Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem
We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of th...
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| Дата: | 2008 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3175 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509220944740352 |
|---|---|
| author | Kashpirovskii, A. I. Torba, S. M. Кашпіровський, О. І. Торба, С. М. |
| author_facet | Kashpirovskii, A. I. Torba, S. M. Кашпіровський, О. І. Торба, С. М. |
| author_sort | Kashpirovskii, A. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:27Z |
| description | We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of the operator A is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse theorems of the theory of approximation in the mean. |
| first_indexed | 2026-03-24T02:37:39Z |
| format | Article |
| fulltext |
UDK 517.9
S. M. Torba (In-t matematyky NAN Ukra]ny, Ky]v),
O. I. Kaßpirovs\kyj (Nac. un-t „Ky[vo-Mohylqns\ka akademiq”)
XARAKTERYZACIQ ÍVYDKOSTI ZBIÛNOSTI ODNOHO
NABLYÛENOHO METODU ROZV’QZUVANNQ
ABSTRAKTNO} ZADAÇI KOÍI*
The method of approximate solution based on the exponent decomposition into orthogonal Lager
polynomials is considered for the Cauchy problem for an operator differential equation. It is proved that
the belonging of an initial value to some space of smooth elements of the operator A is equivalent to the
convergence of some weighted sum of integral residuals. As a corollary, direct and inverse theorems of
the theory of approximation in the mean are obtained.
Rassmotren pryblyΩenn¥j metod reßenyq zadaçy Koßy dlq dyfferencyal\no-operatornoho
uravnenyq, osnovann¥j na razloΩenyy πksponent¥ po ortohonal\n¥m mnohoçlenam Laherra.
Dokazano, çto prynadleΩnost\ naçal\noho znaçenyq opredelennomu prostranstvu hladkyx πle-
mentov operatora A πkvyvalentna sxodymosty nekotoroj vzveßennoj summ¥ yntehral\n¥x ne-
vqzok. Kak sledstvye, poluçen¥ prqm¥e y obratn¥e teorem¥ teoryy pryblyΩenyq v srednem.
1. Vstup. U separabel\nomu hil\bertovomu prostori � rozhlqda[t\sq zadaça
Koßi dlq operatorno-dyferencial\noho rivnqnnq perßoho porqdku
′x + Ax = f t( ), x( )0 = x0 , (1)
de x = x t( ) ta f t( ) — vidpovidno nevidoma ta vidoma �-znaçni funkci], 0 ≤ t <
< ∞, A — samosprqΩenyj dodatnyj operator v �. Napryklad, qkwo � =
= L2(– , )π π , A — samosprqΩenyj operator, porodΩenyj dyferencial\nym vy-
razom – d
dt
2
2 z periodyçnymy krajovymy umovamy u(– )π = u( )π , ′u (– )π = ′u ( )π ,
to rivnqnnq (1) perexodyt\ u vidome rivnqnnq teploprovidnosti
∂
∂
u
t
– ∂
∂
2
2
u
x
= f x t( , ). (2)
Vyvçennq rivnqnnq (1) da[ zmohu z [dyno] toçky zoru doslidΩuvaty abstra-
ktni rivnqnnq paraboliçnoho typu, qki [ uzahal\nennqm rivnqnnq (2).
Qk vidomo [1], rozv’qzok zadaçi Koßi dopuska[ zobraΩennq
x t( ) = U t x( ) 0 + U t s f s ds
t
( – ) ( )
0
∫ , (3)
de
U t( ) = e At– = e dEt– λ
λ
0
∞
∫ (4)
— pivhrupa operatoriv, porodΩena operatorom A, Eλ — spektral\nyj rozklad
operatora A. Oskil\ky spektral\nyj rozklad Eλ , qk pravylo, nevidomyj, to
pryrodno vynyka[ zadaça pobudovy nablyΩen\ pivhrupy U t( ) , a otΩe, i roz-
v’qzkiv rivnqnnq (1).
Odyn iz pidxodiv do nablyΩen\ operatorno] eksponenty polqha[ u rozkladi
funkci] (dyv. [2, c. 225] )
e at– =
1
1 11
0( )
( , )
a
a
a
L t
n
n
n+ +
+
=
∞
∑α α , a > −1 2 , (5)
*
Çastkovo pidtrymano DerΩavnym fondom fundamental\nyx doslidΩen\ Ukra]ny (proekt
#A14.1/003).
