Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval
We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms...
Gespeichert in:
| Datum: | 2007 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3343 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509417755115520 |
|---|---|
| author | Dyukarev, Yu. M. Serikova, I. Yu. Дюкарев, Ю. М. Серикова, И. Ю. Дюкарев, Ю. М. Серикова, И. Ю. |
| author_facet | Dyukarev, Yu. M. Serikova, I. Yu. Дюкарев, Ю. М. Серикова, И. Ю. Дюкарев, Ю. М. Серикова, И. Ю. |
| author_sort | Dyukarev, Yu. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:58Z |
| description | We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions. |
| first_indexed | 2026-03-24T02:40:47Z |
| format | Article |
| fulltext |
UDK 517.5
G. M. Dgkarev, Y. G. Serykova (Xar\kov. nac. un-t)
ORTOHONAL|NÁE NA KOMPAKTNOM YNTERVALE
RACYONAL|NÁE MATRYCÁ-FUNKCYY,
ASSOCYYROVANNÁE S ZADAÇEJ
NEVANLYNNÁ – PYKA V KLASSE S [[[[ a, b ]]]]
We consider the Nevanlinna – Pick interpolation problem with infinitely many interpolation nodes in the
class S [ a, b ] and associate with it rational matrix functions orthogonal on the interval [ a, b ]. A
criterion for the complete indetermination of the infinite Nevanlinna – Pick problem in terms of the
orthogonal rational matrix functions is obtained.
Rozhlqnuto interpolqcijnu zadaçu Nevanlinny – Pika z neskinçennog kil\kistg vuzliv inter-
polqci] u klasi S [ a, b ], z qkog pov’qzano ortohonal\ni na intervali [ a, b ] racional\ni matryci-
funkci]. Otrymano kryterij povno] nevyznaçenosti neskinçenno] zadaçi Nevanlinny – Pika u
terminax ortohonal\nyx racional\nyx matryc\-funkcij.
Klass analytyçeskyx funkcyj S [ a, b ] b¥l vveden M. H. Krejnom v svqzy s
yzuçenyem problem¥ momentov na kompaktnom yntervale [1, s. 527]. Useçennaq
zadaça Nevanlynn¥ – Pyka v matryçnom klasse S [ a, b ] b¥la rassmotrena v ra-
botax [2, 3].
V nastoqwej stat\e rassmatryvaetsq zadaça Nevanlynn¥ – Pyka v klasse
S [ a, b ] s beskoneçn¥m çyslom kompleksn¥x uzlov ynterpolqcyy, s kotoroj
svqz¥vagtsq dva semejstva { Pr, ( n ) } racyonal\n¥x matryc-funkcyj (sm. (15)).
Osnovn¥my rezul\tatamy stat\y qvlqgtsq teorem¥ 3 y 4. Teorema 3 ustanav-
lyvaet ortonormyrovannost\ semejstva { P1, ( n ) } otnosytel\no matryçnoho vesa
( b – t ) dσ ( t ) y { P2, ( n ) } otnosytel\no matryçnoho vesa ( t – a ) dσ ( t ) (sm. (16)).
Teorema 4 daet kryteryj polnoj neopredelennosty zadaçy Nevanlynn¥ – Pyka v
termynax sxodymosty rqdov yz { Pr, ( n ) } (sm. (18)).
Zadaça Nevanlynn¥ – Pyka v klasse S [[[[ a, b ]]]]. Pust\ zadan¥ vewestvenn¥e
çysla a < b y natural\noe çyslo m. Oboznaçym C– = { z ∈ C : Im z < 0 }, C+ =
= { z ∈ C : Im z > 0 }, C± = C– ∪ C+
. Symvolom C
m
× m oboznaçym mnoΩestvo
kompleksn¥x kvadratn¥x matryc porqdka m , symvolom CH — mnoΩestvo πr-
mytov¥x matryc, symvolamy C≥
×m m
y C>
×m m
— mnoΩestva sootvetstvenno ne-
otrycatel\n¥x y poloΩytel\n¥x matryc. Dlq neotrycatel\n¥x (poloΩytel\-
n¥x) matryc budem takΩe yspol\zovat\ oboznaçenyq A ≥ 0 ( A > 0 ). Symvolamy
Im ∈ C
m
× m y 0 m ∈ C
m
× m budem oboznaçat\ edynyçnug y nulevug matryc¥.
