Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators
We construct new examples of operators of generalized translation and convolutions in eigenfunctions of certain self-adjoint differential operators.
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| Дата: | 2005 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська 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/3632 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509755378761728 |
|---|---|
| author | Kosyak, O. V. Nizhnik, L. P. Косяк, А. В. Нижник, Л. П. Косяк, А. В. Нижник, Л. П. |
| author_facet | Kosyak, O. V. Nizhnik, L. P. Косяк, А. В. Нижник, Л. П. Косяк, А. В. Нижник, Л. П. |
| author_sort | Kosyak, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:32Z |
| description | We construct new examples of operators of generalized translation and convolutions in eigenfunctions of certain self-adjoint differential operators. |
| first_indexed | 2026-03-24T02:46:09Z |
| format | Article |
| fulltext |
UDK 517.9
A.�V.�Kosqk, L.�P.�NyΩnyk (Yn-t matematyky NAN Ukrayn¥, Kyev)
OPERATORÁ OBOBWENNOHO SDVYHA Y HYPERHRUPPÁ,
POSTROENNÁE PO SAMOSOPRQÛENNÁM
DYFFERENCYAL|NÁM OPERATORAM
*
We construct new examples of operators of generalized translation and convolutions in eigenfunctions of
some self-adjoint differential operators.
Pobudovano novi pryklady operatoriv uzahal\nenoho zsuvu ta zhortok za vlasnymy funkciqmy
deqkyx samosprqΩenyx dyferencial\nyx operatoriv.
V peryod s 1950 po 1953/hh. G./M./Berezanskym (çastyçno sovmestno s S./H./Krej-
nom) b¥la razrabotana detal\naq obwaq teoryq hyperkompleksn¥x system so
sçetn¥my y neprer¥vn¥my bazysamy (sm. [1] y pryvedennug v nej byblyohra-
fyg). ∏ta teoryq pozvolyla postroyt\ hlubokye obobwenyq harmonyçeskoho
analyza y teoryy poçty peryodyçeskyx funkcyj. Obæekt¥, blyzkye k hyper-
kompleksn¥m systemam, — hyperhrupp¥ — naçaly yzuçat\sq za rubeΩom lyß\
v 1973/h. VaΩn¥m voprosom v teoryy hyperkompleksn¥x system y hyperhrupp
qvlqetsq postroenye razlyçn¥x konkretn¥x prymerov takyx obæektov. Íyro-
kyj klass prymerov osnovan na rassmotrenyy operatorov obobwennoho sdvyha,
kotor¥e svqzan¥ s konkretn¥my dyfferencyal\n¥my y raznostn¥my opera-
toramy [1, 2]. V nastoqwej stat\e pokazano, çto dlq πtyx celej moΩno yspol\-
zovat\ takΩe samosoprqΩenn¥e dyfferencyal\n¥e operator¥ typa Udnova –
Íextera [3], odnovremenno dejstvugwye kak v oblasty, tak y na hranyce.
1. Alhebrayçeskye struktur¥, svqzann¥e s operatoramy pervoho po-
rqdka. 1.1. SamosoprqΩenn¥j operator. Rassmotrym v hyl\bertovom pro-
stranstve H = L2
1
2
1
2
−( ), ⊕ E1 operator A , zadann¥j ravenstvom
A
xϕ
ϕ ϕ
( )
1
2
1
2
1
2
−
+
=
− ′
− −
i x
i
ϕ
ϕ ϕ
( )
1
2
1
2
(1)
na funkcyqx ϕ ∈ W2
1 1
2
1
2
−( ), yz prostranstva Soboleva vsex absolgtno nepre-
r¥vn¥x funkcyj na yntervale −[ ]1
2
1
2
, , u kotor¥x proyzvodnaq ϕ′ yntehryrue-
ma s kvadratom.
Teorema 1. Operator A v hyl\bertovom prostranstve H qvlqetsq sa-
mosoprqΩenn¥m operatorom s çysto dyskretn¥m spektrom. Eho sobstvenn¥e
znaçenyq λn , n ∈ Z , zanumerovann¥e v porqdke vozrastanyq, s uslovyem λ0 =
= 0, λ – n = – λn qvlqgtsq kornqmy xarakterystyçeskoho uravnenyq
ei
λ =
2
2
−
+
i
i
λ
λ
. (2)
Pry bol\ßyx n >> 1 sobstvenn¥e çysla ymegt asymptotyku
λn = π ( 2n – 1 ) + εn + O
n
1
3
, εn = 4
2 1π( )n −
. (3)
Funkcyy ϕ n = ei xnλ , n ∈ Z , obrazugt polnug v prostranstve L2
1
2
1
2
−( ),
systemu funkcyj, ortohonal\nug otnosytel\no skalqrnoho proyzvedenyq
*
V¥polnena pry çastyçnoj fynansovoj podderΩke DFG (proekt 436 UKR113/72) .
