On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$
We prove that the theorem on the incompleteness of polynomials in the space $C^0_w$ established by de Branges in 1959 is not true for the space $L_p (ℝ, dμ)$) if the support of the measure μ is sufficiently dense
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| Sprache: | Russisch Englisch |
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2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509035996905472 |
|---|---|
| author | Bakan, A. G. Бакан, А. Г. Бакан, А. Г. |
| author_facet | Bakan, A. G. Бакан, А. Г. Бакан, А. Г. |
| author_sort | Bakan, A. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:35Z |
| description | We prove that the theorem on the incompleteness of polynomials in the space $C^0_w$ established by de Branges in 1959 is not true for the space $L_p (ℝ, dμ)$) if the support of the measure μ is sufficiently dense |
| first_indexed | 2026-03-24T02:34:42Z |
| format | Article |
| fulltext |
UDK 517.538
A. H. Bakan (Yn-t matematyky NAN Ukrayn¥, Kyev)
O POLNOTE ALHEBRAYÇESKYX POLYNOMOV
V PROSTRANSTVAX L dp( )R, µµ
We prove that the theorem on polynomial incompleteness in the space Cw
0 established by Lui de
Branges in 1959 fails to be true for the space L dp ( , )R µ provided that the support of the measure µ is
sufficiently dense.
Dovedeno, wo vstanovlena Lu] de BranΩem u 1959 roci teorema pro polinomial\nu nepovnotu u
prostori Cw
0 ne [ pravyl\nog dlq prostoru L dp ( , )R µ , qkwo nosij miry µ [ dostatn\o wil\-
nym.
1. Predvarytel\n¥e svedenyq y osnovnoj rezul\tat. 1.1. Teorema de
BranΩa. Pust\ M
+ ( )R oboznaçaet konus koneçn¥x borelevskyx mer na dej-
stvytel\noj osy R , M
∗( )R � mnoΩestvo mer µ ∈ +M ( )R so vsemy koneçn¥-
my momentamy sn( )µ : =
x d xn µ( )
R∫ , n ∈ N0 : = { , , , }0 1 2 … , y neohranyçenn¥m
nosytelem supp µ : = { }: ( , )x x x∈ ∀ > − + >R ε µ ε ε0 0 , M
∗ +( )R � mnoΩe-
stvo mer µ ∈ ∗M ( )R , dlq kotor¥x supp µ ⊂ R+ : = [ 0, + ∞ ) , C( )R � lynej-
noe prostranstvo vsex dejstvytel\noznaçn¥x y neprer¥vn¥x na R funkcyj, P
� mnoΩestvo vsex alhebrayçeskyx mnohoçlenov s dejstvytel\n¥my koπffycy-
entamy, P
s D[ ] � mnoΩestvo tex mnohoçlenov yz P , vse korny kotor¥x
prost¥e y prynadleΩat mnoΩestvu D ⊂ R, B( )R � semejstvo borelevskyx
podmnoΩestv R, P � sovokupnost\ lynejn¥x topolohyçeskyx prostranstv
dejstvytel\noznaçn¥x funkcyj na R, dlq kotor¥x P qvlqetsq plotn¥m pod-
mnoΩestvom, y W
∗( )R � mnoΩestvo poluneprer¥vn¥x sverxu funkcyj w :
R → R
+ s xn
w
< ∞ dlq vsex n ∈ N0, hde f w : = sup ( ) ( )x w x f x∈R . Dlq
w ∈ ∗W ( )R prostranstvo Cw
0 opredelqetsq kak lynejnoe mnoΩestvo vsex
funkcyj f C∈ ( )R , dlq kotor¥x lim ( ) ( )x w x f x→∞ = 0, osnawennoe polu-
normoj ⋅ w . V πtom sm¥sle funkcyy w ∈ ∗W ( )R takΩe naz¥vagtsq vesamy.
Esly µ ∈ +M ( )R y D ∈B( )R , to suΩenyem mer¥ µ na mnoΩestvo D naz¥va-
etsq mera
µ&D, opredelennaq na borelevskoj σ -alhebre B( )R po formule
µ&D A( ) : = µ ( )A DI , A ∈B( )R .
Dlq celoj funkcyy f opredelym M rf ( ) : = sup ( )z r f z= , r ≥ 0, y pust\
Λ f oboznaçaet mnoΩestvo vsex nulej f . Hovorqt, çto f ymeet mynymal\n¥j
πksponencyal\n¥j typ, esly lim log ( )r fr M r→ +∞
−1 = 0. Oboznaçym çerez
E
S
0 [ ]D semejstvo dejstvytel\n¥x na dejstvytel\noj osy transcendentn¥x
cel¥x funkcyj f mynymal\noho πksponencyal\noho typa, vse korny kotor¥x
prost¥e y prynadleΩat mnoΩestvu D ⊂ R, a çerez E
H
0 [ ]D mnoΩestvo tex
© A. H. BAKAN, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3 291
292 A. H. BAKAN
f D∈ES
0 [ ], dlq kotor¥x lim ( ),λ λ λ λ→∞ ∈ ′Λ f
m f = 0 pry lgbom m ∈
∈ N0 .
