ ]
+
[ ]
+
[ ]∑ ( ) + ( ) …( )
), , . (16)
Poznaçymo çerez
[ ]
+
[ ]
=
− −
[ ]
+
[ ]
= +
+ −
[ ]
− +
[ ]=
( ) +
( )∑ ∑1
1
1 2
0
2
2
1 2
0
1 2
1
1 1 1
2 1
2 2
M X x M Y X x Mm t
n
j t
n m
j m j
n
j
j j
n m
j m j
n
, , , +
+ Y Y X x M Y R xj
j j
n m
j
j j
n
j j
n
j j
n
m
m m
m m m m2
3 2
1
12
3 2
2
3
0
1 2 1 2
= +
+ −
= +
−
[ ]
+
[ ]
+
[ ]∑ ∑
… + ( ) + ( )( ) …
−
−
, , .
Lehko baçyty, wo
[ ]
+
[ ]
=
− −
[ ]
+
[ ] [ ]
+
[ ]( ) = + ( )( )∑1
1
1 2
0
2
0
2M x X M Y M xm t
n
j t
n m
j m j
n
j m j
n
, , , ,
[ ] [ ] [ ]( ) = ( )1
1 1M x R xt
n
t
n
, , , (17)
a todi ( )+
[ ] ( )( )R xm
n m
1 1, = m M xm
n! ,
[ ]
+
[ ] ( )1
1 1 .
Skorystavßys\ metodom povno] matematyçno] indukci], z formuly (12) moΩ-
na pokazaty, wo
( )+
[ ] ( )( )R xm s
n m
,1 = m M xs
m s
n! ,
[ ]
+
[ ] ( )1 , s = 1, 2, … , l – m,
de
[ ]
+
[ ]
=
+ − ( + )
[ − ]
+ − +
[ ] [ ]
+ − +
[ ]( ) = ( ) + ( )( )∑s
m s t
n
j t
n m s
j
s
m s j
n
j
s
m s j
nM x X M x Y M x, , ,
1 2
1
1 2 1 2 , (18)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1552 M. M. PAHIRQ
[ ] [ ] [ ]( ) = ( )s
s t
n
s t
nM x R x, , .
Formula (14) nabuva[ vyhlqdu
d
dx
f x Q x f x Q x
n
n n
n
n
+
+
( + )[ ]( ) ( ) = ( ) ( )
1
1
1 +
+ B C f x m M xn
n
m n m
m
l
k m
l
k m
k
n
1 1
1
1
1
[ ]
+
( + − )
= =
[ − ] [ ]∑ ∑( ) ( )! , . (19)
Todi
d
dx
f x Q x f Q x b C m M x
n
n n n
n
n
m
m
l
k m
l
k m
k
n
+
+ +
= =
[ − ] [ ][ ]( ) ( ) ≤ ( ) + ( ) ( )
∑ ∑
1
1 1
1
1
* *
,! . (20)
Zhidno z teoremog 1 [10]
Q x bn
n
n n
n( ) ≤ ( )
( + + ) − ( − + )
+
+ +
+
* 1 1 4 1 1 4
2 1 4
1 1
1
ρ ρ
ρ
. (21)
Znajdemo ocinky
[ − ] [ ]( )k m
k
nM x,1 pry k = m, m + 1, … , l. Pry k = m z (15) ma[mo
[ ]
+
[ ]
=
( ) ≤ ( − − − + )∏0
2 2
1
1
1 2M x
m b
n i m jm i
n
m
j
m
,
*!
.
Pry k = m + 1 iz formuly (17) otrymu[mo
[ ]
+ +
[ ]
= +
− −
[ ]
+
[ ] [ ]
+
[ ]( ) ≤ ( ) ( ) + ( )( )∑1
1 2
2
1 2
0
2
1
2M x X x M x Y M xm i
n
j i
n m
j m j
n
j m j
n
, , , . (22)
Koly m = 0, to z (22) ma[mo
[ ]
+
[ ]
+
[ ]( ) = ( ) ≤ ( − − )1
1 2 1 2 2 2M x R x
d
b
n ii
n
i
n
, ,
*
.
