Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method
For an arbitrary self-adjoint operator B in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator B, the rate of convergence to zero of its best approxima...
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| Дата: | 2005 |
|---|---|
| Автори: | , , , , , |
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| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3629 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509751231643648 |
|---|---|
| author | Gorbachuk, M. L. Hrushka, Ya. I. Torba, S. M. Горбачук, М. Л. Грушка, Я. І. Торба, С. М. |
| author_facet | Gorbachuk, M. L. Hrushka, Ya. I. Torba, S. M. Горбачук, М. Л. Грушка, Я. І. Торба, С. М. |
| author_sort | Gorbachuk, M. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:32Z |
| description | For an arbitrary self-adjoint operator B in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the operator B. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space. |
| first_indexed | 2026-03-24T02:46:05Z |
| format | Article |
| fulltext |
UDK 517.9
M. L. Horbaçuk, Q. I. Hrußka, S. M. Torba
(In-t matematyky NAN Ukra]ny, Ky]v)
PRQMI J OBERNENI TEOREMY
V TEORI} NABLYÛEN| METODOM RITCA*
For an arbitrary self-adjoint operator B in the Hilbert space �, we give direct and inverse theorems
that establish relations between the degree of smoothness of a vector x ∈� with respect to the operator
B, the order of tending to zero of the best approximation of this vector by exponential-type entire
vectors of operator B, and k-module of continuity of x with respect to B. The results obtained are
used in finding a priori estimates of the Rietz approximation of solutions of operator equations in the
Hilbert space.
Dlq dovil\noho samosprqΩenoho operatora B u hil\bertovomu prostori � navedeno prqmi j
oberneni teoremy, wo vstanovlggt\ zv’qzok miΩ stepenem hladkosti vektora x ∈� vidnosno
operatora B, porqdkom prqmuvannq do nulq joho najkrawoho nablyΩennq cilymy vektoramy
eksponencial\noho typu operatora B i k-modulem neperervnosti vektora x wodo operatora B. Re-
zul\taty zastosovano do znaxodΩennq apriornyx ocinok nablyΩenyx za Ritcom rozv’qzkiv opera-
tornyx rivnqn\ u hil\bertovomu prostori.
V teori] nablyΩen\ periodyçnyx funkcij dobre vidomi prqmi j oberneni teore-
my, wo vstanovlggt\ zv’qzok miΩ stepenem hladkosti funkci] vidnosno opera-
tora dyferencigvannq i porqdkom prqmuvannq do nulq ]] najkrawoho nably-
Ωennq tryhonometryçnymy polinomamy.
Meta ci[] roboty — dovesty podibni teoremy u vypadku nablyΩennq vektora,
hladkoho dlq samosprqΩenoho operatora u hil\bertovomu prostori, cilymy vek-
toramy eksponencial\noho typu c\oho operatora i zastosuvaty ci teoremy do
znaxodΩennq apriornyx ocinok nablyΩennq za Ritcom rozv’qzkiv operatornoho
rivnqnnq.
1. Nexaj B — zamknenyj linijnyj operator zi wil\nog oblastg vyznaçennq
D( )B u separabel\nomu hil\bertovomu prostori � nad polem kompleksnyx
çysel.
Poznaçymo çerez C B∞( ) mnoΩynu vsix neskinçenno dyferencijovnyx vek-
toriv operatora B :
C B∞( ) =
n
nB
∈N0
∩ D ( ) , de N0 = N ∪ { }0 .
Dlq çysla α > 0 poklademo
�α α( ) ( ) ( )B x C B c c x k B x ck k= ∈ ∃ = > ∀ ∈ ≤{ }∞ 0 0N .
MnoΩyna �
α( )B [ banaxovym prostorom wodo normy
x
B x
B
n
n
n�α
α( ) sup=
∈N0
.
Todi �( )B = ∪
α
α
>0
� ( )B — linijnyj lokal\no-opuklyj prostir vidnosno topo-
lohi] induktyvno] hranyci banaxovyx prostoriv �
α( )B :
�( )B = lim ( )ind
α
α
→∞
� B .
Elementy prostoru �( )B nazyvagt\sq cilymy vektoramy eksponencial\noho
* Çastkovo pidtrymano DerΩavnym fondom fundamental\nyx doslidΩen\ Ukra]ny (proekt
01.07/027) ta CRDF (proekt UM1-2567-OD-03).
© M. L. HORBAÇUK, Q. I. HRUÍKA, S. M. TORBA, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5 633
634 M. L. HORBAÇUK, Q. I. HRUÍKA, S. M. TORBA
typu operatora B. Pid typom σ( , )x B vektora x B∈�( ) rozumitymemo çyslo
σ( , )x B =
inf : ( )α α> ∈{ }0 x B� = lim sup
n
n nB x
→∞
1
.