V 1924 h. S. N. Bernßtejn [1] sformulyroval problemu o naxoΩdenyy us-
lovyj na ves w ∈ ∗W ( )R , pry kotor¥x alhebrayçeskye mnohoçlen¥ plotn¥ v
prostranstve Cw
0 . V 1959 h. Luy de BranΩ poluçyl reßenye πtoj problem¥ [2]
(sm. takΩe [3 – 5]), kotoroe v 1996 h. b¥lo modyfycyrovano M. L. Sodyn¥m y
P. M. Gdytskym [6] y formulyruetsq v vyde sledugwej teorem¥.
Teorema A. Pust\ dlq w ∈ ∗W ( )R mnoΩestvo Sw : = { }( )x w x∈ >R 0
qvlqetsq neohranyçenn¥m. Alhebrayçeskye mnohoçlen¥ P plotn¥ v Cw
0 toh-
da y tol\ko tohda, kohda dlq lgboj funkcyy B Sw∈EH
0 [ ] ymeet mesto
1
w B
B
( ) ( )λ λλ ′∈
∑
Λ
= + ∞ . (1)
Rassmotrym uslovye (1) bolee podrobno. Esly f S Sw w∈E ES H
0 0[ ] [ ]\ , to su-
westvuet takoe m0 0∈N , çto lim ( ),λ λ λ λ→∞ ∈ ′Λ f
m f0 > 0, y potomu yz op-
redelenyq klassa W
∗( )R poluçaem lim ( ) ( ),λ λ λ λ→∞ ∈ ′Λ f
w f = 0, t. e. raven-
stvo (1) verno dlq B = f . Takym obrazom, trebovanye B Sw∈EH
0 [ ] v teoreme A
moΩno zamenyt\ πkvyvalentn¥m B Sw∈ES
0 [ ].
PredpoloΩym, çto mnoΩestvo Sw dyskretno y suwestvuet takaq funkcyq
E ∈ES
0 [ ]R , çto Sw = ΛE . Tohda yz v¥polnenyq (1) dlq B = E sleduet, çto
dlq lgboj celoj funkcyy G , udovletvorqgwej trebovanyg E G/ ∈P ,
G ∈ES
0 [ ]R y v¥polnqetsq uslovye (1) dlq B = G.
Dejstvytel\no, tak kak E z G z( ) ( )/ = P z( ) = p z pn
n0 + …+ , hde n ∈ N , pk ∈
∈ R , 0 ≤ k ≤ n, p0 ≠ 0, to G z( ) = E z P z( ) ( )/ y suwestvuet takoe çyslo R ≥ 1,
çto P z( ) ≥ 2 1
0
− p z n dlq proyzvol\noho z ∈C , z R≥ . Poπtomu
log ( )M rG ≤ log ( ) log log/M r p n rE − −0 2 pry r ≥ R y, znaçyt, G ymeet myny-
mal\n¥j πksponencyal\n¥j typ. Takym obrazom, G ∈ES
0 [ ]R . Krome toho, po-
skol\ku vnutry kruha UR : = z z R∈ ≤{ }C naxodytsq koneçnoe çyslo nulej
funkcyy E y ΛP ⊂ UR , to
∞ =
1
w E
E RU ( ) ( )\ λ λλ ′∈
∑
Λ
=
1
w G P
G RU ( ) ( ) ( )\ λ λ λλ
′
∈
∑
Λ
≤
≤
2
0w G p n
UG R
( ) ( )\ λ λ λλ ′∈
∑
Λ
≤
2 2
0p w G
G RU ( ) ( )\ λ λλ
′
∈
∑
Λ
≤
≤ 2 2
0p w G
G
( ) ( )λ λλ
′
∈
∑
Λ
,
t. e. uslovye (1) v¥polnqetsq dlq B = G.
∏to oznaçaet, çto dlq v¥polnenyq Cw
0 ∈P ostalos\ potrebovat\ v¥polne-
nyq (1) dlq vsex cel¥x funkcyj yz mnoΩestva D0[ ]ΛE , hde
D0[ ]D : =
g D D g∈ = ∞{ }ES
0 [ ] ( )\card Λ , D ⊂ R , (2)
y card A ∈ ∞N0 U { } oboznaçaet kolyçestvo πlementov mnoΩestva A. MnoΩe-
stvo D0[ ]ΛE ymeet specyal\noe svojstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
O POLNOTE ALHEBRAYÇESKYX POLYNOMOV V PROSTRANSTVAX L dp ( , )R µ 293
g E∈D0[ ]Λ ⇒ P z g z E( ) ( ) [ ]∈D0 Λ ∀ ∈P E gPS[ ]\Λ Λ , (3)
pryçem stepeny polynomov yz mnoΩestva P
S[ ]\Λ ΛE g ne ohranyçen¥ sverxu.