© S. M. TORBA, O. I. KAÍPIROVS|KYJ, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4 557
558 S. M. TORBA, O. I. KAÍPIROVS|KYJ
u rqd za ortohonal\nymy mnohoçlenamy Laherra
L tn( , )α = 1
n
t e t et n t n
!
– – ( )α α +( )
i formal\nij zamini u vyrazi (5) parametra a çy t na operator A. Çastkovi su-
my otrymanoho formal\noho vyrazu doslidΩugt\sq qk nablyΩennq operator-
no] eksponenty.
V roboti M. L. Horbaçuka i V. V. Horodec\koho [3] dlq nablyΩennq pivhrupy
U t( ) u vyrazi (5) operatorom A zamineno parametr t. V qkosti nablyΩen\ roz-
hlqdagt\sq çastkovi sumy
P t AN ( , ) =
1
1 1
0
0t
t
t
L A
k
N n
n+ +
=
∑ ( , ) .
Pry c\omu poslidovnist\ P t AN ( , ) zbiha[t\sq do U t( ) ne dlq vsix vektoriv
f ∈� , a til\ky dlq deqkyx C∞
-vektoriv operatora A. Dlq takyx vektoriv me-
tod nablyΩennq ma[ eksponencial\nu ßvydkist\ zbiΩnosti. Sutt[vyj nedolik
takyx nablyΩen\ pov'qzanyj z vykorystannqm stepeniv vid neobmeΩenoho ope-
ratora A, wo pryvodyt\ do nestijkosti vidpovidnyx çysel\nyx metodiv.
U robotax D. Z. Arova, I. P. Havrylgka, V. L. Makarova, V. B. Vasylyka ta
V.AL. Rqbiçeva [4 – 6] ta v monohrafi] [7] v qkosti nablyΩennq rozhlqda[t\sq
x tN ( ) = e L t T I T xt k
k
N
k
k– (– ) ( , ) ( )γ
γ γγ1 2 0
0
0
=
∑ + , (6)
de γ — dovil\na dodatna stala, Tγ — drobovo-racional\na funkciq vid opera-
tora A,
Tγ = T A( , )γ =
γ
γ
– A
A+
, γ > 0,
qka nazyva[t\sq peretvorennqm Keli operatora A, abo koheneratorom hrupy
U t( ) . NablyΩennq (6) [ polinomial\nym vidnosno t ta Tγ i moΩe buty otryma-
ne z modyfikaci] vyrazu (5) zaminog parametra a na A. NablyΩennq (6) zbiha-
[t\sq dlq vsix vektoriv x ∈� i dlq x0 ∈ D( )Aσ
, σ > 0, ma[ stepenevyj xarak-
ter ßvydkosti prqmuvannq do nulq vidxylu [7]: rivnomirno dlq t ∈ ( , )0 ∞
x tN ( ) – x t( ) ≤ cN –σ ta na dovil\nomu promiΩku p q,[ ] � ( , )0 ∞ x tN ( ) –
– x t( ) ≤ c Np q,
– –σ 1 4
.
U roboti [8] zaproponovano inßyj pidxid do nablyΩennq pivhrupy U t( ) =
= e t A–
, A ≥ I, v qkomu vykorystovu[t\sq rozvynennq funkci] et λ
, λ ∈( ]0 1, , v
rqd Fur’[ – Çebyßova
e
t–
λ = a t Tn n
n
( ) ( )∗
=
∞
∑
0
λ , (7)
de Tn
∗( )λ — zmiweni polinomy Çebyßova [9]. NablyΩennq otrymu[t\sq for-
mal\nog pidstanovkog A–1
zamist\ λ v çastkovi sumy rqdu (7). Pry t > 0 dlq
vidpovidnyx rozv’qzkiv odnoridnoho rivnqnnq vstanovleno eksponencial\nu ocin-
ku
y t y tN( ) – ( ) ≤ C tN xexp −
δ 23
0 .
Rozklad za zmiwenymy polinomamy Qkobi dlq x0 ∈ D( )Aσ
, σ > 0, dozvolyv po-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
XARAKTERYZACIQ ÍVYDKOSTI ZBIÛNOSTI ODNOHO NABLYÛENOHO METODU … 559
krawyty ostanng ocinku pry malyx t [10]:
y t y tN( ) – ( ) ≤ CN N tN A x– – ln exp2 1 2 23
0
σ σδ−
.