Skriz\ u podal\ßomu operator B samosprqΩenyj v � , a E( )∆ — joho spekt-
ral\na mira.
Nexaj G( )λ — majΩe skriz\ skinçenna vymirna funkciq na R. Pid funk-
ci[g G B( ) vid operatora B budemo rozumity
G B( ) : =
−∞
∞
∫ G dE( ) ( )λ λ .
Qk pokazano v [1], dlq bud\-qkoho α > 0 �
α( )B = E −[ ]( )α α, �.
Zhidno z [2] poklademo
ωk t x B, ,( ) = sup
0≤ ≤τ
τ
t
k x∆ , k ∈N , (1)
de
∆h
k = U h k( ) −( )I =
j
k
k j
k
jC U jh
=
−∑ −
0
1( ) ( ), (2)
k ∈N0, h ∈R ∆h h0 1≡ ∈( )+, R ,
a U h( ) = exp ih B( ) — hrupa unitarnyx operatoriv v � z heneratorom i B [3].
Z vyznaçennq ωk t x B, ,( ) bezposeredn\o vyplyva[, wo pry k ∈N :
1) ωk x B0, ,( ) = 0;
2) pry fiksovanomu x funkciq ωk t x B, ,( ) ne spada[ na R+ = 0, ∞[ ) ;
3) ω αk t x B, ,( ) ≤ ( ) , ,1 + ( )α ωk
k t x B , α, t > 0;
4) pry fiksovanomu t ∈ +R funkciq ωk t x B, ,( ) neperervna po x.
Dali bude vstanovleno nerivnist\ typu Bernßtejna – Nikol\s\koho.
Lema 1. Nexaj G( )λ — nevid’[mna parna funkciq na R, nespadna na R+ ,
x B∈�( ) i σ( , )x B ≤ α. Todi
∆h
k k kG B x h G x( ) ( )≤ α α , h > 0, k ∈N0. (3)
Dovedennq. Oskil\ky σ( , )x B ≤ α i 1
2
− ei h kλ = 4
2
2 2sin k hλ
≤ λ2 2k kh ,
λ ∈R , to na osnovi operacijnoho çyslennq dlq operatora B ma[mo
∆h
k G B x( )
2
=
−
∫ −( ) ( )
α
α
λ
λλ1
2 2e G d E x xi h k ( ) , ≤
≤ h G d E x xk k2 2 2
−
∫ ( )
α
α
λλ λ( ) , ≤ h G xk k2 2 2 2α α( ) . (4)
Lemu dovedeno.
Pry k = 0 z lemy 1 oderΩu[mo
G B x G x( ) ( )≤ α . (5)
Naslidok 1. Za umov lemy 1 stosovno x ta σ( , )x B
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
PRQMI J OBERNENI TEOREMY V TEORI} NABLYÛEN| METODOM RITCA 635
∆h
k k kx h x≤ α , h ≥ 0.
Dlq dovedennq dosyt\ v lemi 1 vzqty G( )λ ≡ 1, λ ∈R .
Qkwo � = L2 0 2, π[ ]( ), Bx t( )( ) = i x t′( ) , D ( )B = { x( )⋅ ∈ W2
1 0 2, π[ ]( ): x( )0 =
= x( )2π }, de W2
1 0 2, π[ ]( ) — prostir Sobol[va, to �( )B zbiha[t\sq z mnoΩynog
usix tryhonometryçnyx mnohoçleniv, σ( , )x B — stepin\ mnohoçlena x, �
α( )B
— mnoΩyna vsix tryhonometryçnyx mnohoçleniv, stepeni qkyx ne perevywugt\
α, U h x t( ) ( )( ) = ˜ ( )x t h+ , ωk t x B, ,( ) — k-j modul\ neperervnosti funkci] x t( ), a
nerivnist\ (3) pry G( )λ = λm , k = 0 peretvorg[t\sq na nerivnist\ typu
S.RBernßtejna u prostori L2 0 2, π[ ]( ) [4 ] (tut pid ˜( )x t rozumi[t\sq 2π-pe-
riodyçne prodovΩennq funkci] x t( )).
Dlq dovil\noho x ∈� poklademo, dotrymugçys\ [5, 6],
�
�
r
y B y B r
x B x y( , ) inf
( ): ( , )
= −
∈ ≤σ
, r > 0,
tobto �r x B( , ) — najkrawe nablyΩennq elementa x cilymy vektoramy y eks-
ponencial\noho typu operatora B, dlq qkyx σ( , )y B ≤ r. Pry fiksovanomu x
�r x B( , ) ne zrosta[ i �r x B( , ) → 0, r → ∞. Zrozumilo, wo
�r x B( , ) = x E r r x− −[ ]( ), = x F r x− [ ]( )0, ,
de F( )∆ — spektral\na mira operatora B = B B*
.