Poπtomu v¥polnenye (1) dlq B = g E∈D0[ ]Λ vleçet
1
1 +( ) ′∈
∑
λ λ λλ
mw g
g
( ) ( )Λ
= + ∞ ∀ ∈m N0 ,
çto v sylu neravenstva 1 1 1+( ) +
∈∑ λ ε
λ Λg
< ∞ , ε > 0, πkvyvalentno
lim ( ) ( )
,λ λ
λ λ λ
∈ →∞
′
Λg
m w g = 0 ∀ ∈m N0,
yly, çto to Ωe samoe,
lim
log ( ) ( )
log,λ λ
λ λ
λ∈ →∞
′( )
Λg
w g
= – ∞ . (4)
Esly Ωe Sw ne sovpadaet s mnoΩestvom nulej ny odnoj yz funkcyj mno-
Ωestva E
S
0 [ ]R , to mnoΩestvo Sw B\ Λ beskoneçno dlq lgboj funkcyy B ∈
∈ E
S
0 [ ]Sw , y potomu v πtom sluçae
B Sw∈ES
0 [ ] ⇒ B Sw∈D0[ ] ⇒ PB Sw∈D0[ ] ∀ ∈P Sw BPS[ ]\ Λ , (5)
otkuda toçno tak Ωe, kak y v¥ße, poluçaem, çto uslovye (1) moΩno zamenyt\
πkvyvalentn¥m uslovyem (4) s g = B . Takym obrazom, m¥ dokazaly sledugwug
modyfycyrovannug formu teorem¥ A.
Teorema B. Pust\ dlq w ∈ ∗W ( )R mnoΩestvo Sw : = { }( )x w x∈ >R 0
qvlqetsq neohranyçenn¥m. Alhebrayçeskye mnohoçlen¥ P plotn¥ v Cw
0 toh-
da y tol\ko tohda, kohda
a) lim
log
( )
log ( )
logλ
λ
λ
λ
λ∈
→∞
− ′
ΛF
w
F
1
= + ∞ ∀ ∈F SwD0[ ]
y
b)
λ λ λ∈
∑ ′ΛE
w E
1
( ) ( )
= ∞ ,
esly suwestvuet takaq funkcyq E ∈ E
S
0 [ ]R , çto Sw = ΛE .
1.2. Osnovnoj rezul\tat. Dlq lgboho vesa w ∈ ∗W ( )R s neohranyçenn¥m
Sw yz teorem¥ A sleduet utverΩdenye
Cw
0 ∉P ⇔ ∃ ∈B SwES
0 [ ]:
Cw
B
χΛ
0 ∉P , (6)
hde χD oboznaçaet yndykatornug funkcyg mnoΩestva D ⊂ R . Yn¥my slova-
my, yz nepolnot¥ alhebrayçeskyx polynomov v prostranstve Cw
0
sleduet yx ne-
polnota na suΩenyy πtoho prostranstva na dostatoçno redkoe mnoΩestvo, qv-
lqgweesq mnoΩestvom vsex nulej nekotoroj celoj funkcyy mynymal\noho
πksponencyal\noho typa so vsemy prost¥my nulqmy, prynadleΩawymy mno-
Ωestvu Sw . Podobnoe svojstvo dlq prostranstv L dp( , )R µ b¥lo ustanovleno
v 1998 h. A. A. Boryçev¥m y M. L. Sodyn¥m [7] v sluçae, kohda mera µ dyskret-
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
294 A. H. BAKAN
na y ee nosytel\ udovletvorqet uslovyg
∃ β > 0 : card supp[ , ]−( )r r I µ = O r( )β , r → + ∞ . (7)
Ymenno, yz teorem¥ A y utverΩdenyq A1.5 [7, c. 225, 255] v¥tekaet takoe sled-
stvye.
Sledstvye A. Pust\ 1 ≤ p < ∞ , mera µ ∈ ∗M ( )R dyskretna y ee nosy-
tel\ supp µ udovletvorqet uslovyg (7). Alhebrayçeskye polynom¥ P ne
plotn¥ v prostranstve L dp( , )R µ tohda y tol\ko tohda, kohda suwestvuet
takaq celaq funkcyq E ∈EH
0 [ ]supp µ , çto alhebrayçeskye polynom¥ P ne
plotn¥ v prostranstve
L dp E
( , )R µ&Λ .
PokaΩem, kak πtot rezul\tat v¥tekaet yz sledugwej teorem¥, dokazannoj v
[8, c. 38] (teorema 2.1).
Teorema C. Pust\ µ ∈ ∗M ( )R y 1 ≤ p < ∞ . Alhebrayçeskye polynom¥
P plotn¥ v prostranstve L dp( , )R µ tohda y tol\ko tohda, kohda suwestvu-
gt takaq mera ν ∈ +M ( )R y ves w ∈ ∗W ( )R , çto Cw
0 ∈P y dµ = w dp ν,
t. e. µ ( )A = w x d xp
A
( ) ( )ν∫ dlq lgboho mnoΩestva A ∈B ( )R .
Rassmotrym proyzvol\nug dyskretnug meru µ ∈ ∗M ( )R , opredelennug
formuloj
d xµ( ) = µ δ λλ
λ µ
( )x
S
−
∈
∑ , x ∈ R ,
hde mnoΩestvo Sµ : = supp µ sçetno y neohranyçeno. Esly 1 ≤ p < ∞ , to v
sylu teorem¥ C ymeem L dp( , )R µ ∈P tohda y tol\ko tohda, kohda suwestvugt
takaq mera ν ∈ +M ( )R y ves w ∈ ∗W ( )R , çto Cw
0 ∈P y
d xν( ) = ν δ λλ
λ µ
( )x
S
−
∈
∑ , w x( ) : = w x
S
λ λ
λ
χ
µ
{ }( )
∈
∑ , x ∈ R ,
νλ
λ µ∈
∑
S
< ∞ , µλ = wp
λ λν , wλ , µλ > 0, λ µ∈S .
