Direct and inverse theorems of approximate methods 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 based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator A, we prove direct and inverse...
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| Date: | 2007 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2007
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3349 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509425144430592 |
|---|---|
| author | Torba, S. M. Торба, С. М. |
| author_facet | Torba, S. M. Торба, С. М. |
| author_sort | Torba, S. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:58Z |
| description | We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator A, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean. |
| first_indexed | 2026-03-24T02:40:54Z |
| format | Article |
| fulltext |
UDK 517.9
S. M. Torba (In-t matematyky NAN Ukra]ny, Ky]v)
PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV
ROZV’QZUVANNQ ABSTRAKTNO} ZADAÇI KOÍI
*
We consider an approximate method of the solution of the Cauchy problem for an operator-differential
equation based on the exponent decomposition in the orthogonal Lager polynomials. For the initial
value of finite smoothness with respect to the operator A, we prove direct and inverse theorems of the
theory of approximation in the mean and present examples of the unimprovability of corresponding
estimates in these theorems. We establish the exponential rate of convergence for exponential-type
entire vectors and the subexponential rate of convergence for the Gevrey classes. We also establish the
characterization of both classes of vectors in terms of convergence rate in the mean approximation.
Rassmotren pryblyΩenn¥j metod reßenyq zadaçy Koßy dlq dyfferencyal\no-operatornoho
uravnenyq, osnovann¥j na razloΩenyy πksponent¥ po ortohonal\n¥m mnohoçlenam Lahera.
Dlq naçal\noho znaçenyq koneçnoj hladkosty otnosytel\no operatora A dokazan¥ prqmaq y
obratnaq teorem¥ teoryy pryblyΩenyq v srednem, pryveden¥ prymer¥ neuluçßaemosty soot-
vetstvugwyx ocenok v πtyx teoremax. Dlq cel¥x vektorov πksponencyal\noho typa ustanov-
lena πksponencyal\naq skorost\ sxodymosty, dlq klassov Ûevre — subπksponencyal\naq, a
takΩe xarakteryzacyq sootvetstvugwyx klassov v termynax skorosty sxodymosty v srednem
pryblyΩenyq.
1. Vstup. U roboti M. L. Horbaçuka i V. V. Horodec\koho [1] zaproponovano na-
blyΩenyj polinomial\nyj metod rozv’qzannq zadaçi Koßi dlq dyferencial\no-
operatornoho rivnqnnq v hil\bertovomu prostori, v osnovu qkoho pokladeno roz-
klad deqkyx klasyçnyx funkcij u rqd za ortohonal\nymy polinomamy Lahera i
otrymano ocinky poxybky nablyΩennq v zaleΩnosti vid stupenq hladkosti po-
çatkovyx danyx. O. I. Kaßpirovs\kyj i G. V. Mytnyk [2] zaproponuvaly polino-
mial\nyj metod nablyΩennq, wo ©runtu[t\sq na rozkladi funkcij za deqkymy
inßymy klasamy ortohonal\nyx mnohoçleniv.
U robotax D. Z. Arova, I. P. Havrylgka, V. L. Makarova, V. B. Vasylyka i
V.AL. Rqbiçeva [3 – 6] ta v monohrafi] I. P. Havrylgka, V. L. Makarova [7] zapro-
ponovano inßyj pidxid do rozv’qzannq takyx zadaç, tak zvanyj metod peretvo-
rennq Keli, qkyj ne [ polinomial\nym, i otrymano potoçkovi j intehral\ni ocin-
ky (prqmi teoremy).
U cij roboti rozhlqda[t\sq analohiçnyj pidxid, wo vykorystovu[ rezol\-
ventu vid operatora. Dlq vektoriv skinçenno] hladkosti vidnosno operatora A,
dlq cilyx vektoriv eksponencial\noho typu ta dlq Ωevre[vs\kyx vektoriv dove-
deno prqmi teoremy pro ßvydkist\ zbiΩnosti intehral\noho vidxylu nablyΩe-
noho rozv’qzku. Krim toho, dlq cyx klasiv dovedeno obernenu teoremu. Dlq
klasu vektoriv skinçenno] hladkosti vidnosno operatora A vstanovleno, wo
prqmu teoremu v deqkomu sensi ne moΩna pokrawyty; ce pokrawu[ rezul\taty
I.AP. Havrylgka, V. L. Makarova ta V. L. Rqbiçeva. Bil\ß toho, pokazano, wo
miΩ ocinkamy prqmo] ta oberneno] teorem [ deqkyj „zazor”, bil\ßyj za ln n , i
navedeno pryklad, çomu joho nemoΩlyvo zmenßyty. Dlq klasiv neskinçenno
hladkyx vektoriv C
∞
( A ), cilyx vektoriv eksponencial\noho typu ExpA � ta
dlq klasiv Ωevre[vs\kyx vektoriv G{ β } ta G ( β ) prqmi ta oberneni teoremy da-
gt\ povnu xarakteryzacig vidpovidnyx klasiv u terminax ßvydkosti prqmuvannq
do nulq intehral\noho vidxylu.
2. Prqmi teoremy nablyΩennq. Nexaj A — samosprqΩenyj dodatno vy-
znaçenyj operator, wo di[ v hil\bertovomu prostori �. Dlq operatora A roz-
hlqda[t\sq zadaça Koßi
x′ ( t ) + A x ( t ) = 0, t > 0, x ( 0 ) = x0 . (1)
Vidomo [8], wo rozv’qzok zadaçi Koßi ma[ vyhlqd
*
Pidtrymano DerΩavnym fondom fundamental\nyx doslidΩen\ Ukra]ny (proekt # 14.1/003).
© S. M. TORBA, 2007
838 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 839
x ( t ) = e–
t
A
x0 .
Meta dano] roboty — doslidyty ßvydkist\ zbiΩnosti nablyΩennq rozv’qzku
x ( t ), pobudovanoho za dopomohog rozkladu funkci] e–
a
t
v rqd za mnohoçlenamy
Lahera
Ln ( t, α ) =
1
n
t e t et n t n
!
− + − ( )( )α α
.