Çerez S [ a, b ] oboznaçym mnoΩestvo holomorfn¥x matryc-funkcyj s : C \ [ a,
b ] → C
m
× m takyx, çto
s z s z
z z
( ) − ( )
−
*
≥ 0 ∀z ∈ C±
, s ( x ) ≥ 0 ∀x ∈ R \ [ a, b ].
V [1, s. 528] dokazano, çto matryca-funkcyq prynadleΩyt S [ a, b ] tohda y
tol\ko tohda, kohda ona dopuskaet yntehral\noe predstavlenye vyda
s ( z ) = ( − ) ( )
−∫b z
d t
t z
a
b
σ
. (1)
Zdes\ σ : [ a, b ] →
CH
m m×
— neub¥vagwaq matryca-funkcyq ohranyçennoj va-
ryacyy.
© G. M. DGKAREV, Y. G. SERYKOVA, 2007
764 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
ORTOHONAL|NÁE NA KOMPAKTNOM YNTERVALE … 765
Pust\ zadan¥ beskoneçnaq posledovatel\nost\ poparno razlyçn¥x kompleks-
n¥x çysel Z∞ = { } =
∞z j j 1 ⊂ C + y beskoneçnaq posledovatel\nost\ matryc
{ } =
∞s j j 1 ⊂ C
m
× m. V zadaçe Nevanlynn¥ – Pyka trebuetsq opysat\ vse matryc¥-
funkcyy s ∈ S [ a, b ] takye, çto
s ( zj ) = sj ∀j ∈ N. (2)
MnoΩestvo vsex reßenyj zadaçy (2) oboznaçym F∞
.
Zafyksyruem n ∈ N. Narqdu s zadaçej (2) budem rassmatryvat\ useçennug
zadaçu Nevanlynn¥ – Pyka, v kotoroj trebuetsq opysat\ vse s ∈ S [ a, b ] takye,
çto
s ( zj ) = sj
, 1 ≤ j ≤ n. (3)
MnoΩestvo vsex reßenyj zadaçy (3) oboznaçym Fn
. Qsno, çto F ∞ =
Fnn =
∞
1∩ .
Vvedem oboznaçenyq
Zn = { } =z j j
n
1 ⊂ C+
, Zn = { } =z j j
n
1 ⊂ C–
, Z∞ = { } =
∞z j j 1 ⊂ C–
.
S n-j useçennoj zadaçej (3) svqΩem sledugwye obæekt¥:
s1 ( z ) = s ( z ), s2 ( z ) =
z a
b z
s z
−
−
( ), (4)
T( n ) =
z I
z I
z I
m m m
m m m
m m m m
1
2
0 0
0 0
0 0
…
…
…
� � � �
, s̃
z a
b z
sj
j
j
j=
−
−
, v( n ) =
I
I
m
m
�
,
RT, ( n ) ( z ) = ( T( n ) – z Imn )
–
1 =
( − ) …
… ( − )
−
−
z z I
z z I
m m
m n m
1
1
1
0
0
� � � ,
K1, ( n ) =
s s
z z
i j
i j i j
n
−
−
=
*
, 1
, K2, ( n ) =
˜ ˜*
,
s s
z z
i j
i j i j
n
−
−
=1
,
u1, ( n ) =
s
sn
1
�
, u2, ( n ) =
˜
˜
s
sn
1
�
.