© A./V./KOSQK, L./P./NYÛNYK, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5 659
660 A./V./KOSQK, L./P./NYÛNYK
〈 ϕ , ψ 〉 = ( , )ϕ ψ L2
+ ϕ ψr r⋅ , ϕr = 1
2
1
2
1
2
ϕ ϕ
+ −
, ϕ , ψ ∈ C −( )1
2
1
2
, . (4)
Dokazatel\stvo. Oboznaçym çerez U = col ( u ( x ) , ur ) vektor yz H , postro-
enn¥j po funkcyy u ∈ W2
1 1
2
1
2
−( ), . Vektor u prynadleΩyt oblasty opredele-
nyq operatora A y A u = col ( – i u′ ( x ) , i us ) ∈ H , hde us = u 1
2
( ) – u −( )1
2
. Poka-
Ωem, çto oblast\ opredelenyq operatora A vsgdu plotna v H . Esly b¥ πto
b¥lo ne/tak, to suwestvoval b¥ vektor Ψ = col ( ψ ( x ) , a ) ≠ 0, ψ ∈ L2 , a ∈ C1
,
takoj, çto ( U , Ψ )H = 0 ∀u ∈ W2
1. Dlq funkcyj u ∈ C0
1
2
1
2
∞ −( ), ravenstvo
( U , Ψ )H = 0 perexodyt v ravenstvo ( , )u Lψ
2
= 0 ∀u ∈ C0
∞ . V sylu plotnosty
prostranstva C0
∞ v prostranstve L2 pryxodym k v¥vodu, çto ψ ≡ 0. Tohda ra-
venstvo ( U , Ψ )H = 0 prevrawaetsq v u ar = 0 ∀u ∈ W2
1. Polahaq u ≡ 1,
ymeem a = 0, çto protyvoreçyt uslovyg Ψ = col ( ψ ( x ) , a ) ≠ 0. Poπtomu ob-
last\ opredelenyq operatora A vsgdu plotna v prostranstve H . Operator A
symmetryçeskyj v hyl\bertovom prostranstve H . Dejstvytel\no, dlq vektorov
U = col ( u ( x ) , ur ) y V = col ( v ( x ) , vr ) , hde u , v ∈ W2
1, v sylu opredelenyq ope-
ratora A y formul¥ Hryna
( A U , V )H – ( U , A V )H =
( ) ( ) ( ) ( ), , , ,− ′ + − − ′ −iu iu u i u iL s r E L r s E
v v v v
2 1 2 1 =
=
−
− −
−
+ +[ ]i u u i u us r r s
1
2
1
2
1
2
1
2
v v v v ≡ 0.
PokaΩem teper\, çto operator A samosoprqΩenn¥j v H . Dlq πtoho dostatoç-
no pokazat\, çto oblast\ znaçenyj operatora A ± i I sostavlqet vse prostran-
stvo H , t./e. uravnenye ( A ± i I ) U = Ψ odnoznaçno razreßymo pry lgbom vek-
tore Ψ = col ( ψ ( x ) , a ) ∈ H . Ukazannoe uravnenye svodytsq k dyfferencyal\-
nomu uravnenyg – i u′ ( x ) ± i u ( x ) = ψ ( x ) y hranyçnomu uslovyg i us ± i ur = a .
∏ta zadaça odnoznaçno reßaetsq pry lgb¥x ψ ∈ L2 y a ∈ C, a ee reßenye u ∈
∈ W2
1 lehko predstavyt\ v qvnom vyde.
Dlq naxoΩdenyq sobstvenn¥x vektorov Φλ = col ( ϕλ , ψλ ) y sobstvenn¥x
znaçenyj λ neobxodymo najty netryvyal\noe reßenye uravnenyq A Φλ = λ Φλ .
∏ta zadaça svodytsq k dyfferencyal\nomu uravnenyg − ′iϕλ = λ ϕλ y hranyçno-
mu uslovyg − i sϕλ, = λϕλ,r . Poslednqq zadaça ymeet netryvyal\n¥e reßenyq
ϕλ ( x ) = ei xλ tol\ko v sluçae, esly λ qvlqetsq reßenyem xarakterystyçeskoho
uravnenyq (2). /Uravnenye (2) ymeet odnokratn¥e vewestvenn¥e reßenyq λn , n ∈
∈ Z , kotor¥e moΩno zanumerovat\ v porqdke vozrastanyq, polahaq λ0 = 0. Tohda
λ – n = – λn y dlq bol\ßyx n reßenyq xarakterystyçeskoho uravnenyq (2) yme-
gt asymptotyku (3). Poskol\ku sobstvenn¥e funkcyy Φλn
= col( ), ,ϕ ϕλ λn r
samosoprqΩennoho operatora A obrazugt polnug ortohonal\nug v H syste-
mu, to ( ),Φ Φλ λn m H = 0 pry n ≠ m . ∏to ravenstvo πkvyvalentno uslovyg orto-
honal\nosty funkcyj { } ∈ei x
n Z
nλ otnosytel\no skalqrnoho proyzvedenyq
〈 ⋅ , ⋅ 〉 , zadavaemoho ravenstvom (4). Pry πtom
( ),Φ Φλ λn m H = 〈 〉e ei x i xn mλ λ, = δn m nN,
2 , (5)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
OPERATORÁ OBOBWENNOHO SDVYHA Y HYPERHRUPPÁ, POSTROENNÁE … 661
hde Nn
2 = 1 + [ / ]λn
2 14 1+ − , a δn m, — symvol Kronekera.