VvvaΩatymemo v podal\ßomu, wo α = 0. Na pidstavi rozkladu eksponenty [9,
s.A257]
e–
a
t =
1
1 1
0
0a
a
a
L t
n
n
n+ +
( )
=
∞
∑ , , a > − 1
2
,
ta operatornoho çyslennq dlq samosprqΩenoho operatora A moΩna zapysaty
formal\nyj vyraz dlq operatorno] eksponenty
e–
A
t = A A I L tn n
n
n
( + ) ( )−( + )
=
∞
∑ 1
0
0, , (2)
wo di[ v �. Podibnyj do (2) pidxid, u qkomu zamist\ vyrazu A ( A + I )
–
1
vykorys-
tovu[t\sq peretvorennq Keli ( γ I + A )
–
1
( γ I – A ), rozhlqda[t\sq v monohrafi]
[7] ta v bahat\ox stattqx [3 – 6]. Zaminog à = γ–
1
A – I odyn iz nyx zvodyt\sq do
inßoho. Vykorystovugçy rezul\taty [7], baçymo, wo rqd (2) pry t > 0 zbiha[t\-
sq dlq vsix x ∈ �, pry t = 0 dostatn\og umovog zbiΩnosti [ x ∈ D ( Aσ
), σ > 0.
Dlq zbiΩnosti rqdu (2) razom z formal\nog poxidnog dosyt\ vymahaty, wob
σ ≥ 1 pry t > 0 ta σ > 1 pry t = 0.
Rozhlqnemo toçnyj rozv’qzok
x ( t ) = A A I L t xn n
n
n
( + ) ( )−( + )
=
∞
∑ 1
0
0
0, (3)
i za nablyΩennq viz\memo çastkovu sumu rqdu (3)
xN ( t ) = A A I L t xn n
n
n
N
( + ) ( )−( + )
=
∑ 1
0
0
0, . (4)
Vyraz (4) moΩna interpretuvaty takym çynom: rozhlqda[t\sq poslidovnist\
stacionarnyx zadaç
( A + I ) yp + 1 = A yp
, p = 0, 1, … , y0 = ( A + I )
–
1
x0 ,
zavdqky çomu nestacionarne rivnqnnq (1) „dyskretyzu[t\sq” i zvodyt\sq do po-
slidovnosti stacionarnyx rivnqn\ z nezminnymy pravog ta livog çastynamy, i za
qkymy zapysu[t\sq nablyΩenyj rozv’qzok
xN ( t ) = L t yn n
n
N
( )
=
∑ , 0
0
. (5)
Osnovna vlastyvist\ mnohoçleniv Lahera — ce ]x ortohonal\nist\ za vahog
e–
t
, t > 0. Vidpovidno, slußno rozhlqnuty intehral\nyj vidxyl nablyΩennq
1
1
Potoçkovi ocinky nablyΩennq doslidΩeno v monohrafi] [7], a vidpovidni oberneni teoremy
planu[t\sq rozhlqnuty v odnij iz nastupnyx statej.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
840 S. M. TORBA
z t x t x t e dtN N
t2 2
0
( ) = ( ) − ( ) −
∞
∫ . (6)
Prypustymo, wo vektor x0 ma[ deqkyj stupin\ hladkosti vidnosno operatora
A. Todi ma[ misce taka teorema.
Teorema 1. Nexaj x0 ∈ D ( Aσ
), σ ≥ 0. Todi2
z
e N
o
N
A xN
2
2 1
2 2 2 1 2 1 2 1 0
22 1
2
1 1< ( + ) +
+
+ + + +
σ σ
σ σ σ σ
σ
. (7)
Dovedennq. Vykorystovugçy (5), (6), spektral\ne zobraΩennq samosprq-
Ωenoho operatora ta ortonormovanist\ mnohoçleniv Lahera z vahog e–
t
, zapy-
su[mo
z
A
A I
L t x
A
A I
L t x e dtN
k
k k
k N
k
k k
k N
t2
1 0
1
1 0
10
0 0=
( + )
( )
( + )
( )
+
= +
∞
+
= +
∞∞
−∑ ∑∫ , , , =
=
A
A I
x
A
A I
x
k
k
k
k
k N ( + ) ( + )
+ +
= +
∞
∑ 1 0 1 0
1
, =
=
A
A I
y
A
A I
y
k
k
k
k
k N
−
+
−
+
= +
∞
( + ) ( + )
∑
σ σ
1 0 1 0
1
, =
=
λ
λ
λ
λ
σ
λ
σ
λ
2 2
2 2 0 0
01
2 2
2 2 0 0
10
1 1
k
k
k N
k
k
k N
d E y y d E y y
−
+
∞
= +
∞ −
+
= +
∞∞
( + )
( ) =
( + )
( )∫∑ ∑∫, , ,
de y = Aσ
x0 . Pidraxovugçy sumu pid znakom intehrala, znaxodymo
z d E y yN
N
2
2 2
2 0 0
0
1
1 1
2 1
=
+
+
( )
+∞
∫ λ
λ λ λσ λ , . (8)
Poznaçymo pidintehral\nyj vyraz çerez ϕ ( λ ). Neskladnyj analiz pry umovi
2N > σ pokazu[, wo ϕ ( λ ) zrosta[ pry λ < λmax ta spada[ pry λ > λmax
, de λmax
zadovol\nq[ ocinky
1
4 2
4 3 4 2 1
σ
σ σ
+
( + − − + )N < λmax <
1
4 2
4 3 4 2 1
σ
σ σ
+
( + − + + )N .
Pidstavlqgçy u ϕ ( λ ) vidpovidno livu çy pravu ocinku dlq λmax ta vykorys-
tovugçy nerivnist\ 1
1−
x
x
< e–
1
, x ≥ 1, znaxodymo ocinku maksymal\noho
znaçennq:
ϕ ( λ ) <
( + ) +
+
+ + + +
2 1
2
1 12 1
2 2 2 1 2 1 2 1
σ σ
σ σ σ σe N
o
N
.
Povertagçys\ do (8), otrymu[mo
2
Prqmu teoremu nablyΩennq dovedeno v [6] ta v dysertaci] V. L. Rqbiçeva, ale inßyj metod do-
vedennq dozvolyv pokrawyty stalu z
( + )
+
( + )σ
σ
σ1
2 1
2 1
do
( + )
( + )
( + )
+
σ
σ
σ
σ
1 2
2 1
2 1
2 1
/
e
, tomu cg teoremu ta ]]
dovedennq navedeno v danij roboti.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 841
z
e N
o
N
A xN
2
2 1
2 2 2 1 2 1 2 1 0
22 1
2
1 1< ( + ) +
+
+ + + +
σ σ
σ σ σ σ
σ
.
Teoremu dovedeno.
U monohrafi] [7] navedeno pryklad operatora, dlq qkoho
z
c
N N
N
2
2 1 1≥ + +σ εln
,
tobto nepokrawuvanist\ ocinky (7) u klasi D ( Aσ
) z toçnistg do ln1
+
ε
N. Ale z
prykladu ne moΩna vstanovyty, çy bude spravedlyvog ocinka z ln N. PokaΩe-
mo, wo ma[ misce syl\nißyj fakt, tobto ocinku (7) vzahali ne moΩna pokrawy-
ty. A same, ma[ misce taka teorema.