Opredelenye 1. Useçennaq zadaça Nevanlynn¥ – Pyka (3) naz¥vaetsq vpol-
ne neopredelennoj, esly
K1, ( n ) > 0, K2, ( n ) > 0.
Opredelenye 2. Matryca-funkcyq
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
766 G. M. DGKAREV, Y. G. SERYKOVA
U( n ) ( z ) =
I z b R b K R z u
z b u R b K R z u
m n T n n T n n
n T n n T n n
− ( − ) ( ) ( )
( − ) ( ) ( )
( ) ( ) ( )
−
( ) ( )
( ) ( ) ( )
−
( ) ( )
v*
,
*
, , ,
,
*
,
*
, , ,
2
1
2
1 1
1
1
( − ) ( ) ( )
+ ( − ) ( ) ( )
( ) ( ) ( )
−
( ) ( )
( ) ( ) ( )
−
( ) ( )
z a R b K R z
I z b u R b K R z
n T n n T n n
m n T n n T n n
v v
v
*
,
*
, ,
,
*
,
*
, ,
2
1
1 1
1 =
α β
γ δ
( ) ( )
( ) ( )
( ) ( )
( ) ( )
n n
n n
z z
z z
(5)
naz¥vaetsq rezol\ventnoj matrycej zadaçy (3).
Rassmotrym bloçn¥e matryc¥
J =
0
0
m m
m m
iI
iI
−
, Jπ =
0
0
m m
m m
I
I
.
V [3] rassmotren¥ reßenyq Krejna y Frydryxsa ynterpolqcyonnoj zada-
çyN(3)
sK, ( n ) ( z ) = β α( )
−
( )( ) ( )n nz z1 ∈ Fn
, sF, ( n ) ( z ) = δ γ( )
−
( )( ) ( )n nz z1 ∈ Fn (6)
y poluçen¥ neravenstva
0m < sF, ( n ) ( x ) ≤ s( n ) ( x ) ≤ sK, ( n ) ( x ) ∀s( n ) ∈ Fn ∀x ∈ R \ [ a, b ]. (7)
Opredelenye 3. Matryçn¥j ynterval
I( n ) ( x ) = { A ∈ C
m
× m : sF, ( n ) ( x ) ≤ A ≤ sK, ( n ) ( x ) }, x ∈ R \ [ a, b ],
naz¥vaetsq yntervalom Vejlq v toçke x, assocyyrovann¥m s useçennoj zada-
çej Nevanlynn¥ – Pyka (3).
Oçevydno, çto mnoΩestvo reßenyj ( n + 1 ) -j useçennoj zadaçy soderΩytsq
vo mnoΩestve reßenyj n-j useçennoj zadaçy Fn + 1 ⊂ Fn , n ∈ N. Otsgda y yz
neravenstva (7) dlq vsex n ∈ N ymeem
0m < sF, ( n ) ( x ) ≤ sF, ( n + 1 ) ( x ) ≤ sK, ( n + 1 ) ( x ) ≤ sK, ( n ) ( x ), x ∈ R \ [ a, b ].
V termynax yntervalov Vejlq πty neravenstva moΩno zapysat\ v vyde In + 1 ( x ) ⊂
⊂ In ( x ), n ∈ N.
Teorema 1. Pust\ F∞ oboznaçaet mnoΩestvo reßenyj zadaçy (2), Fn —
mnoΩestvo reßenyj n-j useçennoj zadaçy, a sF, ( n ) y s K, ( n ) — reßenyq Fryd-
ryxsa y Krejna sootvetstvenno. Tohda:
1) suwestvugt ravnomern¥e na kompaktax K ⊂ C \ [ a, b ] predel¥
sK, ( ∞ ) ( z ) : = lim
n→∞
sK, ( n ) ( z ) ∈ F∞
, sF, ( ∞ ) ( z ) : = lim
n→∞
sF, ( n ) ( z ) ∈ F∞
; (8)
2) dlq vsex s ∈ F∞ v¥polnqgtsq neravenstva
0m < sF, ( ∞ ) ( x ) ≤ s ( x ) ≤ sK, ( ∞ ) ( x ), x ∈ R \ [ a, b ]. (9)
Dokazatel\stvo πtoj teorem¥ provodytsq po toj Ωe sxeme, çto y dokaza-
tel\stvo analohyçn¥x rezul\tatov v [4, 5].