Teorema 1. Nexaj G( )λ zadovol\nq[ umovy lemy 1. Todi dlq dovil\noho
x G B∈ ( )D ( )
∀ ∈k N �r x B( , ) ≤
k
G r r
G B x Bk k
+
1
2 ( )
, ( ) ,ω π
, r > 0. (6)
Dovedennq. Vykorystovugçy spektral\ne zobraΩennq dlq operatora B i
monotonnist\ funkci] G( )λ , oderΩu[mo
ωk t G B x B2 , ( ) ,( ) = sup ( )
0
2
≤ ≤
−( )
τ
τ
t
i B k
e G B xI ≥ e G B xitB k
−( )I ( )
2
=
=
−∞
∞
∫ − ( )e G d E x xi t kλ
λλ1
2 2( ) , = 2 1 2k kt G d E x x
−∞
∞
∫ −( ) ( )cos ( ) ,λ λ λ ≥
≥ 2 12k
r
kG r t d E x x( ) cos ,
λ
λλ
≥
∫ −( ) ( ).
Zafiksu[mo r > 0 i viz\memo t
r
∈
0, π
. Todi sin rt ≥ 0. PomnoΩymo obydvi ças-
tyny oderΩano] vywe nerivnosti na sin rt i prointehru[mo po t vid 0 do
π
r
.
Todi
0
2
π
ω
/
, ( ) , sin
r
k t G B x B rt dt∫ ( ) ≥ 2 12
0
k
r
r
kG r t rt d E x x dt( ) cos sin ,
/π
λ
λλ∫ ∫
≥
−( ) ( ) =
= 2 12
0
k
r
r
kG r t rt dt d E x x( ) cos sin ,
/
λ
π
λλ
≥
∫ ∫ −( )
( ) . (7)
Oskil\ky funkciq ωk t G B x B2 , ( ) ,( ) [ monotonno nespadnog, to
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
636 M. L. HORBAÇUK, Q. I. HRUÍKA, S. M. TORBA
0
2
π
ω
/
, ( ) , sin
r
k t G B x B rt dt∫ ( ) ≤
≤
0
2
π
ω π
/
, ( ) , sin
r
k r
G B x B rt dt∫
= 2 2
r r
G B x Bkω π , ( ) ,
. (8)
Vykorystovugçy nerivnist\
0
1
π
θ∫ −( )cos sint t dtk ≥ 2
1
1k
k
+
+
, θ ≥ 1, k ∈N (9)
(dyv. [7]), na osnovi (7) i (8) pryxodymo do spivvidnoßennq
2 2
r r
G B x Bkω π , ( ) ,
≥ 2 1 2
1
2
1
k
r
k
G r
r k
d E x x( ) ,
λ
λ
≥
+
∫ +
( ) =
=
2
1
2 1 2
2
k
r
G r
r k
x B
+
+
( )
( )
( , )� , (10)
rivnosyl\noho (6).
Teoremu dovedeno.
Pry G( )λ = λ m, λ ∈R , m > 0 z teoremy 1 oderΩu[mo takyj naslidok.
Naslidok 2. Nexaj x B m∈ ( )D , m > 0. Todi dlq bud\-qkoho k ∈N
�r x B( , ) ≤
k
r r
B x Bk m k
m+
1
2
ω π , , , r > 0. (11)
U vypadku, koly B — operator dyferencigvannq z periodyçnymy krajovymy
umovamy u prostori � = L2 0 2, π[ ]( ), tobto ( )( )Bx t = ix t′( ) , D ( )B = { ⋅x( ) ∈
∈ W2
1 0 2, π[ ]( ): x( )0 = x( )2π }, nerivnist\ (11) dlq k = 1 navedeno v [8], a dlq
dovil\noho k ∈N — v roboti [7].
Sformulg[mo teper obernenu teoremu u vypadku nablyΩennq vektora x ci-
lymy vektoramy eksponencial\noho typu operatora B.
Teorema 2. Nexaj ω( )t — funkciq typu modulq neperervnosti taka, wo:
1) ω( )t — neperervna i nespadna pry t ∈ +R ;
2) ω( )0 0= ;
3) ∃ >c 0 ∀ >t 0 ω ω( ) ( )2t c t≤ ;
4)
0
1
∫ < ∞ω( )t
t
dt .
Nexaj takoΩ funkciq G( )λ [ parnog, nevid’[mnog i nespadnog pry λ ≥ 0,
pryçomu sup ( )
( )λ
λ
λ>
< ∞
0
2G
G
.