Teorema 2. Nexaj { } =
∞cn n 1 — neobmeΩena nespadna poslidovnist\ dodatnyx
çysel. Todi isnugt\ takyj samosprqΩenyj operator A ta vektor x 0 ∈
∈ D ( Aσ
), wo ne isnu[ stalo] c > 0 tako], wo
z
c
N c
N
N
2
2 1≤ +σ , n ∈ N.
Dovedennq. Pobudu[mo poslidovnist\ indeksiv n1 < n2 < … takym çynom:
ni + 1 — deqkyj indeks takyj, wo ni + 1 > ni ta
cni + 1
> ( i + 1 )
3
.
Takyj indeks isnu[ zavdqky monotonnosti ta neobmeΩenosti { } =
∞cn n 1. Nexaj
{ } =
∞ei i 1 — ortonormovanyj bazys u �. Dig operatora A vyznaçymo tak:
A en = n en
.
Pid oblastg vyznaçennq rozumitymemo D ( A ) = { x ∈ � | A x ∈ � }. Lehko baçyty,
wo zadanyj takym çynom operator A [ samosprqΩenym.
Teper pobudu[mo vektor x0 ∈ D ( Aσ
). Poklademo
x0 =
1
1 n i
e
i
n
i
iσ
=
∞
∑ ∈ �, y0 := Aσ
x0 =
1
1 i
en
i
i
=
∞
∑ ∈ �, (9)
dlq zbiΩnosti rqdiv vykorystovu[t\sq ortonormovanist\ { } =
∞ei i 1. OtΩe, x0 ∈
∈ D ( Aσ
). Formula (8) dlq vektora x0 ta operatora A zvodyt\sq do formuly
z
n
n n n
y eN
i
i
N
i i
n
i
i
2
2 2
2 0
2
1 1
1 1
2 1
=
+
+
( )
+
=
∞
∑ σ , .
Pry N = ni
, vykorystovugçy nerivnist\ 1
1
1
−
+
n
n
≥ e–
1
, ma[mo
z
n
n n n
y e
ce
N i
N
i
i
N
i i
ni
2
2 2
2 0
2 2
2 1 21
1 1
2 1 2
1≥
+
+
( ) ≥
+ −
+σ σ,
˜
, (10)
de c̃ ne zaleΩyt\ vid i.
Prypustymo, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
842 S. M. TORBA
∃ c > 0: z
c
c N
N
N
2
2 1≤ +σ . (11)
Ale todi z nerivnostej (10) ta (11) vyplyva[, wo dlq vsix i
i
c
ce
cni
2 2
2
≥
−˜
> 0,
a ce supereçyt\ vyboru indeksiv ni takym çynom, wob vykonuvalas\ umova
i
c ini
2 1< .
Teoremu dovedeno.
3. Oberneni teoremy nablyΩennq. Dlq vektoriv x0 ∈ D ( Aσ
), σ > 0, v
poperedn\omu punkti otrymano ocinku (7) prqmuvannq do nulq intehral\noho
vidxylu nablyΩennq. Vynyka[ pytannq: çy cq ocinka [ najkrawog? Teorema 2
pokazu[, wo na vsij mnoΩyni D ( Aσ
) ]] pokrawyty ne moΩna. V c\omu punkti
dovedemo obernenu teoremu, tobto za umovy dewo syl\nißo] za (7) ocinky, na-
pryklad umovy
z
c
N N
N
2
2 1 1< + +σ εln
,
obov’qzkovo bude vykonano vkladennq
x0 ∈ D ( Aσ
).
Rozhlqnemo poslidovnist\ dijsnyx çysel { cN } N ∈ N taku, wo:
1) cN > 0;
2) cN monotonno zrostagt\ (ne spadagt\);
3)
1
20 c kk =
∞
∑ < ∞.
Qk pryklad, takog poslidovnistg moΩe buty cN = c ln1
+
ε
N, cN = ln N ( ln ln N )
1
+
ε
towo. Spravedlyvog [ taka teorema.
Teorema 3. Nexaj dlq deqkoho x0 ∈ �, poslidovnosti { cN } N ∈ N , wo zado-
vol\nq[ umovy 1 – 3, ta deqkoho σ > 0 vykonu[t\sq
z
c N
N
N
2
2 1
1 1< +σ .
Todi x0 ∈ D ( Aσ
).
Dovedennq. Dlq toho wob dovesty teoremu, dosyt\ pobuduvaty taku pos-
lidovnist\ funkcij ϕn ( λ ), wo:
ϕn ( λ ) monotonno zrostagt\ pry n → ∞;
ϕ λ λn d E x x( ) ( )
∞
∫ 0 0
0
, ≤ C ∀N ∈ N;
∃ c′ : c′ λ2σ ≤ limn → ∞ ϕn ( λ ).
Pislq c\oho, vykorystovugçy lemu Fatu, peresvidçu[mosq, wo funkciq
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 843
ϕ ( λ ) = lim
n
n
→∞
( )ϕ λ
[ intehrovnog, za teoremog Lebeha intehrovnog vidnosno miry d ( Eλ x0 , x0 ) [ ta-
koΩ funkciq c1 λ2σ
, a intehrovnist\ ostann\o] ekvivalentna naleΩnosti x0 do
D ( Aσ
) .
PokaΩemo qk budu[t\sq ßukana poslidovnist\ ϕn ( λ ). Rozhlqnemo z NN
2 2 1σ +
.
Za umovog teoremy
z N
cN
N
2 2 1 1σ + ≤ . (12)
Qk i pry dovedenni teoremy 1, ma[mo spektral\ne zobraΩennq
z N N d E x xN
N
2 2 1 2 1
2 2
0 0
0
1
1
2 1
σ σ
λ
λ
λ λ
+ +
+∞
=
+
+
( )∫ , .
Poznaçymo çerez ψn ( λ ) pidintehral\nyj vyraz i rozhlqnemo poslidovnist\
funkcij
ϕn ( λ ) = max{ }( ) =ψ λk k
n
1.
Oçevydno, wo ϕn ( λ ) monotonno zrostagt\ pry n → ∞. Dovedemo, wo
∃ C > 0 ∀N ∈ N : ϕ λ λn d E x x( ) ( )
∞
∫ 0 0
0
, ≤ C.