MoΩno dokazat\ (sm. [3]), çto sF, ( ∞ ) y sK, ( ∞ ) qvlqgtsq reßenyqmy vsex use-
çenn¥x zadaç (3). Takym obrazom, mnoΩestvo reßenyj ynterpolqcyonnoj zada-
çy Nevanlynn¥ – Pyka (2) ne pusto.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
ORTOHONAL|NÁE NA KOMPAKTNOM YNTERVALE … 767
Opredelenye 4. Matryçn¥j ynterval
I∞ ( x ) : = [ sF, ( ∞ ) ( x ), sK, ( ∞ ) ( x ) ]
naz¥vaetsq yntervalom Vejlq (2) v toçke x ∈ R \ [ a, b ].
Yz teorem¥ S. A. Orlova (sm. [6]) sleduet, çto dlq vsex x1
, x2 ∈ R \ [ a, b ]
ymeet mesto ravenstvo
rank { sK, ( ∞ ) ( x1 ) – sF, ( ∞ ) ( x1 ) } = rank { sK, ( ∞ ) ( x2 ) – sF, ( ∞ ) ( x2 ) }. (10)
Druhymy slovamy, ranhy predel\n¥x yntervalov Vejlq I∞ ( x ) ne zavysqt ot v¥-
bora toçky x ∈ R \ [ a, b ].
Opredelenye 5. Ynterval Vejlq v toçke x ∈ R \ [ a, b ] naz¥vaetsq nev¥-
roΩdenn¥m, esly sF, ( ∞ ) ( x ) < sK, ( ∞ ) ( x ), t. e.
rank ( sK, ( ∞ ) ( x ) – sF, ( ∞ ) ( x ) ) = m.
Opredelenye 6. Ynterpolqcyonnaq zadaça (2) naz¥vaetsq vpolne neopre-
delennoj, esly dlq vsex x ∈ R \ [ a, b ] predel\n¥e ynterval¥ Vejlq I∞ ( x ) qv-
lqgtsq nev¥roΩdenn¥my matryçn¥my yntervalamy.
Ortohonal\n¥e semejstva racyonal\n¥x matryc-funkcyj. Pust\ dana
vpolne neopredelennaq n-q useçennaq zadaça (3). Tohda
Kr, ( n ) =
K B
B
s s
z z
r n r n
r n
r n r n
n n
, ,
,
* , ,
*
( − ) ( )
( )
−
−
1
=
I
B K I
n m n m m
r n r n m
( − ) ( − ) ×
( ) ( − )
−
1 1
1
1
0
,
*
,
×
×
K
K
I K B
I
r n n m m
m n m r n
n m r n r n
m n m m
,
,
, ,
ˆ
( − ) ( − ) ×
×( − )
( − ) ( − )
−
( )
×( − )
1 1
1
1 1
1
1
0
0 0
, (11)
hde
ˆ
,Kr n =
s s
z z
B K B
r n r n
n n
r n r n r n
, ,
*
,
*
, ,
−
−
− ( ) ( − )
−
( )1
1
, r = 1, 2.