Prymenqq k vesu w teoremu B, ymeem
L dp( , )R µ ∈P ⇔ ∃ ∈ ={ } :νλ λ µµS supp ⊂ ( 0, + ∞ ) : νλ
λ µ∈
∑
S
< ∞ , (8.1)
lim
log log log ( )
logλ
λ
λ λµ ν
λ
λ∈
→∞
− − ′
ΛF
p F
1 1
= + ∞ ∀ ∈F SD0[ ]µ , (8.2)
ν
µ λ
λ
λλ µ
1
1
/
/ ( )
p
p
S E ′∈
∑ = + ∞, esly ∃ ∈ =E S EES
0 [ ] :R µ Λ . (8.3)
Zametym, çto v πtyx uslovyqx moΩno trebovat\ suwestvovanye summyruem¥x
poloΩytel\n¥x posledovatel\nostej v (8.2) y (8.3) otdel\no, tak kak esly dve
poloΩytel\n¥e summyruem¥e posledovatel\nosty { }νλ λ µ
j
S∈ udovletvorqgt
uslovyqm (8.2), (8.3), to posledovatel\nost\ νλ : = max ,{ }ν νλ λ
b c , λ µ∈S , bu-
det udovletvorqt\ uslovyqm (8.1) � (8.3). Bolee toho, vsledstvye yzvestn¥x
svojstv posledovatel\nostej yz l p (sm. [9], hl. 4, p. 4.4, hl. 7, § 1, teorema 1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
O POLNOTE ALHEBRAYÇESKYX POLYNOMOV V PROSTRANSTVAX L dp ( , )R µ 295
ν
µ λ
λ
λλ µ
1
1
/
/ ( )
p
p
S E′∈
∑ < ∞ ∀ ∈{ }νλ λ µS ⊂ ( 0, + ∞ ) , νλ
λ µ∈
∑
S
< ∞ ⇔
⇔ 1
1 1 1µ λλλ µ
/( ) /( )( )p p p
S E− −
∈ ′
∑ < ∞ , esly 1 < p < ∞ ,
y
lim ( )
λ
λ
λµ λ
∈
→∞
′
ΛE
E > 0, esly p = 1,
y potomu levug çast\ uslovyq (8.3) moΩno zamenyt\ ravenstvamy
1
1 1 1µ λλλ µ
/( ) /( )( )p p p
S E− −
∈ ′
∑ = ∞ , esly 1 < p < ∞ ;
(8.4)
lim ( )
λ
λ
λµ λ
∈
→∞
′
ΛE
E = 0, esly p = 1,
kotor¥e uΩe ne zavysqt ot posledovatel\nosty { }νλ λ µ∈S . Yz ostavßyxsq us-
lovyj (8.1) y (8.2) poluçaem lim ,λ λ λµ
ν∈ →∞S = 0 y moΩem zamenyt\ trebova-
nye νλ > 0 v (8.1) πkvyvalentn¥m trebovanyem νλ ≥ γ λ s lgboj fyksyro-
vannoj poloΩytel\noj y summyruemoj posledovatel\nost\g { }γ λ λ µ∈S . Esly
mera µ udovletvorqet uslovyg (7), to moΩno poloΩyt\ γ λ = 1 1+( )− −λ β .
Tohda dlq dostatoçno bol\ßyx λ ∈ΛF budem ymet\ neravenstva
0 > – log
1
νλ
≥ – log
1
γ λ
= – ( ) log1 1+ +( )β λ ,
podstanovka kotor¥x v uslovye (8.2) daet vozmoΩnost\ utverΩdat\, çto ono ne
zavysyt ot v¥bora posledovatel\nosty { }νλ λ µ∈S . Teper\ moΩno sformulyro-
vat\ teoremu, kotoraq qvlqetsq nebol\ßoj modyfykacyej teorem¥ A y ut-
verΩdenyq A1.5 yz [7, c. 225, 255].
Teorema D. Pust\ 1 ≤ p < ∞ , mera µ ∈ ∗M ( )R dyskretna y ee nosy-
tel\ Sµ : = supp µ udovletvorqet uslovyg (7). Alhebrayçeskye polynom¥ P
plotn¥ v prostranstve L dp( , )R µ tohda y tol\ko tohda, kohda
lim
log log ( )
logλ
λ
λµ
λ
λ∈
→∞
− ′
ΛF
p F
1
= + ∞ ∀ ∈F SD0[ ]µ , (9)
y v¥polnqetsq sootvetstvugwee ravenstvo (8.4), esly suwestvuet takaq ce-
laq funkcyq E ∈ES
0 [ ]R , çto ΛE = Sµ .