Pry m = 1 z (22) vyplyva[, wo
[ ]
+
[ ]
= +
−
[ ]
+
[ ] [ ]
+
[ ]( ) ≤ ( ) + ( )( )∑1
2 2
2
3
0
1 2
1
1 2M x X M x Y M xi
n
j i
n
j j
n
j j
n
, , , ≤
≤
2
2 4 34
2
3
4
d
b
n j
d
b
n i n i
j i
n
* *= +
−
∑ ( − − ) = ( − − )( − − ).
Za dopomohog metodu povno] matematyçno] indukci] dovedemo, wo
[ ]
+
[ ]
=
( ) ≤
( − )
( − − + − )∏1
2 2
1
1
1
2 1M x
k b
n i k jk i
n
k
j
m
,
*!
, k = 1, 2, … , l. (23)
U vypadku k = 1, 2 formula (23) vykonu[t\sq. Zrobymo prypuwennq, wo vona
ma[ misce pry k = t. Todi pry k = t + 1 z (22) otrymu[mo
[ ]
+ +
[ ]
= +
− −
[ ]
+
[ ] [ ]
+
[ ]( ) ≤ ( ) + ( )( )∑1
1 2
2
1 2
0
2
1
2M x X M x Y M xt i
n
j i
n t
j t j
n
j t j
n
, , , ≤
≤
d
b t b
n j t l
b
d
t b
n j t lt
l
t
t
l
t
j i
n t
* * * *! !2 2
1
2 2
12
1 2
1
2 1
1
1
2 1( − − + − ) +
( − )
( − − + − )
= == +
− −
∏ ∏∑ =
=
d
t b
n i t jt
j
t
!
*
2 1
1
1
2 1 1( + )
=
+
( − − ( + ) + − )∏ ,
tobto formula (23) ma[ misce i v c\omu vypadku.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
OCINKA ZALYÍKOVOHO ÇLENA INTERPOLQCIJNOHO … 1553
Za dopomohog metodu povno] matematyçno] indukci] dovedemo, wo
[ ]
+ +
[ ]
( + )
=
+
( ) ≤ ( − − ( + ) + − )∏s
m s i
n
s
m s
l
m s
M x
d
m s b
n i m s l,
*
! !2 2
1
2 1 , s = 0, 1, … , l – m.
(24)
Koly s = 0, 1, to formula (24) vykonu[t\sq. Zrobymo prypuwennq, wo dana
formula vykonu[t\sq pry s = k. Todi pry s = k + 1 z (18) otrymu[mo, wo
[ + ]
+ + +
[ ]
= +
− − ( + )
[ ]
+ +
[ ] [ + ]
+ +
[ ]( ) ≤ ( ) + ( )( )∑k
m k i
n
j i
n m k
j
k
m k j
n
j
k
m k j
nM x X M x Y M x1
1 2
2
1 2
2
1
2, , , .
(25)
Koly m = 0, to z (18) otrymu[mo
[ + ]
+ +
[ ]
+ +
[ ]
+
( + )
=
+
( ) ≤ ( ) ≤
( + )
( − − ( + ) + − )∏k
k i
n
k i
n
k
k
j
k
M x R x
d
k b
n i k j1
1 2 1 2
1
2 1
1
1
1
2 1 1, ,
*
!
.
Pry m = 1 z (25) ma[mo, wo
[ + ]
+ +
[ ]
= +
− −
[ ]
+ +
[ ] [ + ]
+ +
[ ]( ) ≤ ( ) + ( )( )∑k
k i
n
j i
n k
j
k
k i
n
j
k
k i
nM x X M x Y M x1
2 2
2
3 2
1 2
1
1 2, , , ≤
≤
j i
n k k
k
l
kd
b
d
k b
n j k l
= +
− −
( + )
=
+
∑ ∏
( − − ( + ) + − )
2
3 2
2 2 1
1
1
2 1 1
* *
!
+
+
1
1
2 1 12
1
2 1
1
1
b
d
k b
n i k l
k
k
l
k
* *!
+
( + )
=
+
( + )
( − − ( + ) + − )
∏ =
=
d
k b
n i k j
k
k
j
k+
( + )
=
+
( + )
( − − ( + ) + − )∏
1
2 2
1
2
1
2 2 1
! *
.