Otsgda sleduet, çto dlq vpolne neopredelennoj n-j useçennoj zadaçy (3)
ˆ
,Kr n > 0, r = 1, 2. Yz (11) ymeem
K
K
r n
r n
,
,
( )
− ( − )
−
=
1 1
1 0
0 0
+
−
−[ ]( − )
−
( ) −
( ) ( − )
−K B
I
K B K Ir n r n
m
r n r n r n m
, ,
, ,
*
,
ˆ1
1
1
1
1
. (12)
Teorema 2. Dlq toho çtob¥ ynterpolqcyonnaq zadaça (2) b¥la vpolne ne-
opredelennoj, neobxodymo, çtob¥ suwestvovaly stroho poloΩytel\n¥e pre-
del¥ dlq vsex x ∈ R \ [ a, b ]
lim * *
,
n
n T r n T nR x K R x
n n→∞
( ) ( )
−
( )[ ]
( ) ( )
( ) ( )v v
1 > 0m
, r = 1, 2, (13)
y dostatoçno, çtob¥ xotq b¥ dlq odnoho x0 ∈ R \ [ a, b ] suwestvovaly stroho
poloΩytel\n¥e predel¥ (13).
Dokazatel\stvo. Yz opredelenyq 6 y formul¥ (10) sleduet, çto dlq vpol-
ne neopredelennosty ynterpolqcyonnoj zadaçy (2) neobxodymo, çtob¥ pry vsex
x ∈ R \ [ a, b ] predel\n¥e ynterval¥ Vejlq b¥ly nev¥roΩden¥, y dostatoçno,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
768 G. M. DGKAREV, Y. G. SERYKOVA
çtob¥ xotq b¥ dlq odnoho x0 ∈ R \ [ a, b ] b¥l nev¥roΩdenn¥m sootvetstvug-
wyj predel\n¥j ynterval Vejlq. Takym obrazom, zafyksyruem toçku x0 ∈
∈ R \ [ a, b ] y dokaΩem utverΩdenye teorem¥ dlq toçky x0 .
Oçevydno, çto
rankR xT nn( )
( ) ( )0 v = m dlq vsex n ∈ N. Sledovatel\no, dlq
vpolne neopredelennoj zadaçy
v v( ) ( )
−
( )( ) ( )
( ) ( )n T r n T nR x K R x
n n
* *
,0
1
0 ∈ C>
×m m , r = 1, 2, ∀n ∈ N.
Yz (12) ymeem
v v( + ) ( + )
−
( + )( + ) ( + )
( ) ( )n T r n T nR x K R x
n n1 0 1
1
0 11 1
* *
, ≥
v v( ) ( )
−
( )( ) ( )
( ) ( )n T r n T nR x K R x
n n
* *
,0
1
0 .
V [3] dokazano, çto
{ }( ) ( )
−
( ) ( )
−
( )( ) − ( ) = ( − )
−
( ) ( )
( ) ( )
s x s x
x a
b a
R x K R xK n F n n T n T nn n, ,
* *
,0 0
1 0
2
0 2
1
0v v +
+
( − )( − )
−
( ) ( )( ) ( )
−
( )( ) ( )
x a x b
b a
R x K R xn T n T nn n
0 0
0 1
1
0v v* *
, . (14)
Slahaem¥e v pravoj çasty posledneho ravenstva qvlqgtsq stroho poloΩy-
tel\n¥my matrycamy y monotonno vozrastagt s rostom n (sm. (12)). Levaq
çast\, v sylu neopredelennosty ynterpolqcyonnoj zadaçy Nevanlynn¥ – Pyka
(2), pry n → ∞ stremytsq k poloΩytel\no opredelennoj matryce { (∞)( )s xK , 0 –
– s xF,(∞)
−( )}0
1
. Otsgda sleduet suwestvovanye y strohaq poloΩytel\nost\ pre-
delov v (13).
Naoborot, pust\ suwestvugt y stroho poloΩytel\n¥ oba predela v (13).
V¥polnym predel\n¥j perexod pry n → ∞ v obeyx çastqx ravenstva (14). Po-
luçym { }(∞) (∞)
−( ) − ( )s x s xK F, ,0 0
1 > 0. Takym obrazom, predel\n¥j ynterval Vej-
lq I∞ ( x0 ) qvlqetsq nev¥roΩdenn¥m y, sledovatel\no, ynterpolqcyonnaq zada-
ça (2) qvlqetsq vpolne neopredelennoj.