Sledstvye A oçevydn¥m obrazom v¥tekaet yz teorem¥ D. Esly dyskretnaq
mera µ ∈ ∗M ( )R ne udovletvorqet uslovyg (7), to uslovye (8.3) stanovytsq ne-
nuΩn¥m, y poπtomu kryteryj L dp( , )R µ ∈P budet sostoqt\ tol\ko yz uslovyj
(8.1), (8.2).
Hlavn¥m rezul\tatom nastoqwej rabot¥ qvlqetsq sledugwaq teorema, ko-
toraq pokaz¥vaet, çto sledstvye A, voobwe hovorq, uΩe nevozmoΩno obobwyt\
na mer¥, ne udovletvorqgwye uslovyg (7), y potomu dlq dyskretn¥x mer s bo-
lee plotn¥m nosytelem sledstvye A perestaet b¥t\ vern¥m v prostranstvax
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
296 A. H. BAKAN
L dp( , )R µ , 2 ≤ p < ∞ . Dlq formulyrovky teorem vvedem neskol\ko nov¥x
oboznaçenyj.
Dlq sçetnoho mnoΩestva A ⊂ ( 0, + ∞ ) y funkcyy
ψ ( x ) : =
1
32
2
1 4
1π
π( )
/
!
m mx
m
e
m m
m−
≥
∑ , x ≥ 0, (10)
opredelym dyskretnug meru µA po formule
d xAµ ( ) : = λ ψ λ δ λλ
λ
e x
A
−
∈
−∑ ( ) ( ), x ∈ R, (11)
y oboznaçym n rA( ) : = card λ λ∈ ≤{ }A r , r ≥ 0.
Teorema 1. Pust\ L : = logk k{ } ≥2, A L⊂ , funkcyq ψ y mera µA op-
redelen¥ formulamy (10) y (11) sootvetstvenno. Tohda µA ∈ ∗ +M ( )R y esly
suwestvugt takye poloΩytel\n¥e postoqnn¥e a, Ca , çto n rA( ) ≤ C ea
r a r−
dlq vsex r ≥ 0, to L dp A( , )R µ ∈P dlq kaΩdoho 1 ≤ p < ∞ . V to Ωe vremq
L x dp
m
L( ), ( )R 1 2+ ∉− µ P dlq lgboho 2 ≤ p < ∞ y m ∈ N .
Oçevydno, çto mera µL uΩe ne udovletvorqet uslovyg (7), tak kak
card supp[ , ]−( )r r LI µ ≥ er − 2 pry r ≥ log 2.
2. Vspomohatel\n¥e rezul\tat¥. Dlq dokazatel\stva teorem¥ 1 neobxo-
dymo napomnyt\ nekotor¥e yzvestn¥e rezul\tat¥ o probleme momentov. KaΩ-
doj mere µ ∈ ∗M ( )R stavytsq v sootvetstvye mnoΩestvo V Vµ µ( )+ vsex tex mer
ν ∈ ∗M ( )R ( )( )M∗ +R , dlq kotor¥x sn( )ν = sn( )µ dlq vsex n ∈N0 . Proble-
ma momentov Hamburhera (Styl\t\esa) sostoyt v naxoΩdenyy dlq posledova-
tel\nosty dejstvytel\n¥x çysel { }γ n n∈N0
takyx mer µ ∈ ∗M ( )R ( )( )M∗ +R ,
çto sn( )µ = γ n dlq vsex n ∈N0 . Esly reßenye suwestvuet y ne qvlqetsq
edynstvenn¥m, to hovorqt, çto sootvetstvugwaq problema momentov qvlqetsq
neopredelennoj. Mer¥ µ , reßagwye takye problem¥, takΩe naz¥vagtsq
neopredelenn¥my. Druhymy slovamy, mera µ ∈ ∗M ( )R ( )( )M∗ +R naz¥vaetsq
neopredelennoj v sm¥sle Hamburhera (Styl\t\esa) ( sokrawenno µ ∈ indet H
(indet S )) , esly Vµ µ\ { } ≠ ∅ ( \ ){ }Vµ µ+ ≠ ∅ , y opredelennoj v sm¥sle Ham-
burhera (Styl\t\esa) (sokrawenno µ ∈ det H (det S )) , esly V Vµ µ( )+ = { }µ .
V 1923 h. M. Ryss [10] ustanovyl prqmug svqz\ meΩdu opredelenn¥my mera-
my v sm¥sle Hamburhera y problemoj polynomyal\noj plotnosty v prostranst-
ve L x d2
21( ), ( )R + µ . On dokazal, çto (sm. [11], utverΩdenye 1.3)
µ ∈ det H ⇔ L x d2
21( ), ( )R + µ ∈ P . (12)
V 1991 h. Kr. Berh y M. Tyll [11] (teorema 3.8) dopolnyly svojstvo (12) sle-
dugwym obrazom:
µ ∈ det S ⇔ L x d2 1( ), ( )R + µ ∈ P y L x x d2 1( ), ( )R + µ ∈ P . (13)
V 1941 h. Uydder [12] opublykoval rezul\tat, poluçenn¥j Boasom, o dosta-
toçn¥x uslovyqx dlq neopredelennosty problem¥ momentov Styl\t\esa. Dlq
posledovatel\nosty poloΩytel\n¥x çysel { }λn n∈N0
Boasom b¥ly vveden¥
uslovyq
λ0 ≥ 1, λ1 ≥ λ0, λ2 ≥ 4 1 1
2( )+ λ , λn ≥ ( )n n
nλ −1 , n = 3, 4, 5, … , (14)
kotor¥e naz¥vagt uslovyqmy Boasa, y b¥lo dokazano (sm. [12, c. 142], hl. 3, teo-
rema 16), çto dlq lgboj posledovatel\nosty, udovletvorqgwej uslovyqm (14),
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
O POLNOTE ALHEBRAYÇESKYX POLYNOMOV V PROSTRANSTVAX L dp ( , )R µ 297
suwestvugt po krajnej mere dva razlyçn¥x reßenyq µ ∈ ∗ +M ( )R problem¥
momentov Styl\t\esa sn( )µ = λ2n, n ∈N0 .