Xoça ostanni dvi ocinky [ krawymy, niΩ ocinky dlq nablyΩen\ (6), dlq nyx
xarakternym [ nasyçennq dlq malyx t > 0. Vodnoças nablyΩennq (6) [ nenasy-
çenym.
U roboti [11] rozhlqnuto pidxid, podibnyj do (6), i vstanovleno prqmi ta
oberneni teoremy dlq zv’qzku ßvydkosti prqmuvannq do nulq intehral\noho vid-
xylu nablyΩenoho rozv’qzku z hladkistg poçatkovoho vektora x0. Meta dano]
roboty polqha[ u vstanovlenni kryterig zv’qzkiv miΩ ßvydkistg prqmuvannq do
nulq intehral\noho vidxylu nablyΩenoho rozv’qzku ta hladkistg poçatkovoho
vektora, zokrema — xarakteryzaci] naleΩnosti x0 do D( )Aσ
, σ > 0, u terminax
ßvydkosti prqmuvannq do nulq intehral\noho vidxylu nablyΩennq e xt A−
0 .
2. Osnovni rezul\taty. Nexaj A — samosprqΩenyj dodatno vyznaçenyj
operator, wo di[ v hil\bertovomu prostori �. Dlq operatora A rozhlqda[t\sq
odnoridna zadaça Koßi
′x t( ) + Ax t( ) = 0, t > 0, x x( )0 0= , (8)
rozv’qzok qko] [1] ma[ vyhlqd
x t( ) = e xt A−
0 .
Qk ce bulo zrobleno v [11], za dopomohog rozkladu funkci] e at−
v rqd za
mnohoçlenamy Laherra [2, c. 225] zapysu[t\sq formal\nyj vyraz dlq operator-
no] eksponenty
e At− = A A I L tn n
n
n
( ) ( , )–( )+ +
=
∞
∑ 1
0
0 , (9)
wo di[ v �. Nahada[mo, wo rqd (9) pry t > 0 zbiha[t\sq dlq vsix x ∈� , a pry
t = 0 dostatn\og umovog zbiΩnosti [ x ∈ D( )Aσ
, σ > 0. Rozhlqnemo toçnyj
rozv’qzok
x t( ) = A A I L t xn n
n
n
( ) ( , )–( )+ +
=
∞
∑ 1
0
00 (10)
i v qkosti nablyΩennq vykorysta[mo çastkovu sumu rqdu (10)
x tN ( ) = A A I L t xn n
n
n
N
( ) ( , )–( )+ +
=
∑ 1
0
00 . (11)
Vyraz (11) moΩna interpretuvaty tak: rozhlqda[t\sq poslidovnist\ stacionar-
nyx zadaç
( )A I yp+ +1 = Ayp , p = 0, 1, … , y0 = ( )–A I x+ 1
0 ,
zavdqky çomu nestacionarne rivnqnnq (8) „dyskretyzu[t\sq” i zvodyt\sq do
poslidovnosti stacionarnyx rivnqn\ z nezminnymy pravog ta livog çastynamy, i
po cyx rivnqnnqx vypysu[t\sq nablyΩenyj rozv’qzok
x tN ( ) = L t yn
n
N
n( , )0
0=
∑ .
Mnohoçleny Laherra L tn( , )0 [ ortonormovanymy na ( , )0 ∞ z vahog e t–
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
560 S. M. TORBA, O. I. KAÍPIROVS|KYJ
tomu ma[ sens rozhlqnuty intehral\nyj vidxyl nablyΩenoho rozv’qzku
zN
2 = x t x t e dtN
t( ) – ( ) –2
0
∞
∫ . (12)
U roboti [11] otrymano prqmi ta oberneni teoremy dlq ßvydkosti prqmuvan-
nq zN
2
do nulq v zaleΩnosti vid hladkosti x0. My pokrawymo ci rezul\taty,
zokrema kryterij naleΩnosti x0 do D( )Aσ
.
Rezul\taty roboty [11] pokazugt\, wo z eksponencial\no] ßvydkosti prqmu-
vannq do nulq zN
2
vyplyva[ naleΩnist\ x0 do mnoΩyny cilyx vektoriv ekspo-
nencial\noho typu. Rozhlqnemo subeksponencial\nyj vypadok.
Nexaj zadano monotonno zrostagçu poslidovnist\ dodatnyx dijsnyx çysel
mn n{ } ∈N
taku, wo
lim
n
n
n
m
e→ ∞ δ = 0 ∀ >δ 0.