Qkwo dlq x ∈� isnu[ m > 0 take, wo
�r x B( , ) < m
G r r( )
ω 1
, r > 0, (12)
to x G B∈ ( )D ( ) i dlq koΩnoho k ∈N isnu[ stala mk > 0 taka, wo
ωk t G B x B, ( ) ,( ) ≤ m t d dk
k
t
k
t1
1
0
∫ ∫+ +
ω τ
τ
τ ω τ
τ
τ( ) ( )
, 0 < t ≤ 1
2
. (13)
Dovedemo spoçatku take tverdΩennq.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
PRQMI J OBERNENI TEOREMY V TEORI} NABLYÛEN| METODOM RITCA 637
Lema 2. Nexaj funkciq ω( )t zadovol\nq[ umovy 1 – 3 teoremy 2. Qkwo
dlq x ∈� isnu[ m > 0 take, wo
�r x B( , ) < m
r
ω 1
, r > 0, (14)
to dlq koΩnoho k ∈N isnu[ stala ck > 0 taka, wo
ω ω τ
τ
τk k
k
t
kt x B c t d( , , )
( )≤ ∫ +
1
1 , 0 < t ≤ 1
2
. (15)
Dovedennq. Z umovy (14) vyplyva[ isnuvannq poslidovnosti ui i{ } =
∞
0 cilyx
vektoriv eksponencial\noho typu takyx, wo σ( , )u Bi ≤ 2i
, a
x u mi i− ≤
ω 1
2
. (16)
Viz\memo dovil\ne h ∈
0 1
2
, i vyberemo çyslo N tak, wob
1
2 1N + < h ≤ 1
2N .
Iz nerivnosti (16) oderΩu[mo
u uj j− −1 ≤ u xj − + x uj− −1 ≤
≤ m jω 1
2
+ m jω 1
2 1−
≤ 2 1
2 1m jω −
≤ 2 1
2
cm jω
. (17)
V sylu monotonnosti ω( )t ma[mo
2
1 2
1 2
1
1
k
k
j
j
u
u
du
/
/
( )
−
∫ +
ω
≥ 2 1
2
1
1 2
1 2
1
1
k
j k
j
j
u
duω
−
∫ +
/
/
=
= 2 1
2
2 1
kj
j
k
k
ω
−( ) ≥ 2 1
2
kj
jω
. (18)
Oskil\ky σ u u Bj j−( )−1, ≤ 2 j
i σ( , )u B0 ≤ 1, to za naslidkom 1
∆h
k ku h u0 0≤ ,
∆h
k
j j
k j k
j ju u h u u−( ) ≤ ⋅ −− −1 12( ) , j ≥ 1.
Formuly (16) – (18) zumovlggt\ nerivnosti
∆h
k
j j
k kj
j
k k
ku u cmh cmh
u
h
du
j
j
−( ) ≤ ⋅
≤−
+
+
−
∫1
1
1 2
1 2
12 2 1
2
2
1
ω ω
/
/
( )
ta
∆h
k
N
ihB k
N
k
N
k
Nx u e x u x u m−( ) ≤ +( ) − ≤ − ≤
1 2 2 1
2
ω ,
vykorystovugçy qki, pryxodymo do spivvidnoßen\
∆h
k x = ∆ ∆ ∆h
k
j
N
h
k
j j h
k
Nu u u x u0
1
1+ − + −
=
−∑ ( ) ( ) ≤
≤ h uk
0 + 2 1
1 1 2
1 2
1
1
k k
j
N
kcmh
u
u
du
j
j
+
=
+∑ ∫
−
/
/
( )ω
+ 2 1
2
k
Nmω
≤
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
638 M. L. HORBAÇUK, Q. I. HRUÍKA, S. M. TORBA
≤ h uk
0 + 2 1
1 2
1
1
k k
kcmh
u
u
du
N
+
+∫
/
( )ω
+ 2k m hω ( ) ≤
≤ h uk
0 + 2 1
1
1
k k
h
kcmh
u
u
du+
+∫ ω( )
+ 2k cm hω ( ) =
= h u cmh
u
u
du cm k
h
h
u
duk k k
h
k
k
k
h
k0
1
1
1
1
12 2
1
+ +
−
+
+ +∫ ∫ω ω( ) ( )
≤
≤ c h
u
u
duk
k
h
k
1
1∫ +
ω( )
,
de
c
u
u
u
du
cm k
k
k
k
k
=
−
∫ +
max ( ) ,
/
0
1 2
1
1
2
1 1
2
ω .
Lemu dovedeno.
ZauvaΩennq 1. Qk vydno z dovedennq, lema [ virnog za dewo slabkißo]
umovy, niΩ vymaha[t\sq v teoremi, a same: dostatn\o, wob dlq elementa x ∈�
isnuvala xoça b odna poslidovnist\ uj j{ } =
∞
0
taka, wo
σ( , )u Bj
j≤ 2 i ∀ ∈j N x u mj j− ≤
ω 1
2
.