Teorema dokazana.
Teorema 2. Dlq toho çtob¥ systema funkcyj { } ∈ei x
n Z
nλ , λ0 = 0, obrazo-
v¥vala polnug v L2
1
2
1
2
−( ), ortohonal\nug otnosytel\no skalqrnoho proyz-
vedenyq (4) systemu funkcyj, neobxodymo y dostatoçno, çtob¥ çysla λn
b¥ly vsemy reßenyqmy uravnenyq (2).
Dokazatel\stvo. Uslovye 〈 〉ei xnλ , 1 = 0 pryvodyt k xarakterystyçeskomu
uravnenyg (2) dlq çysel λn . Esly çysla λn ≠ λm qvlqgtsq reßenyqmy uravne-
nyq (2), to 〈 〉e ei x i xn mλ λ, = 0, t./e. ony ortohonal\n¥ otnosytel\no skalqrnoho
proyzvedenyq (2). Polnota system¥ { } ∈ei x
n Z
nλ
sleduet yz teorem¥/1, poskol\-
ku vektor¥ Φλn
= col( ), cos( / )ei x
n
nλ λ 2 obrazugt polnug ortohonal\nug sys-
temu v prostranstve H .
Teorema dokazana.
Zameçanye�1. Teorem¥ 1, 2 ostagtsq vern¥my, esly skalqrnoe proyzvede-
nye (2) zamenyt\ na bolee obwee
〈 ϕ , ψ 〉 = ϕ ψ,
,
( )
−
L2
1
2
1
2
+ α ϕ ψ2
r r ,
operator A opredelqt\ ravenstvom A col ( u ( x ) , α ur ) = col ( – i u′ ( x ) , i α–1
us ) , a
xarakterystyçeskoe uravnenye (2) zamenyt\ na ei
λ =
2
2
2
2
−
+
i
i
λα
λα
.
1.2. ∏volgcyonnoe uravnenye y operator¥ obobwennoho sdvyha. Rassmot-
rym zadaçu Koßy dlq πvolgcyonnoho uravnenyq v prostranstve H vyda
dU
dt
= i A U, U | t =0 = F. (6)
Poskol\ku operator A qvlqetsq samosoprqΩenn¥m v prostranstve H , zadaça
(6) odnoznaçno razreßyma na vsej osy – ∞ < t < + ∞ pry proyzvol\nom naçal\-
nom uslovyy F ∈ H . Pust\ F = col ( f ( x ) , fr ) , hde f ∈ C −( )1
2
1
2
, . Tohda reßenye
zadaçy (6) ymeet vyd U ( t ) = col u t x u t u t( , ), , ,1
2
1
2
1
2
+ −
, hde funkcyq
u ( t , x ) qvlqetsq reßenyem zadaçy
∂
∂
∂
∂
u
t
u
x
= , ∂
∂t
u t u t u t u t, , , ,1
2
1
2
2 1
2
1
2
+ −
= −
−
,
(7)
u ( 0 , x ) = f ( x ) , u u f f0 1
2
0 1
2
1
2
1
2
, ,
+ −
=
+ −
.
Teorema 3. Reßenye zadaçy (7) predstavymo v vyde u ( t , x ) = ˆ( )f x t+ , hde
funkcyq ˆ( )f x qvlqetsq prodolΩenyem zadannoj na yntervale −[ ]1
2
1
2
, funk-
cyy f na vsg os\. Pry πtom
ˆ( )f x =
− + + + − ≤ ≤ −
− ≤ ≤
− − + +
+ + −
+
− − − − −
−
−
∫
∫
f x e f e f d x
f x x
f x e f e f d
x
r
x
x
x
r
xx
( ) ( ) , ,
( ), ,
( ) ( ) ,
( / ) ( )/
( / ) ( )
/
1 2 4 3
2
1
2
1
2
1
2
1 2 4
2 1 2 2 1
1
1 2
2 1 2 2 1
1 2
1
τ
τ
τ τ
τ τ 11
2
3
2
≤ ≤
x .
(8)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
662 A./V./KOSQK, L./P./NYÛNYK
Esly f ( x ) = ei xnλ , hde λ n — sobstvenn¥e znaçenyq operatora A , to ˆ( )f x =
= ei xnλ
dlq vsex x . Esly f ( x ) = f en
i x
n
nλ∑ — rqd Fur\e funkcyy f ∈ C −( )1
2
1
2
, ,
to ee prodolΩenye f̂ predstavymo tem Ωe rqdom f̂ = f en
i x
n
nλ∑ , x ∈ R
1.