Dlq xarakteryzaci] vektoriv x0, dlq qkyx zn
2 = o
mn
1
, rozhlqnemo dvi funk-
ci]. Perßa vyznaça[t\sq za dopomohog Z-peretvorennq [12]
G1
2( )λ =
1
2 1 1
2
λ
λ
λ+
{ }
+
Z mn = 1
2 1 1
2
1λ
λ
λ+ +
⋅
=
∞
∑
n
n
n
m . (13)
Dlq vyznaçennq druho] rozhlqnemo deqku funkcig g( )ξ , qka [ nepererv-
nog, monotonno zrosta[ i g( )ξ = mn pry ξ = n, n ∈N . Nexaj
G2
2( )λ =
1
2 1
2
1
λ
λ
λ+
{ } +
L g ln =
1
2 1 1
0
2
λ
ξ λ
λ
ξ
ξ
+ +
∞
∫ g d( ) , (14)
de L g{ } — peretvorennq Laplasa vid funkci] g( )ξ .
Todi spravedlyvog [ taka teorema.
Teorema 1. x0 ∈ AD G A( )( ) , de G( )⋅ — odna z funkcij (13), (14), todi i til\-
ky todi, koly rqd
z mn n
n
2
1
⋅
=
∞
∑ (15)
zbiha[t\sq.
Dovedennq. Vidmitymo, wo z (13), (14), monotonnosti poslidovnosti mn{ }
ta funkci] g( )ξ vyplyva[, wo
λ
λ
λ
+
1
2
1
2G ( ) ≤ G2
2( )λ ≤
λ
λ
λ+
1 2
1
2G ( ). (16)
Pry 0 ≤ λ ≤ 1 funkci] G1
2( )λ ta G2
2( )λ [ obmeΩenymy, a pry λ ≥ 1 z (16) vy-
plyva[
1
4 1
2G ( )λ ≤ G2
2( )λ ≤ 4 1
2G ( )λ , otΩe, naleΩnist\ x0 do D G A1( )( ) ekviva-
lentna naleΩnosti x0 do D G A2( )( ) .
Nexaj x0 ∈ A D G A1( )( ). Todi zavdqky ortonormovanosti mnohoçleniv L tn( , )0
z vahog e t–
ma[ misce zobraΩennq (dyv. [11])
zN
2 =
0
2 2
1
2 0 01
1 1
2 1
∞ +
∫ +
+
λ
λ λ λ λ
N
G
d E y y
( )
( , ), (17)
de y0 = G A x1 0( ) . Poznaçymo pidintehral\nyj vyraz çerez φ λN ( ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
XARAKTERYZACIQ ÍVYDKOSTI ZBIÛNOSTI ODNOHO NABLYÛENOHO METODU … 561
PokaΩemo obmeΩenist\ çastkovyx sum rqdu (15). Dlq c\oho rozhlqnemo
z mn n
n
N
2
1
⋅
=
∑ = m d E y yn
n
N
n
=
∞
∑ ∫
1
0 0
0
φ λ λ( ) ( , ) =
=
λ
λ λ λ λ+
+=
∞ +
∑∫ 1 2 1
1
10
2 2
1
2 0 0
n
N n
nm
G
d E y y
( )
( , ). (18)
Ale zavdqky (13)
λ
λ λ+
+=
+
∑ 1 2 11
2 2
n
N n
nm
≤ G1
2( )λ ,
otΩe, (18) zvodyt\sq do vyrazu
n
N
n nz m
=
∑ ⋅
1
2 ≤
0
0 0
∞
∫ d E y y( , )λ = y0
2
.
Navpaky, nexaj zbiha[t\sq rqd (15). Qk i pry dovedenni teoremyA3 v [11], dlq
vstanovlennq vkladennq x0 ∈ AD G A1( )( ) dosyt\ pobuduvaty taku poslidovnist\
funkcij φ λn n( ){ } =
∞
1 , wo:
1) φ λn( ) monotonno zrostagt\ pry n → ∞;
2)
0 0 0
∞
∫ φ λ λn d E x x( ) ( , ) ≤ c ∀ ∈n N ;
3) isnu[ stala ′ >c 0 taka, wo ′c G1
2( )λ ≤ lim ( )
n
n→∞
φ λ , λ ≥ λ0 > 1.