Dovedennq teoremy 2. Zavdqky (12) isnu[ poslidovnist\ un n{ } =
∞
0 taka, wo
σ( , )u Bn ≤ 2n
i
x u m
G
n n n− ≤
( )2
1
2
ω , n ∈N0. (19)
Z nerivnosti (19) ta umov 1, 2 teoremy vyplyva[, wo x un− → 0, n → ∞ , a
tomu vektor x moΩna podaty u vyhlqdi
x u u u
k
k k= + −
=
∞
−∑0
1
1( ).
Oskil\ky σ u u Bk k−( )−1, ≤ 2k , k ∈N , to, beruçy do uvahy (5), oderΩu[mo
G B u G B uk k( ) ( )− −1 ≤ G u uk
k k( )2 1− − ≤
≤ G x u x uk
k k( )2 1− + −( )− ≤ G m
G
m
G
k
k k k k( )
( ) ( )
2
2
1
2 2
1
21 1ω ω
+
− − ≤
≤
2 2
2
1
21 1
G m
G
k
k k
( )
( )− −
ω ≤ 2 1
21cc m kω
≤
2
2
1
2
2 1
cc m u
u
du
k
k
ln
( )
−
− +
∫ ω
,
de çerez c1 poznaçeno sup ( )
( )λ
λ
λ>0
2G
G
. Tomu rqd
k k kG B u G B u=
∞
−∑ −( )1 1( ) ( ) zbiha-
[t\sq. Zamknenist\ operatora G B( ) zumovlg[ vklgçennq x G B∈ ( )D ( ) i riv-
nist\
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
PRQMI J OBERNENI TEOREMY V TEORI} NABLYÛEN| METODOM RITCA 639
G B x G B u G B u G B u
k
k k( ) ( ) ( ) ( )= + −( )
=
∞
−∑0
1
1 .
Zvidsy vyplyva[
G B x G B uj( ) ( )− ≤
k j
k kG B u G B u
= +
∞
−∑ −
1
1( ) ( ) ≤
≤
2
2
1
1 2
2 1
cc m u
u
du
k j k
k
ln
( )
= +
∞
∑ ∫
−
− +
ω
=
2
2
1
0
2cc m u
u
du
j
ln
( )
−
∫ ω
= : c̃ jΩ 2−( ), j ∈N ,
de ˜ :
ln
c
cc m= 2
2
1
, a
Ω( ) :
( )
t
u
u
du
t
= ∫
0
ω
.
NevaΩko perekonatys\, wo funkciq Ω( )t ma[ taki vlastyvosti:
1) Ω( )t [ neperervnog i monotonno nespadnog;
2) Ω( )0 = 0;
3) pry t > 0
Ω Ω( )
( ) ( ) ( )
( )2
2
0
2
0
2
0
2t
u
u
du
u
u
du c
u
u
du c t
t t t
= = ≤ =∫ ∫ ∫ω ω ω
.
Tomu, poklavßy v lemi 2 ω( )t = Ω( )t i vraxuvavßy zauvaΩennq 1, oderΩymo
ωk t G B x B, ( ) ,( ) ≤ c t
u
u
duk
k
t
k
1
1∫ +
Ω( )
=
=
c t
k
u
u
u
u
duk
k
k
t
t
kΩ( )
( )1
1
1
1+
∫ +
ω
≤ m t
u
u
du
u
u
duk
k
t
k
t1
1
0
∫ ∫+ +
ω ω( ) ( )
.
Teoremu dovedeno.
Teorema 2 pokazu[, wo u vypadku, koly ω( )t = tα , t ≥ 0, α > 0, i �r x B( , ) =
= O
r
1
α
,
ωk t x B( , , ) =
O t k
O t t k
O t k
k
k
( ) ,
ln ,
( ) .
pry
pry
pry
<
( ) =
>
α
α
αα
2. Rozhlqnemo rivnqnnq
Ax y= , (20)
de A — dodatno vyznaçenyj samosprqΩenyj operator z dyskretnym spektrom,
y ∈�, x A∈D ( ) — ßukanyj rozv’qzok rivnqnnq (20). Çerez � + poznaçymo po-
povnennq mnoΩyny D ( )A za normog ⋅ + , porodΩenog skalqrnym dobutkom
( , ) ( , )x y Ax y+ = .
Za umov na operator A, zaznaçenyx vywe, rivnqnnq (20) ma[ [dynyj rozv’qzok
x A∈D ( ) i, zhidno z pryncypom Dirixle [9], znaxodΩennq c\oho rozv’qzku ekvi-
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
640 M. L. HORBAÇUK, Q. I. HRUÍKA, S. M. TORBA
valentne vidßukanng vektora u A∈D ( ), na qkomu funkcional
F z Az z y z( ) ( , ) Re( , )= − 2 ,
zadanyj na D ( )A , dosqha[ svoho minimumu.