Teorema dokazana.
Rassmotrym dva semejstva racyonal\n¥x matryc-funkcyj, assocyyrovann¥x
s vpolne neopredelenn¥my useçenn¥my zadaçamy Nevanlynn¥ – Pyka (3)
Pr, ( 1 ) ( z ) = K R zr T,
/
,( )
−
( )( )1
1 2
1 ,
(15)
Pr, ( n ) ( z ) =
ˆ
,
/
,
*
, ,K B K I R zr n r n r n T n n( )
−
( ) ( − )
−
( ) ( )−[ ] ( )1 2
1
1
v , n > 1, r = 1, 2.
Teorema 3. Pust\ k, p ≥ 1, matryca-funkcyq s ∈ Fmax ( k, p ) y σ soder-
Ωytsq v yntehral\nom predstavlenyy (1) matryc¥-funkcyy s . Tohda ymegt
mesto sootnoßenyq obobwennoj ortohonal\nosty
P t b t
t a
b t
d t P t
I k p
k pr k
a
b r
r p
m m
m m
, ,
*
, ,
, ,( )
−
( )
×
×
( )( − ) −
−
( ) ( ) =
=
≠
∫
1
0
σ r = 1, 2. (16)
Dokazatel\stvo. Yz yntehral\noho predstavlenyq (1) sleduet ravenstvo
Kr, ( k ) = R t b t
t a
b t
d t R tT k k
a
b r
k T k,
*
,
*
( ) ( )
−
( ) ( )( ) ( − ) −
−
( ) ( )∫ v v
1
σ , r = 1, 2. (17)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
ORTOHONAL|NÁE NA KOMPAKTNOM YNTERVALE … 769
Pust\ k = p > 1. Tohda
P t b t
t a
b t
d t P t K B K Ir k
a
b r
r p r k r k r k, ,
*
,
/
,
*
,
ˆ
( )
−
( ) ( )
−
( ) ( − )
−( )( − ) −
−
( ) ( ) = −[ ]∫
1
1 2
1
1σ ×
×
R t b t
t a
b t
d t R t
K B
I
T k k
r
k T k
a
b
r k r k
,
*
,
* , ,
( ) ( )
−
( ) ( )
( − )
−
( )( ) ( − ) −
−
( ) ( )
−
∫ v v
1
1
1
σ ×
× ˆ ˆ ˆ
,
/
,
/
,
*
, ,
, ,
,
/K K B K I K
K B
I
Kr k r k r k r k r k
r k r k
r k
− −
( ) ( − )
−
( )
( − )
−
( ) −= −[ ] −
1 2 1 2
1
1 1
1
1 2 =
= ˆ ˆ ˆ
,
/
, ,
/K K Kr k r k r k
− −1 2 1 2 = Im × m
.
Zdes\ vtoroe ravenstvo sleduet yz (17), a tret\e — yz (11).
Pust\ teper\ k > p > 1. Ymeem
P t b t
t a
b t
d t P t K B K Ir k
a
b r
r p r k r k r k, ,
*
,
/
,
*
,
ˆ
( )
−
( )
−
( − )
−( )( − ) −
−
( ) ( ) = −[ ]∫
1
1 2
1
1σ ×
×
R t b t
t a
b t
d t R t
I
T k k
r
k T k
a
b
pm pm
k p m pm
,
*
,
*
( ) ( )
−
( ) ( )
×
( − ) ×
( ) ( − ) −
−
( ) ( )
∫ v v
1
0
σ ×
×
−
= [ ]
( − )
−
( ) − −
( − ) × ( )
×
( − ) ×
K B
I
K K K
I
r p r p
r p r k k m m r k
pm pm
k p m pm
, ,
,
/
,
/
,
ˆ ˆ ˆ1
1
1 2 1 2
10
0
×
×
−
=( − )
−
( ) −
×
K B
I
Kr p r p
r k m m
, ,
,
/ˆ1
1
1 2 0 .