Dokazatel\stvo. Yz uravnenyq ∂
∂
∂
∂
u
t
u
x
= sleduet u ( t , x ) = ˆ( )f x t+ . Yz na-
çal\noho uslovyq u ( 0 , x ) = f ( x ) poluçaem ˆ( )f x = f ( x ) pry –
1
2
≤ x ≤ 1
2
. Yz
ostavßehosq sootnoßenyq (7) ymeem
ˆ ˆ ˆ ˆ′ +
+ +
= − ′ −
+ −
f t f t f t f t1
2
2 1
2
1
2
2 1
2
. (9)
Yntehryruq (9) pry 1
2
≤ t ≤ 3
2
y pry –
3
2
≤ t ≤ –
1
2
s uçetom ˆ( )f x = f ( x ) , polu-
çaem sootnoßenyq (8).
Esly vektor naçal\n¥x uslovyj F = col( ), cos( / )ei x
n
nλ λ 2 ≡ Φλn
v zadaçe
Koßy (6) qvlqetsq sobstvenn¥m vektorom operatora A s sobstvenn¥m znaçe-
nyem λn , to reßenye ymeet vyd U ( t ) = ei tn
n
λ
λΦ . Poπtomu funkcyq u ( t , x ) =
= = +e e f x ti t i xn nλ λ ˆ( ) y, sledovatel\no, ˆ( ) ( , )f x u x ei xn= =0 λ . Takym obrazom, pro-
dolΩenye ei xnλ̂ = ei xnλ pry vsex x . Poπtomu ˆ( )f x = f̂ en
i x
n
nλ∑ = f en
i x
n
nλ∑ .
Teorema dokazana.
Opredelenye 1. Operator Tt obobwennoho sdvyha, sootvetstvugwyj za-
daçe (7), opredelqetsq ravenstvom
T
t
f ( x ) = u ( t , x ) = ˆ( )f x t+ , (10)
hde f̂ — prodolΩenye funkcyy f yz yntervala −[ ]1
2
1
2
, na vsg os\ sohlasno
teoreme 3.
Teorema 4. Esly λ n — sobstvenn¥e znaçenyq operatora A , t.0e. reßenyq
xarakterystyçeskoho uravnenyq (2), to
T et i xnλ = e ei t i xn nλ λ . (11)
Esly f ( x ) = f en
i x
n
nλ∑ qvlqetsq rqdom Fur\e funkcyy f otnosytel\no sys-
tem¥ funkcyj { } ∈ei x
n Z
nλ , ortohonal\n¥x otnosytel\no skalqrnoho proyz-
vedenyq (4), t.0e.
fn = f x e
N
i x
n
n( ), λ 1
2 , (12)
to
T
t
f ( x ) = f e en
i t i x
n
n nλ λ∑ . (13)
Semejstvo operatorov T
t obrazuet unytarnug odnoparametryçeskug hruppu
operatorov otnosytel\no skalqrnoho proyzvedenyq (4):
T Tt t1 2 = T t t1 2+ , T
0 = I , 〈 T
t
ϕ , T
t
ψ 〉 = 〈 ϕ , ψ 〉. (14)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
OPERATORÁ OBOBWENNOHO SDVYHA Y HYPERHRUPPÁ, POSTROENNÁE … 663
Dokazatel\stvo. Poskol\ku prodolΩenye, sohlasno teoreme 3, funkcyy
ei xnλ sovpadaet s nej samoj, v sylu (10) poluçaem (11). Ravenstvo (13) poluça-
etsq prymenenyem lynejnoho operatora T
t k razloΩenyg v rqd Fur\e
funkcyy f ( x ) po systeme { } ∈ei x
n Z
nλ s uçetom ravenstva (11). Poslednee
ravenstvo v (14) sleduet yz ravenstva Parsevalq dlq rqdov Fur\e po polnoj
ortohonal\noj systeme funkcyj
〈 f , g 〉 = f g
N
n n
nn
1
2∑ (15)
y qvnoho vyda koπffycyentov Fur\e funkcyy T
t
f , ( T
t
f )n = f en
i tnλ , v¥tekag-
wyx yz (13).
Teorema dokazana.
1.3. Svertka.
Opredelenye 2. Svertka dvux neprer¥vn¥x funkcyj f , g ∈ C −( )1
2
1
2
, opre-
delqetsq çerez skalqrnoe proyzvedenye 〈 ⋅ , ⋅ 〉 (4) y operator obobwennoho
sdvyha T
t ravenstvom
( f ∗ g ) ( t ) = 〈 T
t
f , g∗ 〉, (16)
hde g∗( x ) = g x( )− — ynvolgcyq.
Teorema 5. Dannaq v opredelenyy 2 svertka qvlqetsq assocyatyvn¥m y
kommutatyvn¥m umnoΩenyem.
Dokazatel\stvo. Opredelym svertku dvux xarakterov ei xnλ y ei xmλ .