Nexaj ek k{ } =
∞
1 — povna linijno nezaleΩna systema vektoriv iz D ( )A (tak
zvana koordynatna systema) i
H n ne e= …{ }l o. . , ,1 .
Poznaçymo çerez xn vektor, na qkomu F z( ) nabuva[ minimal\noho znaçennq na
H n . Vektor xn nazyva[t\sq nablyΩenym za Ritcom rozv’qzkom rivnqnnq (20).
Qk vidomo, nezaleΩno vid vyboru koordynatno] systemy poslidovnist\ xn
zbiha[t\sq do x u prostori � + (a tym paçe, i v � ). Wo stosu[t\sq nev’qzky
Rn = Ax yn − , to vona ne zavΩdy prqmu[ do nulq v �. Ale qkwo koordynatnu
systemu ek k{ } =
∞
1 vybraty tak, wob vona utvorgvala ortonormovanyj vlasnyj
bazys qkoho-nebud\ samosprqΩenoho dodatno vyznaçenoho operatora B, sporid-
nenoho z A v tomu rozuminni, wo D ( )A = D ( )B , to Rn → 0, n → ∞ (dyv. [9]), a
tomu j velyçyny rn = x xn − + prqmugt\ do nulq pry n → ∞ . Prote do-
slidΩennq povedinky na neskinçennosti cyx velyçyn, wo zaleΩat\ vid vyboru
ek k{ } =
∞
1 i pravo] çastyny rivnqnnq (20), vyqvylos\ dosyt\ vaΩkog zadaçeg, ne
rozv’qzanog j ponyni. Deqki okremi rezul\taty dlq operatoriv, porodΩenyx
krajovymy zadaçamy dlq zvyçajnyx dyferencial\nyx rivnqn\, bulo oderΩano v
çyslennyx robotax bahat\ox avtoriv (dyv. ohlqd [10]). Wo Ω stosu[t\sq ab-
straktnoho vypadku, to deqki çastynni sytuaci] rozhlqnuto v [11]. U roboti [6]
uperße otrymano prqmi j oberneni teoremy za umovy, wo x C B∈ ∞( ), a takoΩ
dano ocinky velyçyny Rn , qkwo hladkist\ vektora x [ skinçennog, tobto
x Bk∈D ( ) . NyΩçe dlq x Bk∈D ( ) povnistg xarakteryzu[t\sq velyçyna rn .
Nadali prypuskatymemo, wo vykonugt\sq taki umovy:
1°. Operator A [ samosprqΩenym dodatno vyznaçenym.
2°. Koordynatnog systemog v metodi Ritca [ ortonormovanyj bazys samo-
sprqΩenoho dodatno vyznaçenoho operatora B z dyskretnym i prostym spekt-
rom Be ek k k=( )λ , sporidnenoho z A.
Poznaçymo çerez xn nablyΩenyj za Ritcom rozv’qzok rivnqnnq (20) vidnos-
no koordynatno] systemy ek k{ } =
∞
1 i poklademo x̃n =
k
n
k kx e e=∑ 1
( , ) . Oskil\ky
operatory A i B dodatno vyznaçeni i samosprqΩeni i D ( )A = D ( )B , to z
nerivnosti Hajnca [12] vyplyva[, wo dlq dovil\noho α ∈( , )0 1 D ( )Aα = D ( )Bα
,
a otΩe, operatory B A1 2 1 2/ /−
, A B1 2 1 2/ /−
vyznaçeni j obmeΩeni na vs\omu pro-
stori � i dlq dovil\noho x A∈D ( )
c1
1−
+x ≤ x + ≤ c2 x + , (21)
de x + = B x1 2/ , c1 = B A1 2 1 2/ /− , c2 = A B1 2 1 2/ /−
.
Lema 3. Dlq bud\-qkyx n ∈N i x B∈D ( ) spravdΩu[t\sq nerivnist\
x xn− +˜ ≤ x xn− + ≤ c3 x xn− +˜ , (22)
de c3 = B A1 2 1 2/ /− A B1 2 1 2/ /−
.
Dovedennq. Oskil\ky B x e e
k
n
k k
1 2
1
/ ( , )=∑( ) =
k
n
k kB x e e=∑ ( )1
1 2/ , , to
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
PRQMI J OBERNENI TEOREMY V TEORI} NABLYÛEN| METODOM RITCA 641
x xn− +˜ = B x x e e
k
n
k k
1 2
1
/ ( , )−
=
∑ =
= B x B x e e
k
n
k k
1 2
1
1 2/ / ,− ( )
=
∑ ≤ B x B xn
1 2 1 2/ /− = x xn− + .