Formul¥ (16) dokazan¥ pry p > 0. Dlq p = 0 ony oçevydn¥.
Teorema dokazana.
Teorema 4. Dlq toho çtob¥ zadaça Nevanlynn¥ – Pyka (2) b¥la vpolne ne-
opredelennoj, neobxodymo, çtob¥ pry vsex x ∈ R \ [ a, b ] sxodylys\ rqd¥
P x P xj j
j
1 1
1
,
*
,( ) ( )
=
∞
( ) ( )∑ , P x P xj j
j
2 2
1
,
*
,( ) ( )
=
∞
( ) ( )∑ , (18)
y dostatoçno, çtob¥ rqd¥ (18) sxodylys\ xotq b¥ pry odnom x0 ∈ R \ [ a, b ].
Dokazatel\stvo. Ymegt mesto ravenstva
v v( ) ( ) ( )
−
( ) ( ) ( ) ( )
=
∞
( ) ( ) = ( ) ( )∑k T k r k T k k r j r j
j
R x K R x P x P x*
,
*
, , ,
*
,
1
1
, r = 1, 2.
Dejstvytel\no,
v v( ) ( ) ( )
−
( ) ( )( ) ( )k T k r k T k kR x K R x*
,
*
, ,
1 =
=
v( ) ( )
( − )
−
( − )
−
( )
( − )
−( )
+
−
−[ ]
k T k
r k r k r k
r k r k r kR x
K K B
I
K B K I*
,
* , , ,
, ,
*
,
ˆ1
1
1
1
1
10
0 0
×
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
770 G. M. DGKAREV, Y. G. SERYKOVA
×
R x R x K R x P x P xT k k k T k r k T k k r k r k,
*
,
*
, , ,
*
,( ) ( ) ( − ) ( − ) ( − )
−
( − ) ( − ) ( ) ( )( ) = ( ) ( ) + ( ) ( )v v v1 1 1
1
1 1 = …
… = P x P xr j r j
j
,
*
,( ) ( )
=
∞
( ) ( )∑
1
.
Zdes\ m¥ vospol\zovalys\ (12). Otsgda y yz teorem¥ 2 sleduet utverΩdenye
teorem¥.
1. Krejn M. H., Nudel\man A. A. Problema momentov Markova y πkstremal\n¥e zadaçy. – M.:
Nauka, 1973. – 552 s.
2. Dgkarev G. M., Çoke Ryvero A. E. Zadaça Nevanlynn¥ – Pyka v klasse S [ a, b ] // Yzv. vuzov.
Matematyka. – 2003. – S. 36 – 45.
3. Dyukarev Yu. M., Serikova I. Yu. Friedrichs and Krein solutions of the Nevanlinna – Pick
interpolation problem in the class S [ a, b ] // Zb. prac\ In-tu matematyky NAN Ukra]ny. – 2004. –
1, # 3. – S. 55 – 66.
4. Dgkarev G. M. O kryteryqx neopredelennosty matryçnoj problem¥ momentov Stylt\esa
// Mat. zametky. – 2004. – 75, # 1. – S. 71 – 88.
5. Dgkarev G. M. Zadaça Nevanlynn¥ – Pyka dlq styl\t\esovskyx matryc-funkcyj // Ukr.
mat. Ωurn. – 2004. – 56, # 2. – S. 366 – 380.
6. Orlov S. A. Hnezdqwyesq matryçn¥e kruhy, analytyçesky zavysqwye ot parametra, y
teorem¥ ob ynvaryantnosty ranhov radyusa predel\n¥x matryçn¥x kruhov // Yzv. AN SSSR.
Ser. mat. – 1976. – 40, # 3. – S. 593 – 644.