Dlq c\oho zauvaΩymo, wo
ϕn ( λ ) < 22 1
2
0
2
σ ψ λ+
=
[ ]
( )
∑ k
k
nlog
. (13)
Spravdi, dlq bud\-qkoho k ≤ n poznaçymo çerez r = [ log2 k ] najbil\ße cile
çyslo take, wo 2r ≤ k. Todi matymemo
ψ λ
ψ λ
λ
λ
σ
σk
r
k
r
r
k( )
( )
≤
+
<
+ − ⋅
+
2
2 1 2 2 2
2 1
2 1
2 ,
oskil\ky za oznaçennqm k < 2 ⋅ 2r
. Vidpovidno, koΩne z ψk ( λ ), 2r ≤ k < 2r
+
1
, ne
perevywu[ 22 1
2
σ ψ λ+ ( )r , wo dovodyt\ (13). Z (12) ta (13) znaxodymo
ϕ λ ψ λλ
σ
λ
σ
n
k
n
k
n
d E x x d E x x
c
k
k
( ) ( ) < ( ) ( ) ≤
∞
+
∞
=
[ ]
+
=
[ ]
∫ ∫∑ ∑0 0
0
2 1
2 0 0
00
2 1
20
2 2
12 2
, ,
log log
< C
zavdqky tretij umovi na poslidovnist\ ck . Perßi dvi umovy na poslidovnist\ ϕn
vykonano. Dovedemo, wo vykonu[t\sq j tretq, tobto isnu[ stala c̃ > 0 taka, wo
˜ limc
n
nλ ϕ λ ϕ λσ2 ≤ ( ) = ( )
→∞
, λ ≥ λ0 > 2
(dosyt\ vymahaty, wob umova vykonuvalasq ne dlq vsix λ > 0, a lyße dlq do-
syt\ velykyx λ ). Dlq c\oho rozhlqnemo deqke λ′, poklademo
n0 =
2 1
2
σ λ+ ′
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844 S. M. TORBA
ta ocinymo ψn 0
( λ′ ). Ma[mo
ψn 0
( λ′ ) =
2 1
2 1
1
2 1 4
2 1 2
2 1
2
2 2 1
2 1
2σ λ λ
λ λ
σ λ
σ
σ λ σ
σ
σ+ ′
′
′ +
′ +
≥ ′
+
+ ′
+ +
+e
,
a tomu j
ϕ ( λ ) ≥ ϕn 0
( λ ) ≥ ψn 0
( λ ) ≥
σ λ
σ
σ
σ
2 1
2 1
2
4
+
+ ′
e
,
wo i potribno bulo dovesty.
Vynyka[ pytannq: a çy vsi umovy v teoremi 3 [ sutt[vymy? MoΩlyvo, inßyj
metod dovedennq dozvolyt\ dovesty obernenu teoremu z ocinkog, wo z toçnistg
do stalo] zbiha[t\sq z (7)? PokaΩemo, wo ce ne tak. Dlq c\oho rozhlqnemo po-
slidovnist\ { } =
∞cn n 1, wo zadovol\nq[ taki umovy: 1 ta 2, qk u teoremi 3, ta umovy
3′ )
1
21c ii =
∞
∑ = ∞,
4) ∃ C0 > 0 ∀i :
c
c
i
i
2
2
1+
≤ C0 .
Umovy 1, 2, 3′, 4 zadovol\nqgt\ bahato poslidovnostej, napryklad, cn =
= const, cn = ln n abo cn = ln n ln ln n.
Ma[ misce taka teorema.
Teorema 4. Nexaj poslidovnist\ { } =
∞cn n 1 zadovol\nq[ umovy 1, 2, 3′ ta 4.
Todi isnugt\ samosprqΩenyj operator A ta vektor x0 taki, wo x0 ∉ D ( Aσ
),
ale
z
c
c N
N
N
2
2 1≤ +σ
dlq deqkoho c > 0.
Dovedennq. Nexaj { } =
∞ei i 1 — ortonormovanyj bazys u �. Operator A vy-
beremo takyj, qk i v dovedenni teoremy 2. Za poçatkovyj viz\memo vektor x0
,
zadanyj rqdom
x0 =
1 1
221
2c
e
i
i
i
i
=
∞
∑ σ .
Lehko baçyty, wo x0 ∈ �, ale x0 ∉ D ( Aσ
), oskil\ky rqd
Aσ
x0 =
1
21
2c
e
i
i
i =
∞
∑
[ rozbiΩnym u � zavdqky umovi 3′ na poslidovnist\ { cn }.
Qk i pry dovedenni teoremy 2, moΩemo zapysaty
z
cN i
N
i
i i
i
2
2 2
1
1 2
2
1
1
2 1
1
2 1
1
2
1= −
+
+
+
=
∞
+∑ σ . (14)
Nexaj σ ⋅ 2n 0 ≤ N < σ ⋅ 2n 0
+
1
. Ocinku sumy (14) provedemo okremo dlq i ≥ n0
ta i < n0
. Pry i = n ≥ n0 1
1
2 1
2 2
−
+
+
n
N
< 1, otΩe,
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PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 845
2
2 1
1
2 1
1
2
1
2
1
2
2
2 2
1 2 2
2
20 0
n
n
N
n n n n n nN N+
+
< ( )
+
+ − ( − )σ σ
σ
σ
σ σ
=
=
1
2
2
2 1
2 2 1
2 10( + )( − )
+
+σ
σ σ
σ
σ
n n N
.
Vraxovugçy monotonnist\ poslidovnosti { cn }, znaxodymo
1
1
2 1
1
2 1
1
2
1
2 2
1 2
20
−
+
+
+
=
∞
+∑ i
N
i n
i i c i
σ <
<
1 1
2
2 1
2
2 1
2 2 1
2 1
2
2 1
0 0
0
0
c N
c
c Nn n
n n
n n ( )
≤+ −
+
+
=
∞
+∑ σ
σ σ
σ σ
σ ˜
, c̃ =
−
+
+
2
2 1
2 2 1
2 1
σ σ
σ
σ
. (15)
Zalyßylos\ ocinyty sumu dodankiv z i < n0
. Vykorysta[mo ocinky
1 1
2 2
2
2
2
2
0
20
0
0 01
1 0
c c
c
c
c
c
C
ci n
n
n
i
i n
n i
= … ≤
−
+
−
(16)
zhidno z vlastyvistg 4 poslidovnosti { cn }.