Beruçy do uvahy, wo nablyΩennq za Ritcom xn [ najkrawym nablyΩennqm
vektora x u normi ⋅ + , ma[mo
x xn− + = B x xn
1 2/ ( )− ≤
≤ B A1 2 1 2/ /− A x xn
1 2/ ( )− = c1 x xn− + ≤ c1 x xn− +˜ =
= c1
1 2A x xn
/ ( ˜ )− ≤ c c1 2
1 2B x xn
/ ( ˜ )− = c3 x xn− +˜ .
Lemu dovedeno.
Vraxovugçy, wo
�λn
B x B x xn
1 2/ , ˜( ) = − +
i
� �λ λ ηn n
B x B B x B1 2 1 2/ /, ,( ) = ( )+ , 0 < η < λn +1 – λn ,
nerivnosti (21), (22) i teoremu 1 z G( )λ = λ α −1 2/ , α ≥ 1, pryxodymo do nastup-
noho tverdΩennq.
Teorema 3. Qkwo x B∈D( )α
, α ≥ 1, to dlq bud\-qkoho k ∈N
x xn− + ≤ c0
1
1 2
1
1
2
k
B x Bk
n
k
n
+
+
−
+λ
ω π
λα
α
/ , , ,
de c c c0 2 3= , c2 i c3 — stali, wo fihurugt\ v nerivnostqx (21), (22).
Oskil\ky pry x B∈D( )α
ω π
λ
α
k
n
B x B
+
1
, , → 0, n → ∞,
to dlq x B∈D( )α
lim /
n n nx x
→∞ +
−
+− =λα
1
1 2 0 . (23)
Navedemo pryklady operatoriv A i B, dlq qkyx rivnist\ (23) pry α > 1 ne
zumovlg[ vklgçennq x B∈D( )α
. Poklademo � = L2 0, π[ ]( ), A = B = − d
dt
2
2 ,
D ( )A = D ( )B = { x( )⋅ ∈ W2
2 0, π[ ]( ) : x( )0 = x( )π = 0}, λk B( ) = k2 , ek =
= 2
π
sin kt , x = x t( ) = 2
2π k kx kt=
∞∑ sin , de xk = 1
2 1 2 1 2k kα + / /ln
, k ∈N \{}1 .
Rivnist\
k
k
k k=
∞
+∑
2
4
4 1
α
α ln
=
k k k=
∞
∑
2
1
ln
= ∞
pokazu[, wo x B∉D ( )α
. Ale oskil\ky
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
642 M. L. HORBAÇUK, Q. I. HRUÍKA, S. M. TORBA
x xn− +
2
= x xn− +˜ 2
=
k n
k
k k= +
∞
+∑
1
2
4 1α ln
≤
≤
1
1
1
4 1ln( )n t
dt
n+
∞
−∫ α =
1
4 2 14 2( ) ln( )α α− +−n n
,
to
lim ( )/
n n nB x x
→∞
−
+−λα 1 2 ≤ lim
ln( )n
n
n n→∞
−
−− +
2 1
2 1
1
4 2
1
1
α
αα
= 0.
Prote, qk vyplyva[ z teoremy 3, nerivnosti (21) i lemy 3, virnog [ taka te-
orema.
Teorema 4. Nexaj ω( )t zadovol\nq[ umovy teoremy 2. Qkwo dlq x B∈D ( ) ,
n ∈N , α > 1 vykonu[t\sq nerivnist\
x x cn n
n
− ≤
+ +
− −
+
λ ω
λ
α
1
1 2
1
1( / )
,
de c ≡ const, to x B∈D ( )α
.
Zaznaçymo, wo na osnovi nerivnosti (21), ⋅ + v teoremax 3 i 4 moΩna zaminy-
ty na ⋅ + .
Z ci[] Ω teoremy bezposeredn\o vyplyva[ takyj naslidok.
Naslidok 3. Nexaj dlq x B∈D ( ) , n ∈N , α > 1, ε > 0 vykonu[t\sq neriv-
nist\
x x cn n− ≤+ +
− + −λ α ε
1
1 2( / )
.
Todi x B∈D ( )α
.
ZauvaΩennq 2. Qkwo za nablyΩenyj za Ritcom rozv’qzok rivnqnnq (20) vzq-
ty vektor xn , na qkomu funkcional F z( ) dosqha[ svoho minimumu na �n =
=
�λ1
⊕
�λ2
⊕ … ⊕
�λn
, de �λ j
— vlasnyj pidprostir operatora B, wo vid-
povida[ vlasnomu çyslu λ j , to u prypuwenni 2° moΩna vidkynuty umovu pro-
stoty spektra.