Poskol\ku T et i xnλ = e ei t i xn mλ λ , a [ ]ei xmλ ∗ = ei xmλ , yz (16) sleduet
e e ti x i xn mλ λ∗( )( ) = e e ei t i x i xn n mλ λ λ, = e Ni t
nm n
nλ δ 2 . (17)
Pust\ zadan¥ funkcyy f , g ∈ C −( )1
2
1
2
, . Predstavlqq πty funkcyy rqdamy
Fur\e f ( x ) = f en
i x
n
nλ∑ , g ( x ) = g en
i x
n
nλ∑ y uçyt¥vaq ravenstvo (17), ymeem
( f ∗ g ) ( x ) = f g N en n n
i x
n
n−∑ 2 λ
. (18)
Yz predstavlenyq (18) sleduet, çto koπffycyent¥ Fur\e svertky v¥raΩagtsq
çerez proyzvedenye koπffycyentov Fur\e svertoçn¥x mnoΩytelej, t./e.
svertka (16) kommutatyvna.
Predstavlqq koπffycyent¥ Fur\e povtorn¥x svertok ( f ∗ g ) ∗ h y
f ∗ ( g ∗ h ) çerez proyzvedenyq koπffycyentov Fur\e svertoçn¥x mnoΩytelej,
poluçaem
[( f ∗ g ) ∗ h ]n = f g h Nn n n n
−4 = [ f ∗ ( g ∗ h ) ]n .
Takym obrazom, svertka (16) assocyatyvna.
Teorema dokazana.
Yz opredelenyq 2 svertky, qvnoho vyda skalqrnoho proyzvedenyq (4), qvnoho
vyda operatora sdvyha T t
f ( x ) = ˆ( )f x t+ y qvnoho vyda (8) prodolΩenyq f̂
funkcyy f lehko poluçyt\ v¥raΩenye dlq svertky dvux funkcyj
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
664 A./V./KOSQK, L./P./NYÛNYK
( f ∗ g ) ( x ) = f g c x d d( ) ( ) ( , , )
/
/
/
/
ξ η ξ η ξ η
−−
∫∫
1 2
1 2
1 2
1 2
,
hde c x( , , )ξ η = ( )( )− −χ χξ η[ / , ] [ / , ]*1 2 1 2 x , a χ ξ[ / , ]−1 2 — xarakterystyçeskaq
funkcyq yntervala −
1
2
, ξ .
∏tu svertku moΩno predstavyt\ takΩe v qvnom vyde
( f ∗ g ) ( x ) = f x s g s ds f x s g s ds
x
x
( ) ( ) ( ) ( )
/
/
/
/
− − − −
− −
−
∫ ∫
1 2
1 2
1 2
1 2
1 +
+ 4 2 1e f g d dx
Dx
− − + +∫∫ ( ) ( ) ( )ξ η ξ η ξ η +
+ e f g f e g d g e f dx
r r r
x
r
x
− +
−
−
+
−
−
⋅ + +
∫ ∫2 1
1 2
1 2
1
1 2
1 2
2 2η ξη η ξ ξ( ) ( )
/
/
/
/
,
hde x > 0, a Dx = { ( ξ , η ) : –1/2 ≤ ξ , η ≤ 1/2, ξ + η ≤ x – 1 }.
1.4. Strukturn¥e konstant¥. Sovokupnost\ xarakterov { } ∈ei x
n Z
nλ
obrazuet ortohonal\nug systemu otnosytel\no svertoçnoho umnoΩenyq (17).
Ob¥çnoe proyzvedenye dvux xarakterov moΩno v¥razyt\ çerez yx lynejnug
kombynacyg v vyde
e ei x i xn mλ λ = c en m k
i x
k
k
, ,
λ∑ ,
hde cn m k, , — strukturn¥e konstant¥. Poskol\ku dannoe predstavlenye qvlq-
etsq rqdom Fur\e funkcyy e ei x i xn mλ λ , to
cn m k, , = 1
2N
e e e
k
i x i x i xn m kλ λ λ, .
Pry λn + λm – λk ≠ 0
cn m k, , = ( )
( )
( )
/
− +( ) +( ) +( )[ ] +
+ −
+ + + −
1 4 4 4
21 2 2 2 1 2
2
n m k
n m k
n m n m
k n m kN
λ λ λ λ λ λ λ
λ λ λ
.
Krome πtoho, cn k, ,0 = c n k0, , = δn k, , cn n, ,− 0 = 1.
2. Alhebrayçeskye struktur¥, svqzann¥e s dyfferencyal\n¥my ope-
ratoramy vtoroho porqdka. 2.1. SamosoprqΩenn¥j operator. Rassmotrym
v prostranstve H = L2 ( 0 , 1 ) ⊕ E1
operator B , opredelqem¥j ravenstvom
B col ( u ( x ) , u ( 1 )) = col ( – u″ ( x ) , u′ ( 1 ) ) (19)
na funkcyqx u ( x ) yz sobolevskoho prostranstva W2
2 0 1( , ) , udovletvorqgwyx
hranyçnomu uslovyg u′ ( 0 ) = 0.