Dlq perßoho mnoΩnyka vykorysta[mo ocinku
1
1
2 1
1
1
2 1
2 2 2 2 1
2
0
0−
+
< −
+
< ( )
+ ( + )
−
−
−
i
N
i
n i i
n i
e
σ
σ
. (17)
Ostannq neobxidna ocinka [ takog:
1
2 1
1
2
2
2 2
2 2
2
2 2
1 2 1
2 2
2 1 2
2 2 1 2 1
2 1
0
0 0
0
0
0
i i
n i
n n i
n i
n
n i
N+
−
+ −
( − )
( + )
+ ( − )( + )
++
=
+
( ) <σ
σ σ
σ σ
σ σ σ
σ
σ
σ σ
σ
σ
σ
. (18)
Po[dnugçy (16) – (18), pryxodymo do ocinky
2
2 1
1
2 1
1
2
1 2
2
2 2
1 2
21
1 2 2 1
2
2 1
2
0
2 1
1
10 0
0
i
i
N
i i
i
n
i i
i
n
c c N
e C
i n
i
+
+
< ( ) ( )
+
+
=
− +
+
− +
=
−
∑ ∑σ
σ σ
σ
σ σσ
. (19)
Zavdqky zbiΩnosti rqdu ( ) ( )− +
=
∞
∑ e C
n n n
n
σ σ2
0
2 1
1
2 , z (15) ta (19) otrymu[mo
∃ C̃ > 0 : z
C
c N
CC
c N
CC
c N
N
Nn n
2
2
2 1
0
2
2 1
0
2 1
0 0 1
≤ ≤ ≤+ + +
+
˜ ˜ ˜
σ σ σ ,
de v ostannix dvox nerivnostqx vykorystano vlastyvosti 2, 4 poslidovnosti { cn }.
Ostannq velyçyna i [ ßukanog.
Teoremu dovedeno.
4. Prqmi ta oberneni teoremy nablyΩennq v klasax neskinçenno dyfe-
rencijovnyx vektoriv operatora A. Perejdemo do rozhlqdu vypadku neskin-
çenno] hladkosti poçatkovoho vektora x0
,
x0 ∈ C
∞
( A ) =
D
=
( )
∞
An
n 1
∩ .
Qk pokazano v [10], qkwo A — zamknenyj wil\no vyznaçenyj operator, wo ma[
prynajmni odnu rehulqrnu toçku, to C A∞( ) = �. Vidpovidno, u vypadku, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
846 S. M. TORBA
rozhlqda[t\sq v danij statti, mnoΩyna vektoriv neskinçenno] hladkosti [ wil\-
nog, a zadaça nablyΩennq — zmistovnog.
Teoremy 1 ta 3 dozvolqgt\ otrymaty nastupnu xarakteryzacig vektoriv z
C
∞
( A ) u terminax ßvydkosti prqmuvannq do nulq intehral\noho vidxylu.
TverdΩennq. Vektor x0 ∈ C
∞
( A ) todi i lyße todi, koly
∀k ∈ N : lim
N
N
kz N
→∞
+2 2 1 = 0.
Dlq otrymannq konkretnyx prqmyx ta obernenyx teorem dlq neskinçenno
hladkyx poçatkovyx vektoriv vydilymo vuΩçi pidklasy v C
∞
( A ). Dotrymugçys\
[10], dlq β ≥ 0 poznaçymo
Cα 〈 nn β
〉 ( A ) = { x ∈ C
∞
( A ) | ∃ c > 0 : || An
|| ≤ c αn
nn β
}.
U podal\ßomu v poznaçenni budemo nextuvaty operatorom i pysaty prosto
Cα 〈 nn β
〉. Cα 〈 nn β
〉 — banaxovyj prostir vidnosno normy
x
A x
nC n
n
n
n nn
α
β
α β〈 〉 = sup .
Vidpovidno, rozhlqdagt\sq
C C n C n
n
n n
n{ }
> →∞
= 〈 〉 = 〈 〉β α
β
α α
α
β
0
∪ lim ind
ta
C C n C n
n
n n
n( )
> →∞
= 〈 〉 = 〈 〉β α
β
α α
α
β
0
∩ lim proj
— induktyvna ta proektyvna hranyci banaxovyx prostoriv.
U konkretnyx vypadkax, napryklad dlq prostoru C ( [ a, b ] ) abo L2 ( [ a, b ] ),
A — operatora dyferencigvannq, mnoΩyny C
nn{ }β ta C
nn( )β retel\no vyvçeno
u bahat\ox pracqx. Napryklad, C
d
dxnn{ }
ta C
d
dxnn( )
— prostory vidpo-
vidno analityçnyx i cilyx funkcij, C
d
dx{ }
1 — prostir cilyx funkcij ekspo-
nencial\noho typu, C
d
dxnn{ }
β ta C
d
dxnn( )
β , β > 1, — klasy Ûevre typu
Rum’[ i B\orlinha zastosovugt\sq pry doslidΩenni bahat\ox vaΩlyvyx zadaç.
Spoçatku rozhlqnemo vypadok β = 0. U takomu vypadku prostir ExpA � : =
: = C{1} ( A ) nazyva[t\sq prostorom cilyx vektoriv eksponencial\noho typu
3
. Qk
vyplyva[ z rezul\tativ roboty [11], dlq samosprqΩenoho operatora cej prostir
[ wil\nym u �. Krim toho, v [12] navedeno xarakteryzacig prostoriv Cα 〈 1 〉:
dlq dovil\noho α > 0
Cα 〈 1 〉 ( A ) = E ( [ 0, α ] ) �, (20)
de E ( ∆ ) — spektral\na mira operatora A.
Ma[ misce nastupna teorema.
Teorema 5. Vektor x0 ∈ Cα 〈 1 〉 todi i til\ky todi, koly
z c x eN
N2
0
2≤ ( ) −, γ γ
, N ≥
α
2
, (21)
de γ = ln /( )( + )1 α α .
3
Pry β = 0 prostir C(1) ( A ) = Ker A [ tryvial\nym i tomu ne rozhlqda[t\sq.
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PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 847
ZauvaΩymo, wo u statti [6] teoremu 5 otrymano u çastkovomu vypadku α =
= 1 / 2 , a zvorotnu çastynu, krim fiksovanoho α, we j za umovy dyskretnosti
spektra operatora A ta zv’qzku najmenßoho vlasnoho znaçennq z α.
Dovedennq. Nexaj x0 ∈ Cα 〈 1 〉. Todi formulu (8) z ohlqdu na umovu (20)
zvodymo do tako] (skriz\ u podal\ßomu u (8) poklademo σ = 0):
z d E x xN
N
2
2 2
0 0
0
1
1
2 1
=
+
+
( )
+∞
∫ λ
λ λ λ , =
=
λ
λ λ λ
α
+
+
( )
+
∫ 1
1
2 1
2 2
0 0
0
N
d E x x, . (22)
Poznaçymo çerez ϕ ( λ ) pidintehral\nyj vyraz. Znajdemo maksymum velyçyny
ϕ ( λ ) na promiΩku [ 0, α ]. V dovedenni teoremy 1 pokazano, wo funkciq ϕ ( λ )
monotonno zrosta[ do λmax
, de λmax zadovol\nq[ ocinky
2N + 1 < λmax < 2N + 2.