Pislq c\oho, vykorystovugçy lemu Fatu, peresvidçu[mosq, wo funkciq
φ λ( ) = lim ( )
n
n→∞
φ λ
[ intehrovnog. Za teoremog Lebeha intehrovnog vidnosno miry d E x x( , )λ 0 0 ta-
koΩ bude funkciq ′c G1( )λ , a intehrovnist\ ostann\o] ekvivalentna naleΩnosti
x0 do D G A1( )( ).
Dlq pobudovy poslidovnosti φ λn n( ){ } =
∞
1 rozhlqnemo çastkovu sumu rq-
duA(15). Analohiçno (18) ma[mo
n
N
n nz m
=
∑ ⋅
1
2 =
m
d E x xn
n
N n
2 1 110
2 2
0 0λ
λ
λ λ+ +
=
∞ +
∑∫ ( , ) . (19)
Poznaçymo pidintehral\nyj vyraz çerez φ λN ( ) i pokaΩemo, wo poslidovnist\
φ λn n( ){ } =
∞
1 [ ßukanog. Oçevydno, wo φ λn( ) monotonno zrostagt\ pry n →
→ ∞. Zi zbiΩnosti rqdu (15) vyplyva[, wo
∃ >c 0: φ λ λn d E x x( ) ( , )0 0
0
∞
∫ ≤ c ∀ ∈n N .
Zalyßa[t\sq pokazaty, wo dlq φ λN ( ) vykonu[t\sq j tretq umova. A pry λ ≥
≥ 1 vona vyplyva[ z (19), tomu wo
∀ ≥λ 1 ∃ Nλ ∀ >N Nλ : φ λN ( ) =
mn
n
N n
2 1 11
2 2
λ
λ
λ+ +
=
+
∑ ≥
≥
1
4 2 1 11
2
mn
n
N n
λ
λ
λ+ +
=
∑ ≥ 1
8 2 1 11
2
mn
n
n
λ
λ
λ+ +
=
∞
∑ =
G1
2
8
( )λ
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
562 S. M. TORBA, O. I. KAÍPIROVS|KYJ
Teoremu dovedeno.
Dovedena teorema za formog [ analohom teoremy pro uzahal\neni hranyçni
znaçennq zadaçi (8) z monohrafi] V. I. Horbaçuk, M. L. Horbaçuka [13, c. 84].
Z teoremyA1 vyplyva[, wo nablyΩennq (11) [ nenasyçenymy, oskil\ky toç-
nist\ nablyΩennq pokrawu[t\sq v zaleΩnosti vid hladkosti poçatkovoho zna-
çennq x0.
PokaΩemo, qk za dopomohog teoremyA1 moΩna otrymaty xarakteryzacig na-
leΩnosti vektoriv x0 do D( )Aσ
u terminax ßvydkosti prqmuvannq do nulq
zN
2
. Dlq c\oho rozhlqnemo mn = n
2σ
i, vidpovidno, g( )ξ = ξ σ2
.
Ma[ misce taka teorema.
Teorema 2. x0 ∈A D( )Aσ
, σ > 0, todi i til\ky todi, koly rqd
z nn
n
2
1
2⋅
=
∞
∑ σ
(20)
zbiha[t\sq.
Dovedennq. Z teoremyA1 vyplyva[, wo dosyt\ dovesty ekvivalentnist\ na-
leΩnosti x0 do D( )Aσ
ta x0 do D G A2( )( ) , de G2( )λ vyznaça[t\sq po g( )ξ =
= ξ σ2
za dopomohog (14). ZauvaΩymo, wo dosyt\ dovesty isnuvannq stalyx ′c ,
′′c takyx, wo pry λ ≥ 1
′c λ σ2 ≤ G2
2( )λ ≤ ′′c λ σ2
. (21)
Dlq G2( )⋅ ma[mo
G2
2( )λ = 1
2 1 1
2
0
2
λ
ξ λ
λ
ξσ
ξ
+ +
∞
∫ d =
=
1
2 1 2
1
0
2
2 1 2 1λ λ
λ
σ
σ σ+ +
∞
+ +∫ t e
dt
t–
ln
= 1
2 1
2 1
2 1 12 1 2 1λ
σ
λ
σ σ+
+
+
+ +
Γ( )
ln
. (22)
Skorystavßys\ nerivnostqmy
1
2λ
≤ ln 1 1+
λ
≤ 1
λ
, spravedlyvymy pry λ ≥ 1, z
(22) otrymu[mo (21).