Lemma 1. Pust\ funkcyq ψ opredelena formuloj (10). Tohda dlq lgboho
b > 0 funkcyq x xbψ( ) qvlqetsq yntehryruemoj na R+ , velyçyn¥
σ ( b ) : =
ψ (log )
log
k
k
kb
k =
∞
∑
1
, γ ( b ) : =
ψ (log )
log
t
t
t dtb
1
∞
∫ (15)
qvlqgtsq koneçn¥my y
γ ( )b
e4 ≤ σ ( )b ≤ e bγ ( ) . (16)
Dokazatel\stvo. Pust\ N ∈N y
ψN ( x ) : =
1
32
2
1 4
1π
π( )
/
!
m mx
m
N e
m m
m−
=
∑ .
Tohda dlq proyzvol\noho β ≥ 2 s pomow\g yntehral\noho predstavlenyq ham-
ma-funkcyy y formul¥ umnoΩenyq dlq nee (sm. [13], hl. 1, § 1.2) poluçaem
ψ β
N u u du( ) /4 1
0
−
∞
∫ = 4 4 1
0
ψ β
N u u du( ) −
∞
∫ =
1
2
2
0 1
1
1
π
π β
+∞ −
=
−∫ ∑ ( )
/
!
m mu
m
N e
m m
u du
m
=
=
1 2
21m
m m
mm
N m
m!
( )( )
=
∑ π β
π β
Γ
=
1
1 0
1
m
r
mm
N
r
m
!= =
−
∑ ∏ +
Γ β .
Poskol\ku funkcyq Γ ( )2 + x vozrastaet pry x ≥ 0 (sm. [14], hl. 6), poslednqq
summa ne prev¥ßaet eΓ ( )β+ −1 1 y po teoreme Beppo Levy dlq proyzvol\noho
b > 0 budem ymet\ trebuemug summyruemost\ funkcyy x xbψ( ) na poloΩy-
tel\noj poluosy. Krome toho, yz posledneho ravenstva sleduet spravedlyvost\
ocenok
eΓ ( )β − 1 ≤ γ β
4
1−
≤ eΓ ( )β+ −1 1, β ≥ 2. (17)
Oçevydno, çto funkcyq e xx− ψ( ) ub¥vaet na R+ . Poπtomu na kaΩdom yn-
tervale vyda [ ]log , log( )k k + 1 , k ≥ 1, v¥polnqetsq neravenstvo
e xx− ψ( ) ≤ C e kk
k− + +log( ) (log( ))1 1ψ , x k k∈ +[ ]log , log( )1 ,
hde
Ck =
e k
e k
k
k
−
− + +
log
log( )
(log )
(log( ))
ψ
ψ1 1
, k ∈N .
DokaΩem, çto C1 ≤ e4 y Ck ≤ e dlq proyzvol\noho k ≥ 2, t. e. ymegt
mesto neravenstva
ψ (log )x
x
≤ e4 2
2
ψ (log )
, x ∈[ , ]1 2 ,
(18)
ψ (log )x
x
≤ e
k
k
ψ (log( ))+
+
1
1
, x k k∈ +[ , ]1 , k ≥ 2.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
298 A. H. BAKAN
Yz oçevydn¥x sootnoßenyj
ψ ( )0 =
1
32
2
1π
π( )
!
m
m m m≥
∑ ≤
e 2 1
32
π
π
−
ψ ( )1 =
1
32
2
1π
π( )
!
m m
m
e
m m
−
≥
∑ ≥
1
32
1
2
2
2
π
π
π
π
e
e
e
e − −
sleduet
C1 =
ψ
ψ
( )
(log )log
0
22e− ≤
eψ
ψ
( )
( )
0
1
≤ 2
1
1
2
2
2
π
π
π
π
e
e
e
e
−
− −
< e4 ,
çto oznaçaet spravedlyvost\ levoho neravenstva v (18). Teper\ ocenym sverxu
postoqnn¥e Ck pry k ≥ 2. Pry m ≥ 1 ymeem
log( ) log log ( ) log/ /k k m k km m+ − + + −[ ]1 11 4 1 4 ≤ log , log log/ /1 5 3 21 4 1 4+ −[ ]m m m =
= log , /
log
log
1 5
1
4
1
1 1 4
2
3
+ −∫ t
dtm ≤ log ,
log ,
log /1 5
1 5
4 21 1 4+ − m ≤ log ,
log ,
log
1 5
1 5
4 2
+ < 1,
otkuda
− −log log /k m km1 4 ≤ 1 1 11 4− + − +log( ) log ( )/k m km ,
y potomu
e kk−log (log )ψ =
1
32
2
1 4
1π
π( ) log log /
!
m k m k
m
e
m m
m− −
≥
∑ ≤
≤
1
32
2 1 1 1
1
1 4
π
π( ) log( ) log ( )/
!
m k m k
m
e
m m
m− + − +
≥
∑ = ee kk− + +log( ) (log( ))1 1ψ ,
çto y dokaz¥vaet prav¥e neravenstva v (18).