OtΩe, pry N ≥ α / 2 maksymum funkci] ϕ ( λ ) na vidrizku [ 0, α ] dosqha[t\sq
pry λ = α, i moΩemo zapysaty
ϕ ( λ ) ≤ ϕ ( α ) =
α
α α α
γ
γ
+
+
=
+
=
+ −( + )
−
1
1
2 1 2 1
2 2 2 2
2
N N
Ne
ce , (23)
de γ = ln
1 + α
α
, a c =
e−
+
2
2 1
γ
α
.
Z (22) ta (23) otrymu[mo ßukane
z ce d E x x c x eN
N N2 2
0 0
0
0
2 2≤ ( ) =− −∫γ
λ
α
γ, .
Nexaj teper dlq x0 vykonano ocinku (21). Dovedemo, wo x0 ∈ Cα 〈 1 〉. Pry-
pustymo, wo na deqkomu promiΩku ( l, l + ε )
d E x x
l
l
( )
+
∫ λ
ε
0 0, = M > 0. (24)
Vyxodqçy z (8), zapysu[mo ocinku
z d E x xN
N
l
l
2
2 2
0 01
1
2 1
≥
+
+
( )
++
∫ λ
λ λ λ
ε
, . (25)
Pry N > ( l + ε ) / 2 vnaslidok monotonnoho zrostannq ϕ ( λ ) na ( 0, 2N ) ma[mo
ocinku
λ
λ λ
γ γ
+
+
≥
+
+
=
+
+ + − −
1
1
2 1 1
1
2 1 2 1
2 2 2 2 2 2N N Nl
l l
e e
l
l l
,
de γl = ln
l
l
+ 1
. Z (24), (25) ta ostann\o] ocinky znaxodymo
z
Me
l
eN
Nl
l2
2
2
2 1
≥
+
−
−
γ
γ
. (26)
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848 S. M. TORBA
Oskil\ky, za prypuwennqm, dlq vsix N vykonano (21), to, porivnggçy (21) z
(26), pryxodymo do vysnovku, wo
– γl ≤ – γ ⇔ l ≤ α. (27)
Zavdqky vyboru ε qk dovil\noho dodatnoho z (27) vyplyva[, wo x0 ∈ E ( [ 0,
α ] ) �, tobto (za kryteri[m (20)) x0 ∈ Cα 〈 1 〉.
Teoremu dovedeno.
Pry α > 0 vyraz ln
1 + α
α
nabuva[ vsix moΩlyvyx znaçen\ vid 0 do + ∞, to-
mu z teoremy 5 vyplyva[ takyj naslidok.
Naslidok. NablyΩennq xN ( t ) ma[ eksponencial\nu ßvydkist\ zbiΩnosti u
seredn\omu do x ( t ) todi i lyße todi, koly x 0 [ cilym vektorom eksponenci-
al\noho typu operatora A.
Perejdemo teper do prostoriv C
nn{ }β ta C
nn( )β , β > 0. Dlq c\oho vykorys-
ta[mo xarakteryzacig naleΩnosti vektora do prostoru Cα 〈 nn β
〉 u terminax
zbiΩnosti spektral\nyx intehraliv. Navedemo neobxidni vidomosti, vykorystovu-
gçy rezul\taty z [10]. Pry β > 0 poslidovnist\ { nn β
} n ∈ N ma[ vlastyvist\
∀α > 0 ∃ c = c ( α ) > 0 ∀n ∈ N : nn β ≥ c αn
. (28)
Umova (28) zabezpeçu[ vkladennq ExpA ⊂ Cα 〈 nn β
〉 pry dovil\nomu α > 0 i, vid-
povidno, netryvial\nist\ ostannix prostoriv.
Rozhlqnemo ρ ( λ ) = exp ( β e
–
1
λ1
/
β
). Za funkci[g ρ ( λ ) budu[t\sq sim’q
hil\bertovyx prostoriv
�t 〈 ρ 〉 = D ( ρ ( t A ) ), ( f, g ) �t
〈 ρ 〉 = ( ρ ( t A ) f, ρ ( t A ) g ).
Qk pokazano v roboti [10], normy u prostorax Cα 〈 nn β
〉 ta �t 〈 〉ρ pov’qzani
miΩ sobog spivvidnoßennqmy
∀t > 0 ∀s > 1:
x m c s x
t ts
nC n� 〈 〉 〈 〉≤ ( )
( )−
ρ β
2
0
2 2 2
1
(29)
ta
x m xC nt
n
t
〈 〉
−
〈 〉≤
−
β ρ
2
0
2 2
1�
. (30)
Nastupna teorema da[ xarakteryzacig klasiv Ûevre typu Rum’[ G{ β } ( A ) ta
typu B\orlinha G( β ) ( A ), β > 0:
Teorema 6. Vektor x0 ∈ G{ β } ( A ) todi i lyße todi, koly
∃ c > 0 ∃ c′ > 0 : z c cNN
2 1 12≤ ′ (− )( + )exp / β
;
x0 ∈ G( β ) ( A ) todi i lyße todi, koly
∀c > 0 ∃ c′′ > 0 : z c cNN
2 1 12≤ ′′ (− )( + )exp / β
.
Dlq dovedennq teoremy 6 sformulg[mo i dovedemo dvi lemy.
Lema 1. Nexaj x0 ∈ Cα 〈 nn β
〉. Todi
∀t > α ∃ c > 0 ∃ N0 ∀N > N0 : z c c NN t
2 1 12≤ (− )( + )exp / β
,
de ct = t e− ( + ) − ( + )( + )1 1 1 1/ /β β β β .
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PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 849
Dovedennq. Nexaj x0 ∈ Cα 〈 nn β
〉. Vyberemo s =
t + α
2
, todi α < s < t. Qk
vyplyva[ z (29),
ρ λ λ
2 1
0 0
0
( ) ( )−
∞
∫ s d E x x, < ∞,
de ρ ( λ ) = exp ( β e
–
1
λ1
/
β
). Zvidsy robymo vysnovok, wo vyznaçenyj element y0 =
= ρ ( s–
1
A ) x0 ∈ �. OtΩe, qk i pry dovedenni spivvidnoßennq (8), moΩemo zapysaty
z
s
d E y yN
N
2
2 2
2 1 0 0
0
1
1
2 1
1=
+
+ ( )
( )
+
−
∞
∫ λ
λ λ ρ λ λ , . (31)
Poznaçymo çerez ψ ( λ ) pidintehral\nyj vyraz i znajdemo joho maksymum po λ.