3. Poklademo � = L2 0, π[ ]( ), D ( )A = { x( )⋅ ∈ W2
2 0, π[ ]( ) : ′x ( )0 = ′x ( )π = 0},
( )( ) ( ) ( ) ( )Ax t x t q t x t= − ′′ + , q t( ) > 0 , q C( ) ,⋅ ∈ [ ]( )0 π .
Operator B vyznaçymo takym çynom:
D ( )B = D ( )A , ( )( ) ( ) ( )Bx t x t x t= − ′′ + .
Operatory A i B [ samosprqΩenymy j dodatno vyznaçenymy v L2 0, π[ ]( ).
Spektr B sklada[t\sq z vlasnyx znaçen\ λk B( ) = k2 + 1, k ∈N0, qkym
vidpovidagt\ vlasni funkci]
2
π
cos( )kt , wo utvorggt\ ortonormovanyj bazys
u prostori L2 0, π[ ]( ).
Nexaj k ∈N i g C k( ) ,⋅ ∈ [ ]( )2 0 2π . NevaΩko perekonatysq, wo D ( )Ak +1 =
= D ( )Bk +1
todi i til\ky todi, koly g j2 1 0+( )( ) = g j2 1+( )( )π = 0, j = 0, … , k. Qkwo
y( )⋅ ∈ C k2 1 0 2( ) ,− [ ]( )π i y j2 1 0+( )( ) = y j2 1+( )( )π = 0, to y t Ak( ) ( )∈D , a tomu
rozv’qzok zadaçi
− ′′ + =x t g t x t y t( ) ( ) ( ) ( ) , (24)
′ = ′ =x x( ) ( )0 0π , (25)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
PRQMI J OBERNENI TEOREMY V TEORI} NABLYÛEN| METODOM RITCA 643
naleΩyt\ do mnoΩyny D ( )Ak +1
= D ( )Bk +1
i iz spivvidnoßennq (23) bezposered-
n\o vyplyva[ take tverdΩennq.
Teorema 5. Qkwo g C k( ) ,⋅ ∈ [ ]( )2 0 2π , g j2 1 0+( )( ) = g j2 1+( )( )π = 0, j = 0, … , k,
a y( )⋅ ∈R C k2 1 0 2( ) ,− [ ]( )π , y j( )( )2 1 0+ = y j( )( )2 1+ π = 0, j = 0, … , k – 1, to dlq na-
blyΩenoho za Ritcom rozv’qzku zadaçi (24), (25) vykonu[t\sq spivvidnoßennq
x x o
n
n W k− =
[ ]( ) +2
2 0 2 1
1
, π .
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// Uspexy mat. nauk. – 1968. – 23, v¥p. 4. – S. 118 – 178.
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zamknutoho operatora y yx prymenenye k voprosam approksymacyy // Uspexy mat. nauk. –
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ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
|
| id | umjimathkievua-article-3629 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:46:05Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/18/aa195ec8655ed645b0a20e8dd7512818.pdf |
| spelling | umjimathkievua-article-36292020-03-18T20:00:32Z Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method Прямі й обернені теореми в теорії наближень методом Рітца Gorbachuk, M. L. Hrushka, Ya. I. Torba, S. M. Горбачук, М. Л. Грушка, Я. І. Торба, С. М. For an arbitrary self-adjoint operator B in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the operator B. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space. Для довільного самоспряженого оператора $B$ у гільбертовому просторі $\mathfrak{Y}$ наведено прямі й обернені теореми, що встановлюють зв'язок між степенем гладкості вектора $X \in \mathfrak{Y}$ відносно оператора $B$, порядком прямування до нуля його найкращого наближення цілими векторами експоненціального типу оператора $B$ і $k$-модулем неперервності вектора $x$ щодо оператора $B$. Результати застосовано до знаходження апріорних оцінок наближених за Рітцом розв'язків операторних рівнянь у гільбертовому просторі. Institute of Mathematics, NAS of Ukraine 2005-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3629 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 5 (2005); 633–643 Український математичний журнал; Том 57 № 5 (2005); 633–643 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3629/3988 https://umj.imath.kiev.ua/index.php/umj/article/view/3629/3989 Copyright (c) 2005 Gorbachuk M. L.; Hrushka Ya. I.; Torba S. M. |
| spellingShingle | Gorbachuk, M. L. Hrushka, Ya. I. Torba, S. M. Горбачук, М. Л. Грушка, Я. І. Торба, С. М. Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method |
| title | Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method |
| title_alt | Прямі й обернені теореми в теорії наближень методом Рітца |
| title_full | Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method |
| title_fullStr | Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method |
| title_full_unstemmed | Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method |
| title_short | Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method |
| title_sort | direct and inverse theorems in the theory of approximation by the ritz method |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3629 |
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