Koly m = 2, to
[ + ]
+ +
[ ]
= +
− −
[ ]
+ +
[ ] [ + ]
+ +
[ ]( ) ≤ ( ) + ( )( )∑k
k i
n
j i
n k
j
k
k i
n
j
k
k i
nM x X M x Y M x1
3 2
2
5 2
2 2
1
2 2, , , ≤
≤
j i
n k k
k
l
k
d
b
d
k b
n j k l
= +
− −
( + )
=
+
∑ ∏
( − − ( + ) + − )
2
5 2
2 2 2
1
2
2
2 2 1
* *! !
+
+
1
1
2 2 12
1
2 1
1
2
b
d
k b
n i k l
k
k
l
k
* *!
+
( + )
=
+
( + )
( − − ( + ) + − )
∏ =
=
d
k b
n i k j
k
k
j
k+
( + )
=
+
( + )
( − − ( + ) + − )∏
1
2 3
1
3
1 2
2 3 1
! !
*
.
Zrobymo prypuwennq, wo (24) vykonu[t\sq pry m = t. Todi pry m = t + 1 z (25)
ma[mo
[ + ]
+ + +
[ ]
= +
− − ( + )
[ ]
+ + +
[ ] [ + ]
+ + +
[ ]( ) ≤ ( ) + ( )( )∑k
k t i
n
j i
n k t
j
k
k t j
n
j
k
k t j
nM x X M x Y M x1
2 2
2
3 2
1 2
1
1 2, , , ≤
≤
j i
n k t k
k t
l
k td
b
d
k t b
n j k t l
= +
− − ( + )
( + + )
=
+ +
∑ ∏
( + )
( − − ( + + ) + − )
2
3 2
2 2 1
1
1
1
2 1 1
* *
! !
+
+
1
1
2 1 12
1
2 1
1
1
b
d
k t b
n i k t l
k
k t
l
k t
* *
! !
+
( + + )
=
+ +
( + )
( − − ( + + ) + − )
∏ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1554 M. M. PAHIRQ
=
d
k t b
n i k t j
k
k t
j
k t+
( + + )
=
+ +
( + ) ( + )
( − − ( + + ) + − )∏
1
2 2
1
2
1 1
2 2 1
! !
*
.
Formula (24) virna i v c\omu vypadku, otΩe vona virna pry dovil\nyx s ta m. Iz
(24) vyplyva[, wo
[ ]
+
[ ]
( + )
=
+
( ) ≤ ( − ( + ) + )∏s
m s
n
s
m s
l
m s
M x
d
m s b
n m s l,
*
! !1 2
1
2 . (26)
Z (3), (20), (21) ta (26) otrymu[mo (11).
Teorema 3. Qkwo funkciq f ( x ) ∈ C (
n
+
1
)( [ α, β ] ), za znaçennqmy funkci] v
toçkax mnoΩyny Λ pobudovanyj ILD Tile (10), çastynni çysel\nyky ta zna-
mennyky qkoho zadovol\nqgt\ umovu Sl[ßyns\koho – Prin©shejma, tobto 0 <
< x xi− −1 ≤ d, bi ≥ d + 1, i = 1, 2, … , n, d ≠ 1, to ma[ misce ocinka
f x
P x
Q x
d d
n d
f bn
n
n n
n
n n
n( ) − ( )
( )
≤ ( − )
( + ) ( − )
( )
( + + ) − ( − + )
+
+ + + +
+
1 1 1 1
1
1
1 1
1 1 4 1 1 4
2 1 4!
* * ρ ρ
ρ
+
+ C
b i
n m i jn
m
m
l
m
i
i
l m
j
m i
+
= =
−
=
+
∑ ∑ ∏( − ( + ) + )
1
1
2
0 1
1
2
* !
ρ
, (27)
de d, ρ, b*
, b*, f * vyznaçeni v umovi teoremy 2.
Dovedennq. Zhidno z teoremog 2 iz [10], pry vykonanni vkazanyx umov vyko-
nu[t\sq nerivnist\
Q x
d
dn
n
( ) ≥ −
−
+1 1
1
.
Todi, iz umov dano] teoremy ta teoremy 2 otrymu[mo nerivnist\ (27).
1. Havrylgk I. P., Makarov V. L. Metody obçyslen\: U 2 ç. – Ky]v: Vywa ßk., 1995. – Ç. 1. –
367 s.