Rivnqnnq ψ ′ ( λ ) = 0 zvodyt\sq do rivnqnnq
e s
s s
s
N− −
− −
−( ) = ′( )
( )
= +
+
+
+
−1 1 1
1 1
1
1
1
1
2
1
2 1
1
2
λ λρ λ
ρ λ λ λ
β/
. (32)
Dlq doslidΩennq rivnqnnq (32) skorysta[mosq tym, wo u livij çastyni
mistyt\sq monotonno zrostagça funkciq, vodnoças prava çastyna mistyt\ sumu,
qka monotonno spada[ pry λ → + ∞. Pryxodymo do vysnovku, wo rivnqnnq (32)
ma[ [dynyj korin\ λmax . Ocinymo promiΩok [ λ1 , λ2 ], qkomu naleΩyt\ λmax .
Rozhlqnemo
λ1 : = ( − − )− −N N e s N e s e s1 2/ βγ βγ γ γ βγ γ βγ βγ γ
, (33)
λ2 : = N e sβγ βγ γ
, (34)
de γ =
1
1β +
. Pry c\omu β γ =
β
β + 1
< 1. Pidstavlqgçy λ1 u rivnqnnq (32), zna-
xodymo
N N N e s N e s+
+
−
+
= −
+
+ +
+
−
+
1
1 2 1
1 2
1
1 2 1
1 2 11
1
1 1 1
1
1λ
λ
λ λ λ
λ
λ
βγ βγ γ βγ βγ γ/ /
>
>
N N e s
N e s
− ( − )− − −1 2
1
1
/ βγ βγ γ
βγ βγ γ
λ
>
> ( − − ) = ( )− − − − − −N N e s N e s e s e s1 2 1 1
1
1/ /βγ βγ γ γ βγ γ γ βγ γ βλ .
Dlq druho] toçky ma[mo
N N N N+
+
−
+
= + +
+
−
−
+
1
1 2 1
1
1 2 12
2
2 2 2 2
2
2λ
λ
λ λ λ λ
λ
λ
=
=
N
N e s
N
N e s e sβγ βγ γ
γ βγ γ βλ
λ λ
λ
λ
λ+ −
( + )
−
+
< = ( )− − − −2
2 2
2
2
1 1
2
1
1 2 1
/
,
oskil\ky z (34) vyplyva[, wo pry N > eβ s vykonu[t\sq N > λ2
. OtΩe, λmax ∈
∈ [ λ1 , λ2 ].
Pidstavlqgçy u vyraz ψ ( λ ) vidpovidno λ1 ta λ2
, znaxodymo
ψ ( λ ) < 1
1
1
1
2 1
2
2
2 2
1
1 1
1
1−
+
+
− ( )
+
− −( )
λ λ
β λ β
N
e sexp /
. (35)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
850 S. M. TORBA
Perßyj mnoΩnyk ocing[mo takym çynom:
1
1
1
2 2
12
2 2
2
−
+
< − +
+
+
λ λ
N N
exp <
< exp exp− +
< − ( )( )− −2 2
2
2 2
2 1
1N N
c Nt e
λ λ
γ βγ
,
oskil\ky s < t i, vidpovidno, s–
1 > t–
1
. Analohiçno
1
2 1
2 2
1
1 1
1
1
2
1
λ
β λ ββ γ βγ
+
− ( ) < − ( )( ) ( )− − − −exp exp/e s c Nt e ,
a otΩe (vraxovugçy (31)),
z c c sA x Nt eN
2
1 2 0
2 1 1 1 12 1< ( ) − ( ) ( + )( )− ( + ) − ( + )ρ ββ β βexp / /
.
Lemu dovedeno.
Lema 1 [ prqmog teoremog nablyΩennq. Dovedemo teper lemu, qka [ oberne-
nog teoremog nablyΩennq.
Lema 2. Nexaj
∃ c > 0, c1 > 0 : z
c
N
cNN
2 1
2
1 12≤ (− )( + )exp / β
. (36)
Todi x0 ∈ Cα 〈 nn β
〉, de α vyznaça[t\sq vyrazom
α = e c− + −( + )( + )β β ββ1 1 1
. (37)
Dovedennq. Sxema dovedennq analohiçna sxemi dovedennq teoremy 3. Pobu-
du[mo poslidovnist\ funkcij ϕn ( λ ) taku, wo:
ϕn ( λ ) monotonno zrostagt\ pry n → ∞;
ϕ λ λn d E x x( ) ( )
∞
∫ 0 0
0
, ≤ C ∀N ∈ N;
∃ c′ : ′ ( )( )− −c eexp /2 1 1 1β α λ β ≤ limn → ∞ ϕn ( λ ), de α zada[t\sq vyrazom (37).
Pislq c\oho vkladennq x0 ∈ Cα 〈 nn β
〉 vyplyva[ z (30). PokaΩemo, qk pobudu-
vaty ßukanu poslidovnist\. Rozhlqnemo z N cNN
2 1 1 12β β β/ /exp( + ) ( + )( ). Za umovog
teoremy
z N cN
c
N
N
2 1 1 1 1
2 12β β β
β β
/ /
/exp( + ) ( + )
( + ) ( + )( ) ≤ . (38)
Qk i pry dovedenni teoremy 3, ma[mo spektral\ne zobraΩennq
z N cNN
2 1 1 12β β β/ /exp( + ) ( + )( ) =
=
λ
λ λ
β β β
λ+
+
( ) ( )( + ) ( + )
∞
∫ 1
1
2 1
21 1 1
0 0
0
N cN d E x x/ /exp , .
Poznaçymo çerez ψn ( λ ) pidintehral\nyj vyraz i rozhlqnemo poslidovnist\ fun-
kcij
ϕn ( λ ) = max{ }( ) =ψ λk k
n
1.
Oçevydno, wo ϕn ( λ ) monotonno zrostagt\ pry n → ∞. ObmeΩenist\ intehraliv
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
PRQMI TA OBERNENI TEOREMY NABLYÛENYX METODIV … 851
ϕ λ λn d E x x( ) ( )
∞
∫ 0 0
0
, ≤ C, n ∈ N,
vyplyva[ z oçevydno] ocinky
ϕn ( λ ) = max{ }( ) =ψ λk k
n
1 ≤ ψ λk
k
n
( )
=
∑
1
,
umovy (38) ta zbiΩnosti rqdu
1
2 11kk
n
( + ) ( + )=∑ β β/ . OtΩe, perßi dvi umovy dlq po-
slidovnosti ϕn vykonano. Dovedemo j tretg. Z umovy (37) znaxodymo
c = α βγ βγ− − ( + )e 1 , γ =
1
1β +
.
Rozhlqnemo dovil\ne λ > 1. Poznaçymo
N = [ ]( + ) − −λ αβ β β1 1 1/ /e + 1,
de [ ⋅ ] poznaça[ cilu çastynu çysla. Todi, vykorystovugçy vlastyvist\ cilo]
çastyny [ x ] + 1 > x, otrymu[mo
exp exp/( ) = ( ) ( + )( + ) − −( )2 2 11 1 1cN N eβ γ βγα β >
> exp /( )( ) ( + )− −2 11 1 1λα ββe , (39)
λ
λ
λ
λ+
= − + ( + )
+
1
1
2 2
2 2N
Nexp ln .