Poluçeno 26.01.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
|
| id | umjimathkievua-article-3343 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:40:47Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f9/6cf7fe61ced283f9238de74fd4daebf9.pdf |
| spelling | umjimathkievua-article-33432020-03-18T19:51:58Z Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval Ортогональные на компактном интервале рациональные матрицы-функции, ассоциированные с задачей Неванлинны – Пика в классе S [a, b] Dyukarev, Yu. M. Serikova, I. Yu. Дюкарев, Ю. М. Серикова, И. Ю. Дюкарев, Ю. М. Серикова, И. Ю. We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions. Розглянуто інтерполяційну задачу Неванлінни &#8211; Піка з нескінченною кількістю вузлів інтерполяції у класі S [a, b], з якою пов'язано ортогональні на інтервалі [a, b] раціональні матриці-функції. Отримано критерій повної невизначеності нескінченної задачі Неванлінни &#8211; Піка у термінах ортогональних раціональних матриць-функцій. Institute of Mathematics, NAS of Ukraine 2007-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3343 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 6 (2007); 764–770 Український математичний журнал; Том 59 № 6 (2007); 764–770 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3343/3430 https://umj.imath.kiev.ua/index.php/umj/article/view/3343/3431 Copyright (c) 2007 Dyukarev Yu. M.; Serikova I. Yu. |
| spellingShingle | Dyukarev, Yu. M. Serikova, I. Yu. Дюкарев, Ю. М. Серикова, И. Ю. Дюкарев, Ю. М. Серикова, И. Ю. Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval |
| title | Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval |
| title_alt | Ортогональные на компактном интервале рациональные матрицы-функции, ассоциированные с задачей Неванлинны – Пика в классе S [a, b] |
| title_full | Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval |
| title_fullStr | Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval |
| title_full_unstemmed | Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval |
| title_short | Rational matrix functions associated with the Nevanlinna-Pick problem in the class S [a, b] and orthogonal on a compact interval |
| title_sort | rational matrix functions associated with the nevanlinna-pick problem in the class s [a, b] and orthogonal on a compact interval |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3343 |
| work_keys_str_mv | AT dyukarevyum rationalmatrixfunctionsassociatedwiththenevanlinnapickproblemintheclasssabandorthogonalonacompactinterval AT serikovaiyu rationalmatrixfunctionsassociatedwiththenevanlinnapickproblemintheclasssabandorthogonalonacompactinterval AT dûkarevûm rationalmatrixfunctionsassociatedwiththenevanlinnapickproblemintheclasssabandorthogonalonacompactinterval AT serikovaiû rationalmatrixfunctionsassociatedwiththenevanlinnapickproblemintheclasssabandorthogonalonacompactinterval AT dûkarevûm rationalmatrixfunctionsassociatedwiththenevanlinnapickproblemintheclasssabandorthogonalonacompactinterval AT serikovaiû rationalmatrixfunctionsassociatedwiththenevanlinnapickproblemintheclasssabandorthogonalonacompactinterval AT dyukarevyum ortogonalʹnyenakompaktnomintervaleracionalʹnyematricyfunkciiassociirovannyeszadačejnevanlinnypikavklassesab AT serikovaiyu ortogonalʹnyenakompaktnomintervaleracionalʹnyematricyfunkciiassociirovannyeszadačejnevanlinnypikavklassesab AT dûkarevûm ortogonalʹnyenakompaktnomintervaleracionalʹnyematricyfunkciiassociirovannyeszadačejnevanlinnypikavklassesab AT serikovaiû ortogonalʹnyenakompaktnomintervaleracionalʹnyematricyfunkciiassociirovannyeszadačejnevanlinnypikavklassesab AT dûkarevûm ortogonalʹnyenakompaktnomintervaleracionalʹnyematricyfunkciiassociirovannyeszadačejnevanlinnypikavklassesab AT serikovaiû ortogonalʹnyenakompaktnomintervaleracionalʹnyematricyfunkciiassociirovannyeszadačejnevanlinnypikavklassesab |