Poskol\ku pry lgbom b > 0 funkcyq log ( )b x1 + vozrastaet na R+ , dlq
proyzvol\noho natural\noho N ≥ 3
ψ(log )
log
k
k
k
k
N
b
=
∑
2
≤
ψ(log )
log
k
k
x dx
k
N
k
k
b
=
+
∑ ∫
2
1
( )18
≤
( )18
≤ e
x
x
x dx
k
N
k
k
b
=
+
∑ ∫
2
1 ψ(log )
log ≤ e γ ( b ) ,
otkuda sledugt koneçnost\ σ ( b ) pry lgbom b > 0 y ocenka σ ( b ) ≤ e γ ( b ) .
Dlq poluçenyq levoho neravenstva v (16) neobxodymo snova yspol\zovat\ mono-
tonnost\ funkcyy log ( )b x1 + pry x ≥ 0 y neravenstva (18):
σ ( b ) ≥
ψ(log )
log
k
k
x dx
k k
k
b
=
∞
−
∑ ∫
2 1
( )18
≥
1
4
2 1
e
x
x
x dx
k k
k
b
=
∞
−
∑ ∫ ψ(log )
log =
1
4e
bγ ( ) .
Lemma 1 dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
O POLNOTE ALHEBRAYÇESKYX POLYNOMOV V PROSTRANSTVAX L dp ( , )R µ 299
Lemma 2. Posledovatel\nost\ çysel { / }( )σ 1 2
0
+ ∈n n N udovletvorqet us-
lovyqm Boasa (14).
Dokazatel\stvo. Pry b ≥ 2 y n ∈ N yz ocenok (17) poluçaem
γ b n+ −
2
4
1 ≥ e b nΓ( )+ −2 1 = ( )( )e b n b nΓ + − + − −2 1 2 1 1 ≥
≥ ( )( )e b n b nΓ + − + −−2 1 2 11 ≥ γ b n b n+ − −
+ −2 2
4
1
2 1
,
t. e.
γ b n
4 2
1+ −
≥ γ b n b n
4
1
2
1
2 1
+ − −
+ −
, n ∈ N , b ≥ 2. (19)
Oboznaçym
γ n : = γ 1
2
+
n
, σn : = σ 1
2
+
n
, n ∈ N0 .
Tohda yz svojstva (17) s β = +8 2n y (19) s b = 8 ymeem
e nΓ( )8 2 1+ − ≤ γ n ≤ e nΓ( )9 2 1+ − , n ∈ N0 ; γ n ≥ γ n
n
−
+
1
7 2 , n ∈ N . (20)
Yspol\zuq (20) y (16), naxodym
σn ≥
1
4e
nγ ≥
1
4 1
7 2
e
n
nγ −
+ ≥
1 1
4 7 2 1
7 2
e e n n
n
+ −
+σ ,
t. e. posledovatel\nost\ { }σn n∈N0
ymeet sledugwye svojstva:
σ0 ≥
e
e
Γ( )8
4
1−
, σn ≥
1
11 2 1
7 2
e n n
n
+ −
+σ , n ∈ N , (21.1)
e
e
nΓ( )8 2
4
1+ −
≤ σn ≤ e e nΓ( )9 2 1+ −( ), n ∈ N0 . (21.2)
DokaΩem teper\, çto yz (21.1) y (21.2) sledugt svojstva (14) posledovatel\nos-
ty { }σn n∈N0
.
Yz (21.1) ymeem
σ0 ≥ eΓ( )8 5− = e7 5!− > e2 > 1, (22.1)
y potomu levoe neravenstvo v (14) v¥polnqetsq.
Yz pravoho neravenstva v (21.1) pry n = 1 ymeem σ1 ≥ σ0
9 13/e , a yz (22.1)
sleduet, çto σ0
9 13/e ≥ σ0 . Poπtomu σ1 ≥ σ0
9 13/e ≥ σ0, çto dokaz¥vaet vtoroe
sleva neravenstvo v (14).
Dlq proverky tret\eho sleva neravenstva v (14) zametym, çto 4 1 1
2( )+ σ ≤
≤ 16 1
2σ < e4
1
2σ , y v sylu pravoho neravenstva v (21.1) σ2 ≥ σ1
11 15/e pry n = 2.
No yz (21.2) ymeem σ1 ≥ eΓ( )10 5− > e16 9/ , t. e. σ1
11 15/e > e4
1
2σ , otkuda
σ2 ≥ σ1
11 15/e > e4
1
2σ > 4 1 1
2( )+ σ ,
çto y trebovalos\ dokazat\.