Qk naslidky z teoremyA2, pokaΩemo, qk otrymaty analoh prqmo] teoremy
(zAtoçnistg do stalo]) ta oberneno] teoremy z roboty [11].
Naslidok 1 (prqma teorema). Nexaj x0 ∈AD( )Aσ
, σ > 0. Todi1
zN
2 = o
N
1
2 1σ +
.
Dovedennq. Iz zobraΩennq (17) baçymo, wo zn
2 ≥ zm
2
pry n < m, a tomu pry
2k ≤ n ≤ 2 1k +
, k ∈N , vykonu[t\sq
2 22 2
2
2σ σ⋅ ( )k z k ≥ n zn
2 2σ ≥
2
2
1 2
2 2
2
1
k
z k
+( )
+
σ
σ , (23)
tobto iz zbiΩnosti rqdu (20) vyplyva[ zbiΩnist\ rqdu
2 2
1
1 2
2
2
1
k
k
k z k⋅ ( )
=
∞
+∑ +
σ
= 1
2
2
2
2 1
2
2
k
k z k
=
∞
+∑ ( ) σ
,
1
ZauvaΩymo, wo naslidok 1 dewo pokrawu[ teoremuA1 z [11], de vstanovleno, wo zN
2 ≤ cN – –2 1σ
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
XARAKTERYZACIQ ÍVYDKOSTI ZBIÛNOSTI ODNOHO NABLYÛENOHO METODU … 563
tomu 2
2 1
2
2k z k( ) +σ
→ 0. Z uraxuvannqm perßo] nerivnosti (23) baçymo, wo j
n zn
2 1 2σ + ≤ 2 22 1 2 1
2
2σ σ+ +( )k z k → 0, n → ∞,
wo slid bulo dovesty.
Rozhlqnemo, qk ce bulo zrobleno u [11], poslidovnist\ dijsnyx çysel
cN N{ } ∈N
taku, wo:
1) cN > 0;
2) cN monotonno zrostagt\ (ne spadagt\);
3)
k c k
=
∞∑ 0
2
1 < ∞.
Qk pryklad, takog poslidovnistg moΩe buty cN = c N⋅ +ln1 ε
, cN = ln N ×
× ln ln N( ) +1 ε
towo. Todi ma[ misce takyj naslidok.
Naslidok 2 (obernena teorema). Nexaj dlq deqkoho x0 ∈�, poslidovnosti
cN N{ } ∈N
, wo zadovol\nq[ umovy (1) – (3), ta deqkoho σ > 0 vykonu[t\sq
zN
2 <
1 1
2 1c NN
σ + .
Todi x0 ∈A D( )Aσ
.
Dovedennq. Spravedlyvist\ naslidku vyplyva[ z teoremyA2, umov 1 – 3 na
poslidovnist\ cN N{ } ∈N
ta nerivnosti, wo vykonu[t\sq dlq vsix natural\nyx k:
1
2
2 11
NcNk
k +
∑
–
≤ 1
2 22
2 11
k c kk
k +
∑
–
= 1
2c k
.
1. Krejn S. H. Lynejn¥e dyfferencyal\n¥e uravnenyq v banaxovom prostranstve. – M.: Nau-
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2. Suetyn P. K. Klassyçeskye ortohonal\n¥e mnohoçlen¥. – 3-e yzd. – M.: Fyzmatlyt, 2005.
– 480 s.
3. Horbaçuk M. L., Horodeckyj V. V. O polynomyal\nom pryblyΩenyy reßenyj dyfferen-
cyal\no-operatorn¥x uravnenyj v hyl\bertovom prostranstve // Ukr. mat. Ωurn. – 1984. –
36, # 4. – S. 500 – 502.
4. Arov D. Z., Gavrilyuk I. P. A method for solving initial value problems for linear differential equa-
tions in Hilbert space based on the Cayley transform // Numer. Func. Anal. and Optimiz. – 1993. –
14, # 5, 6. – P. 456 – 473.
5. Gavrilyuk I. P., Makarov V. L. The Cayley transform and the solution of an initial value problem
for a first order differential equation with an unbounded operator coefficient in Hilbert space // Ibid.
– 1994. – 15, # 5, 6. – P. 583 – 598.