Teorema 6. Operator B qvlqetsq samosoprqΩenn¥m operatorom v
hyl\bertovom prostranstve H s çysto dyskretn¥m spektrom. Eho sobstven-
n¥e znaçenyq λn
2 , zanumerovann¥e v porqdke vozrastanyq, qvlqgtsq kornqmy
xarakterystyçeskoho uravnenyq
tan λn = – λn , λ0 = 0, λn ≥ 0. (20)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
OPERATORÁ OBOBWENNOHO SDVYHA Y HYPERHRUPPÁ, POSTROENNÁE … 665
Funkcyy ϕn = cos λn x , n = 0, 1, … , obrazugt polnug v prostranstve H sys-
temu funkcyj, ortohonal\nug otnosytel\no skalqrnoho proyzvedenyq
〈 ϕ , ψ 〉 = ( , )ϕ ψ L2
+ ϕ ψ( ) ( )1 1⋅ , ϕ , ψ ∈ C ( 0 , 1 ) . (21)
Pry πtom
〈 cos λn x , cos λm x 〉 = δn m nN,
2 , (22)
hde Nn
2 = 1
2
1 2+( )cos λn = 1
2
1 1
1 2+
+
λn
, n > 0 y N0
2 = 2.
Dokazatel\stvo teorem¥ provodytsq analohyçno dokazatel\stvu teorem¥ 1.
2.2. Obobwenn¥j sdvyh.
Opredelenye 3. Lynejn¥j operator obobwennoho sdvyha T
t
na bazyse
cos λn x , hde λ n — korny xarakterystyçeskoho uravnenyq (20), opredelqetsq
ravenstvom
T
t
cos λn x = cos λn t cos λn x . (23)
Opredelenye 4. ProdolΩenye f̂ neprer¥vnoj funkcyy f ∈ C ( 0 , 1 ) na vsg
os\ opredelqetsq znaçenyem rqda Fur\e funkcyy f po polnoj ortohonal\noj
otnosytel\no skalqrnoho proyzvedenyq (21) systeme funkcyj { } ≥cosλn nx 0
ˆ( )f x = f xn n
n
cosλ
=
∞
∑
0
, (24)
hde fn = 〈 f ( x ) , cos λn x 〉 1
2Nn
— koπffycyent¥ Fur\e funkcyy f .
Teorema 7. Operator obobwennoho sdvyha T
t
dejstvuet na proyzvol\nug
neprer¥vnug funkcyg f ∈ C ( 0 , 1 ) po pravylu
T
t
f ( x ) = 1
2
ˆ( ) ˆ( )f x t f x t+ + −[ ], (25)
hde f̂ — prodolΩenye funkcyy f na vsg os\ sohlasno opredelenyg 4.
Dokazatel\stvo. Pust\ f ( x ) = f xn nn
cosλ=
∞∑ 0
— rqd Fur\e funkcyy f ∈
∈ C ( 0 , 1 ) . Prymenqq k πtomu rqdu operator T
t, s uçetom (23) ymeem
T
t
f ( x ) = f t xn n n
n
cos cosλ λ
=
∞
∑
0
=
= 1
2 0
f x t x tn n n
n
cos ( ) cos ( )λ λ+ + −[ ]
=
∞
∑ = 1
2
ˆ( ) ˆ( )f x t f x t+ + −[ ].
Teorema 8. Pust\ U ( t ) = col ( u ( t , x ) , u ( t , 1)) ∈ H — reßenye zadaçy Koßy
d U
dt
2
2 + B U = 0, U ( 0 ) = col ( f ( x ) , f ( 1)) ,
dU
dt t=0
= 0. (26)
Tohda u ( t , x ) = T
t
f ( x ) , hde T
t
— operator obobwennoho sdvyha.
Dokazatel\stvo. Esly f ( x ) = cos λn x , hde λn
2
— sobstvennoe znaçenye
samosoprqΩennoho operatora B , to u ( t , x ) = cos λn t cos λn x , t./e. v¥polnqetsq
ravenstvo (23). Poluçennoe metodom sobstvenn¥x funkcyj reßenye zadaçy (26)
predstavymo v vyde
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
666 A./V./KOSQK, L./P./NYÛNYK
u ( t , x ) = 1
2
ˆ( ) ˆ( )f x t f x t+ + −[ ], (27)
çto pryvodyt k predstavlenyg (24) dlq operatorov T
t. S druhoj storon¥, pod-
stavlqq (27) v (26), dlq funkcyy f̂ poluçaem dyfferencyal\noe uravnenye
ˆ ( ) ˆ( )′ + + +f t f t1 1 = ˆ ( ) ˆ( )′ − + −f t f t1 1 .
Yntehryruq πto uravnenye, ymeem
ˆ( )f x + 1 = – f ( 1 – x ) + 2e–x
f ( 1 ) + 2 1
0
e f dx
x
− − −∫ ( ) ( )τ τ τ , 0 ≤ x ≤ 1.
Teorema dokazana.
2.3. Svertka.
Opredelenye 5. Opredelym svertku dvux funkcyj f , g ∈ C ( 0, 1 ) ra-
venstvom
( f ∗ g ) ( x ) = f y T g xy( ), ( )− . (28)
Teorema 9. Svertka (28) assocyatyvna y kommutatyvna.