2. Pryvalov A.A. Teoryq ynterpolyrovanyq funkcyj: V 2-x kn. – Saratov: Yzd-vo Sarat. un-
ta, 1990. – 423 s.
3. Bejker, ml. DΩ., Hrejvs-Morrys P. Approksymacyy Pade. – M.: Myr, 1986. – 502 s.
4. Skorobahat\ko V. Q. Teoryq vetvqwyxsq cepn¥x drobej y ee prymenenye v v¥çyslytel\noj
matematyke. – M.: Nauka, 1983. – 312 s.
5. Thiele T. N. Interpolationsprechnung. – Leipzig: Commisission von B. G. Teubner, 1909. – XII +
175 S.
6. Mykeladze Í. E. Çyslenn¥e metod¥ matematyçeskoho analyza. – M.: HYTTL, 1953. – 527 s.
7. Pahirq M. M. Pro efektyvnist\ nablyΩennq funkcij deqkymy typamy interpolqcijnyx
lancghovyx drobiv // Mat. metody ta fiz.-mex. polq. – 2003. – 46, # 4. – S. 57 – 64.
8. Perron O. Lehre von den Kettenbrüchen. Band I. – Stuttgart: Teubner, 1954. – 194 S.
9. Pahirq M. M. Interpolqciq funkcij lancghovym drobom ta hillqstym lancghovym drobom
special\noho vydu // Nauk. visnyk UΩhorod. un-tu. Ser. mat. – 1994. – Vyp. I. – S. 72 – 79.
10. Pahirq M. M. Zadaça interpolqci] funkcij lancghovymy drobamy // Nauk. visnyk UΩhorod.
un-tu. Ser. matem. i inform. – 2005. – Vyp. 10 – 11. – S. 77 – 87.
11. Hildebrand F. B. Introduction to numerical analysis. 2nd ed. – New York: Dover Publications,
Inc, 1987. – 669 p.
12. Pahirq M. M. Interpolqciq funkcij laqcghovym drobom Tile // Zbirnyk nauk. prac\ z
obçyslgval\no] matematyky. – UΩhorod, 1997. – S. 21 – 26.
OderΩano 18.05.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
|
| id | umjimathkievua-article-3268 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:39:19Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b5/cefed328003324f152d538b8dad55ab5.pdf |
| spelling | umjimathkievua-article-32682020-03-18T19:49:31Z Evaluation of the remainder term for the Thiele interpolation continued fraction Оцінка залишкового члена інтерполяційного ланцюгового дробу Тіле Pahirya, M. M. Пагіря, М. М. We present an estimate of the remainder term for the Thiele interpolation continued fraction. Получена оценка остаточного члена для интерполяционной цепной дроби Тиле. Institute of Mathematics, NAS of Ukraine 2008-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3268 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 11 (2008); 1548–1554 Український математичний журнал; Том 60 № 11 (2008); 1548–1554 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3268/3282 https://umj.imath.kiev.ua/index.php/umj/article/view/3268/3283 Copyright (c) 2008 Pahirya M. M. |
| spellingShingle | Pahirya, M. M. Пагіря, М. М. Evaluation of the remainder term for the Thiele interpolation continued fraction |
| title | Evaluation of the remainder term for the Thiele interpolation continued fraction |
| title_alt | Оцінка залишкового члена інтерполяційного ланцюгового дробу Тіле |
| title_full | Evaluation of the remainder term for the Thiele interpolation continued fraction |
| title_fullStr | Evaluation of the remainder term for the Thiele interpolation continued fraction |
| title_full_unstemmed | Evaluation of the remainder term for the Thiele interpolation continued fraction |
| title_short | Evaluation of the remainder term for the Thiele interpolation continued fraction |
| title_sort | evaluation of the remainder term for the thiele interpolation continued fraction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3268 |
| work_keys_str_mv | AT pahiryamm evaluationoftheremaindertermforthethieleinterpolationcontinuedfraction AT pagírâmm evaluationoftheremaindertermforthethieleinterpolationcontinuedfraction AT pahiryamm ocínkazališkovogočlenaínterpolâcíjnogolancûgovogodrobutíle AT pagírâmm ocínkazališkovogočlenaínterpolâcíjnogolancûgovogodrobutíle |