6. Makarov V. L., Vasylyk V. B., Rqbyçev V. L. Neuluçßaem¥e po porqdku ocenky skorosty
sxodymosty metoda preobrazovanyq Kπly dlq pryblyΩenyq operatornoj πksponent¥ // Ky-
bernetyka y system. analyz. – 2002. – # 4. – S. 180 – 185.
7. Havrylgk Y. P., Makarov V. L. Syl\no pozytyvn¥e operator¥ y çyslenn¥e alhorytm¥ bez
nas¥wenyq toçnosty. – Kyev: Yn-t matematyky NAN Ukrayn¥, 2004. – 500 s.
8. Kaßpirovs\kyj O. H., Mytnyk G. V. Aproksymaciq rozv’qzkiv operatorno-dyferencial\nyx
rivnqn\ za dopomohog operatornyx polinomiv // Ukr. mat. Ωurn. – 1998. – 50, # 11. – S. 1506
– 1516.
9. Paßkovskyj S. V¥çyslytel\n¥e prymenenyq mnohoçlenov y rqd¥ Çeb¥ßeva. – M.: Nauka,
1983. – 452 s.
10. Kaßpirovs\kyj O. H. Aproksymaciq hladkyx rozv’qzkiv operatorno-dyferencial\nyx riv-
nqn\ // Nauk. zap. NaUKMA. Fiz.-mat. nauky. – 2002. – 20. – S. 16 – 21.
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daçi Koßi // Ukr. mat. Ωurn. – 2007. – 59, # 6. – S. 838 – 852.
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Press, New York, NY: IEEE Press, 2000. – 1335 p.
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OderΩano 07.11.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
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| id | umjimathkievua-article-3175 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:39Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/46/ca88e82c366a59fffdcc189d6bc82146.pdf |
| spelling | umjimathkievua-article-31752020-03-18T19:47:27Z Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem Характеризація швидкості збіжності одного наближеного методу розв'язування абстрактної задачі Коші Kashpirovskii, A. I. Torba, S. M. Кашпіровський, О. І. Торба, С. М. We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of the operator A is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse theorems of the theory of approximation in the mean. Рассмотрен приближенный метод решения задачи Коши для дифференциально-операторного уравнения, основанный на разложении экспоненты по ортогональным многочленам Лагерра. Доказано, что принадлежность начального значения определенному пространству гладких элементов оператора A эквивалентна сходимости некоторой взвешенной суммы интегральных невязок. Как следствие, получены прямые и обратные теоремы теории приближения в среднем. Institute of Mathematics, NAS of Ukraine 2008-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3175 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 4 (2008); 557–563 Український математичний журнал; Том 60 № 4 (2008); 557–563 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3175/3097 https://umj.imath.kiev.ua/index.php/umj/article/view/3175/3098 Copyright (c) 2008 Kashpirovskii A. I.; Torba S. M. |
| spellingShingle | Kashpirovskii, A. I. Torba, S. M. Кашпіровський, О. І. Торба, С. М. Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem |
| title | Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem |
| title_alt | Характеризація швидкості збіжності одного наближеного методу розв'язування абстрактної задачі Коші |
| title_full | Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem |
| title_fullStr | Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem |
| title_full_unstemmed | Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem |
| title_short | Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem |
| title_sort | characterization of the rate of convergence of one approximate method for the solution of an abstract cauchy problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3175 |
| work_keys_str_mv | AT kashpirovskiiai characterizationoftherateofconvergenceofoneapproximatemethodforthesolutionofanabstractcauchyproblem AT torbasm characterizationoftherateofconvergenceofoneapproximatemethodforthesolutionofanabstractcauchyproblem AT kašpírovsʹkijoí characterizationoftherateofconvergenceofoneapproximatemethodforthesolutionofanabstractcauchyproblem AT torbasm characterizationoftherateofconvergenceofoneapproximatemethodforthesolutionofanabstractcauchyproblem AT kashpirovskiiai harakterizacíâšvidkostízbížnostíodnogonabliženogometodurozv039âzuvannâabstraktnoízadačíkoší AT torbasm harakterizacíâšvidkostízbížnostíodnogonabliženogometodurozv039âzuvannâabstraktnoízadačíkoší AT kašpírovsʹkijoí harakterizacíâšvidkostízbížnostíodnogonabliženogometodurozv039âzuvannâabstraktnoízadačíkoší AT torbasm harakterizacíâšvidkostízbížnostíodnogonabliženogometodurozv039âzuvannâabstraktnoízadačíkoší |