Dokazatel\stvo. V¥çyslym svertku xarakterov
( cos λn ( ⋅ ) ∗ cos λm ( ⋅ ) ) ( x ) = cos cos cosλ λ λn m my y x dy
0
1
∫ +
+ cos λn cos λm cos λm x = δ λn m n nx N, cos ⋅ 2 .
Dal\nejßee dokazatel\stvo povtorqet dokazatel\stvo teorem¥ 5.
Yspol\zovav qvn¥j vyd operatorov sdvyha T
t, pryvedem qvn¥j vyd dlq
svertky:
( f ∗ g ) ( x ) = 1
2 0
f y g x y dy
x
( ) ( )−∫ + 1
2 0
1
f x y g y f y g x y dy
x
( ) ( ) ( ) ( )+ + +[ ]
−
∫ –
– 1
2
1 1
2
1 1
22
2
f x s g x s ds
x
x
− +
− −
−
∫
/
/
+
+ exp( ) ( ) ( )exp( )2 − − −∫∫x f y g s x s dy ds
Dx
+ f g x( ) ( )exp( )1 1 − +
+ f g y x y dy
x
( ) ( )exp( )1 1
1
1
− −
−
∫ + g f y x y dy
x
( ) ( )exp( )1 1
1
1
− −
−
∫ ,
hde
Dx = { ( s , y ) ∈R2 | s ≤ 1 , y ≤ 1, s + y ≥ 2 – x }.
2.4. Strukturn¥e konstant¥. Systema { } =
∞cosλn nx 0 qvlqetsq xarakte-
ramy dlq obobwenn¥x sdvyhov (23). V¥razym ob¥çnoe proyzvedenye dvux xarak-
terov çerez yx lynejnug kombynacyg
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
OPERATORÁ OBOBWENNOHO SDVYHA Y HYPERHRUPPÁ, POSTROENNÁE … 667
cos λn x cos λm x = c xn m k k
k
, , cosλ∑
s pomow\g strukturn¥x konstant cn m k, , . Poskol\ku πto predstavlenye qvlq-
etsq rqdom Fur\e, to
cn m k, , = 1
2Nk
〈 cos λn x cos λm x , cos λk x 〉.
Yspol\zuq formulu
cos a cos b cos c = 1
4
cos( ) cos( ) cos( ) cos( )a b c a b c a b c a b c+ + + + − + − + + − + +( ) ,
naxodym
( cos λn x cos λm x , cos λk x )H = cos cos cosλ λ λn m kx x x dx
0
1
∫ + cos λn cos λm cos λk =
= 1
4
sin( ) sin( )λ λ λ
λ λ λ
λ λ λ
λ λ λ
n m k
n m k
n m k
n m k
+ +
+ +
+ + −
+ −
+
+
sin( ) sin( )λ λ λ
λ λ λ
λ λ λ
λ λ λ
n m k
n m k
n m k
n m k
− +
− +
+ − + +
− + +
+
+ cos λn cos λm cos λk .
Yspol\zuq xarakterystyçeskoe uravnenye (20), pry n , m , k ≠ 0 poluçaem
cn m k, , =
2 1 1 1 1
2
2 2 2 2 2 2 1 2
2 4 4 4 2 2 2 2 2 2
( )
/
− +( ) +( ) +( )[ ]
+ + − + +( )[ ]
+ + −n m k
n m k n m k
k n m k n m n k m kN
λ λ λ λ λ λ
λ λ λ λ λ λ λ λ λ
.
Krome toho, cn m, ,0 = 1
2
2δn m nN, , c cm k m k m k0 0, , , , ,= = δ .
3. V¥vod¥. V¥ße b¥ly pryveden¥ dva prymera postroenyq operatorov
obobwennoho sdvyha, osnovann¥x na sobstvenn¥x funkcyqx samosoprqΩenn¥x
dyfferencyal\n¥x operatorov. ∏ta konstrukcyq s abstraktnoj toçky zrenyq
sostoyt v suΩenyy bazysa, na kotorom opredelen¥ operator¥ sdvyha. Pust\ Ω
— topolohyçeskoe prostranstvo, a C ( Ω ) — prostranstvo neprer¥vn¥x funk-
cyj na Ω y pust\ na C ( Ω ) zadan¥ operator¥ obobwennoho sdvyha T
t
[2].
Pust\ funkcyy χ ( x , λ ) , x ∈ Ω , λ ∈ Λ , obrazugt semejstvo xarakterov, t./e.