Vykorystovugçy nerivnosti ln ( 1 + 1 / λ ) < 1 / λ pry λ > 1 ta [ x ] ≤ x, z ostan-
n\oho vyrazu oderΩu[mo
exp ln exp /− + ( + )
≥ − ( )( )− −λ
λ
λα β1
2 2 2 1 1 1N e (40)
i ostanng ocinku
N e
c
β β β β β β β
λ
λ α
λ
/ / / /
˜
( + ) ( + ) − − ( + )
+
> ( )
+
>
1 1 1 1 1
2 1 2 1
> 0. (41)
Vraxovugçy (39) – (41), pryxodymo do vysnovku, wo
ϕ ( λ ) = lim
n
n
→ ∞
( )ϕ λ ≥ ϕN ( λ ) ≥ ψN ( λ ) > ˜ exp /c e( )− −( )2 1 1 1β λα β
,
a ce j harantu[ vkladennq x0 ∈ Cα 〈 nn β
〉.
Lemu dovedeno.
Dovedennq teoremy 6. Pry t ∈ ( 0, ∞ ) vyraz ct = t e− ( + ) − ( + )( + )1 1 1 1/ /β β β β
nabuva[ vsix moΩlyvyx znaçen\ z ( 0, ∞ ). Tak samo j α = e c− + −( + )( + )β β ββ1 1 1
nabuva[ vsix dodatnyx dijsnyx znaçen\ pry c ∈ ( 0, ∞ ).
Krim toho, nevaΩko pomityty, wo z nerivnosti
z c cNN
2
1
1 12≤ (− )( + )exp / β
vyplyva[
∀ c′ > c, ∃ ′c1 > 0 : z
c
N
c NN
2 1
2
1 12≤ ′ (− ′ )( + )exp / β
,
a tomu teorema 6 vyplyva[ z lem 1 ta 2.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
852 S. M. TORBA
1. Horbaçuk M. L., Horodeckyj V. V. O polynomyal\nom pryblyΩenyy reßenyj dyfferency-
al\no-operatorn¥x uravnenyj v hyl\bertovom prostranstve // Ukr. mat. Ωurn. – 1984. – 36,
# 4. – S. 500 – 502.
2. Kaßpirovs\kyj O. I., Mytnyk G. V. Aproksymaciq rozv’qzkiv operatorno-dyferencial\nyx
rivnqn\ za dopomohog operatornyx polinomiv // Tam Ωe. – 1998. – 50, # 11. – S. 1506 – 1516.
3. Arov D. Z., Gavrilyuk I. P. A method for solving initial value problems for linear differential
equations in Hilbert space based on the Cayley transform // Numer. Func. Anal. and Optim. – 1993.
– 14, # 5, 6. – P. 456 – 473.
4. 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.
5. Gavrilyuk I. P., Makarov V. L. Representation and approximation of the solution of an initial value
problem for a first order differential equation in Banach space // J. Anal. and Appl. (ZAA). – 1996. –
15, # 2. – P. 495 – 527.
6. Makarov V. L., Vasylyk V. B., Rqbyçev V. L. Neuluçßaem¥e po porqdku ocenky skorosty
sxodymosty metoda preobrazovanyq Kπly dlq pryblyΩenyq operatornoj πksponent¥ //
Kybernetyka 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. Gorbachuk M. L., Gorbachuk V. I. Boundary-value problems for operator-differential equations. –
Dordrecht: Kluwer, 1991. – 364 p.
9. Suetyn P. K. Klassyçeskye ortohonal\n¥e mnohoçlen¥. – M.: Nauka, 1979. – 416 s.
10. Horbaçuk V. Y., Knqzgk A. V. Hranyçn¥e znaçenyq reßenyj dyfferencyal\no-operatorn¥x
uravnenyj // Uspexy mat. nauk. – 1989. – 44, v¥p. 3. – S. 55 – 90.
11. Rad¥no Q. V. Prostranstva vektorov πksponencyal\noho typa // Dokl. AN BSSR. – 1983. –
27, # 9. – S. 215 – 229.
12. Horbaçuk M. L. Pro analityçni rozv’qzky dyferencial\no-operatornyx rivnqn\ // Ukr. mat.
Ωurn. – 2000. – 52, # 5. – S. 596 – 607.
OderΩano 16.03.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
|
| id | umjimathkievua-article-3349 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:40:54Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ad/51481b1c0f719689d33fc34bf3a3c6ad.pdf |
| spelling | umjimathkievua-article-33492020-03-18T19:51:58Z Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem Прямі та обернені теореми наближених методів розв'язування абстрактної задачі Коші Torba, S. M. Торба, С. М. We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator A, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean. Рассмотрен приближенный метод решения задачи Коши для дифференциально-операторного уравнения, основанный на разложении экспоненты по ортогональным многочленам Лагера. Для начального значения конечной гладкости относительно оператора A доказаны прямая и обратная теоремы теории приближения в среднем, приведены примеры неулучшаемости соответствующих оценок в этих теоремах. Для целых векторов экспоненциального типа установлена экспоненциальная скорость сходимости, для классов Жевре — субэкспоненциальная, а также характеризация соответствующих классов в терминах скорости сходимости в среднем приближения. Institute of Mathematics, NAS of Ukraine 2007-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3349 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 6 (2007); 838–852 Український математичний журнал; Том 59 № 6 (2007); 838–852 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3349/3442 https://umj.imath.kiev.ua/index.php/umj/article/view/3349/3443 Copyright (c) 2007 Torba S. M. |
| spellingShingle | Torba, S. M. Торба, С. М. Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem |
| title | Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem |
| title_alt | Прямі та обернені теореми наближених методів розв'язування абстрактної задачі Коші |
| title_full | Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem |
| title_fullStr | Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem |
| title_full_unstemmed | Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem |
| title_short | Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem |
| title_sort | direct and inverse theorems of approximate methods for the solution of an abstract cauchy problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3349 |
| work_keys_str_mv | AT torbasm directandinversetheoremsofapproximatemethodsforthesolutionofanabstractcauchyproblem AT torbasm directandinversetheoremsofapproximatemethodsforthesolutionofanabstractcauchyproblem AT torbasm prâmítaoberneníteoreminabliženihmetodívrozv039âzuvannâabstraktnoízadačíkoší AT torbasm prâmítaoberneníteoreminabliženihmetodívrozv039âzuvannâabstraktnoízadačíkoší |