T
t
χ ( x , λ ) = χ ( t , λ ) χ ( x , λ ) . Rassmotrym podprostranstvo Ω0 ⊂ Ω , y pust\ na
prostranstve C ( Ω0 ) zadano skalqrnoe proyzvedenye 〈 ⋅ , ⋅ 〉 . Krome toho, pust\
suwestvuet semejstvo xarakterov { } =
∞χ λ( , )x n n 1, obrazugwee polnug ortoho-
nal\nug systemu otnosytel\no πtoho skalqrnoho proyzvedenyq. Tohda kaΩdug
funkcyg f ( x ) ∈ C ( Ω0 ) moΩno predstavyt\ rqdom Fur\e f ( x ) = f xn
n
n∑ χ λ( , ) ,
x ∈ Ω0 . Poskol\ku xarakter¥ opredelen¥ takΩe pry x ∈ Ω , rqd Fur\e oprede-
len pry vsex x ∈ Ω y predstavlqet estestvennoe prodolΩenye f̂ funkcyy f ,
zadannoj na Ω0 . Poπtomu operator¥ obobwennoho sdvyha T
t, opredelenn¥e na
C ( Ω ) , poroΩdagt operator¥ obobwennoho sdvyha Tt
0 na C ( Ω0 ) po formule
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
668 A./V./KOSQK, L./P./NYÛNYK
T ft
0 =
( )ˆ
( )T ft
CÛ Ω0
,
t./e. snaçala funkcyg f estestvenno prodolΩaem s Ω0 na Ω , dalee prymenq-
em operator T
t, a poluçennug funkcyg T ft ˆ
rassmatryvaem kak funkcyg na
Ω0 . Po-vydymomu, na πtom puty moΩno poluçyt\ y rqd druhyx soderΩatel\n¥x
prymerov operatorov obobwennoho sdvyha.
1. Berezanskyj0G.0M., KalgΩn¥j0A.0A. Harmonyçeskyj analyz v hyperkompleksn¥x systemax.
– Kyev: Nauk. dumka, 1992. – 352/s.
2. Levytan0B.0M. Teoryq operatorov obobwennoho sdvyha. – M.: Nauka, 1973. – 312/s.
3. Ercolano J., Schechter M. Spectral theory for operators generated by elliptic boundary problems
with eigenvalue parameter in boundary conditions, I, II // Communs Pure and Appl. Math. – 1965. –
18. – P. 83 – 105; 397 – 414.
Poluçeno 03.02.2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
|
| id | umjimathkievua-article-3632 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:09Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/85/9b1371d326b86f44a6a51535c0c5c185.pdf |
| spelling | umjimathkievua-article-36322020-03-18T20:00:32Z Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators Операторы обобщенного сдвига и гипергруппы, построенные по самосопряженным дифференциальным операторам Kosyak, O. V. Nizhnik, L. P. Косяк, А. В. Нижник, Л. П. Косяк, А. В. Нижник, Л. П. We construct new examples of operators of generalized translation and convolutions in eigenfunctions of certain self-adjoint differential operators. Побудовано нові приклади операторів узагальненого зсуву та згорток за власними функціями деяких самоспряжених диференціальних операторів. Institute of Mathematics, NAS of Ukraine 2005-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3632 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 5 (2005); 659–668 Український математичний журнал; Том 57 № 5 (2005); 659–668 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3632/3994 https://umj.imath.kiev.ua/index.php/umj/article/view/3632/3995 Copyright (c) 2005 Kosyak O. V.; Nizhnik L. P. |
| spellingShingle | Kosyak, O. V. Nizhnik, L. P. Косяк, А. В. Нижник, Л. П. Косяк, А. В. Нижник, Л. П. Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators |
| title | Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators |
| title_alt | Операторы обобщенного сдвига и гипергруппы, построенные по самосопряженным дифференциальным операторам |
| title_full | Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators |
| title_fullStr | Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators |
| title_full_unstemmed | Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators |
| title_short | Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators |
| title_sort | operators of generalized translation and hypergroups constructed from self-adjoint differential operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3632 |
| work_keys_str_mv | AT kosyakov operatorsofgeneralizedtranslationandhypergroupsconstructedfromselfadjointdifferentialoperators AT nizhniklp operatorsofgeneralizedtranslationandhypergroupsconstructedfromselfadjointdifferentialoperators AT kosâkav operatorsofgeneralizedtranslationandhypergroupsconstructedfromselfadjointdifferentialoperators AT nižniklp operatorsofgeneralizedtranslationandhypergroupsconstructedfromselfadjointdifferentialoperators AT kosâkav operatorsofgeneralizedtranslationandhypergroupsconstructedfromselfadjointdifferentialoperators AT nižniklp operatorsofgeneralizedtranslationandhypergroupsconstructedfromselfadjointdifferentialoperators AT kosyakov operatoryobobŝennogosdvigaigipergruppypostroennyeposamosoprâžennymdifferencialʹnymoperatoram AT nizhniklp operatoryobobŝennogosdvigaigipergruppypostroennyeposamosoprâžennymdifferencialʹnymoperatoram AT kosâkav operatoryobobŝennogosdvigaigipergruppypostroennyeposamosoprâžennymdifferencialʹnymoperatoram AT nižniklp operatoryobobŝennogosdvigaigipergruppypostroennyeposamosoprâžennymdifferencialʹnymoperatoram AT kosâkav operatoryobobŝennogosdvigaigipergruppypostroennyeposamosoprâžennymdifferencialʹnymoperatoram AT nižniklp operatoryobobŝennogosdvigaigipergruppypostroennyeposamosoprâžennymdifferencialʹnymoperatoram |