Zafyksyruem teper\ proyzvol\noe n ≥ 3. Tohda trebuemoe v (14) neraven-
stvo
σn ≥ ( )n n
nσ −1 (22.2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
300 A. H. BAKAN
v sylu (21.1) budet sledstvyem v¥polnenyq neravenstva σn
n ne−
+ +
1
7 2 11 2/ ≥ ( )n n
nσ −1 ,
yly, çto to Ωe samoe,
σn−1 ≥ n en n n n/( ) ( )/( )7 11 2 7+ + + . (22.3)
Dlq dokazatel\stva (22.3) zametym, çto n n/( )7 + ≤ 1, ( ) ( )/11 2 7+ +n n ≤ 3 y
blahodarq (21.2) σn−1 ≥ e nΓ( )6 2 5+ − . Poπtomu (22.3) budet sledstvyem neraven-
stva
e nΓ( )6 2+ ≥ ne8 = e n8+log , (22.4)
kotoroe v¥polnqetsq v sylu sootnoßenyj
Γ( )6 2+ n = ( ) ( )5 2 5 2+ +n nΓ ≥ ( ) ( )5 2 11+ n Γ ≥
≥ ( log( )) ( )5 2 1 11+ +n Γ > 8 + logn .
Takym obrazom, (22.2) v¥polnqetsq y posledovatel\nost\ { }σn n∈N0
udovlet-
vorqet vsem uslovyqm Boasa (14).
Lemma 2 dokazana.
3. Dokazatel\stvo teorem¥ 1. Poskol\ku po formule (11) (sm. takΩe (15))
x d xn
Lµ ( )
0
+∞
∫ =
ψ (log )
log
k
k
kn
k
+
=
∞
∑ 1
2
= σ ( n + 1 ) , n ∈ N0 ,
to µA ∈ ∗ +M ( )R dlq proyzvol\noho A ⊂ L . V sylu lemm¥ 2 y upomqnutoj
v¥ße teorem¥ Boasa (sm. [12, c. 142], hl. 3, teorema 16) budem ymet\ ne menee
dvux razlyçn¥x reßenyj ν ∈ ∗ +M ( )R problem¥ momentov Styl\t\esa
x d xn ν( )
0
+∞
∫ = σ ( n + 1 ) , n ∈ N0 .
Tak kak µL qvlqetsq reßenyem πtoj problem¥, to µL ∈ indet S . V sylu (13) y
0 ∉ supp µL πto oznaçaet, çto L x x d L2 1( ), ( )R + ∉µ P . No yz toho, çto mnoΩe-
stvo supp µL ne qvlqetsq mnoΩestvom nulej ny odnoj yz funkcyj mnoΩestva
E
S
0 [ ]R , sohlasno utverΩdenyg A1.2 yz [7, c. 250] poluçaem, çto L2( ,R
( ) )1 2+ − ∉x dm
Lµ P dlq proyzvol\noho m ∈ N y tem bolee
L x dp
m
L( ), ( )R 1 2+ − ∉µ P , m ∈ N , 2 ≤ p < ∞ . (23)
Rassmotrym teper\ takoe nepustoe mnoΩestvo A ⊂ L , çto n rA( ) ≤ C ea
r a r− ,
r ≥ 0, pry nekotor¥x a, Ca > 0. Tohda dlq proyzvol\noho n ∈ N0
sn A( )µ = x d xn
Aµ ( )
0
+∞
∫ =
k A
nk
k
k
∈
+∑ ψ (log )
log 1 ≤
≤ ψ( )
log
0
1
k A
n k
k∈
+
∑ = ψ( ) ( )0 1
0
x e dn xn x
A
+ −
+∞
∫ .
No
0 < x e dn xn x
A
+ −
+∞
∫ 1
0
( ) =
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
O POLNOTE ALHEBRAYÇESKYX POLYNOMOV V PROSTRANSTVAX L dp ( , )R µ 301
= x e n x n x n x e x e dxn x
A A
n x n x+ − +∞ − + −
+∞
− + −[ ]∫1
0
1
0
1( ) ( ) ( ) =
= x n x e x n dxn
A
x( ) ( )−
+∞
− +[ ]∫ 1
0
≤ x n x e dxn
A
x+ −
+∞
∫ 1
0
( ) ≤
≤ C x e dxa
n a x+ −
+∞
∫ 1
0
=
2 2 4
2 4
C n
a
a
n
Γ( )+
+ ,
y potomu yz asymptotyçeskoj formul¥ dlq hamma-funkcyy (sm. [13, c. 62])
sleduet, çto mera µA udovletvorqet uslovyg Karlemana v sm¥sle opredele-
nyq 1 v [15, c. 222], t. e. s n A
n
n 2
1 4
1
( ) /µ −
≥∑ = + ∞ . V sylu yzvestnoj teorem¥
Kr. Berha y Y. Krystensena [16] (sm. takΩe [15, c. 222], teorema A) πto ozna-
çaet, çto L dp A( , )R µ ∈ P dlq proyzvol\noho 1 ≤ p < ∞ .
Teorema 1 dokazana.
1. Bernstein S. Le problème de l’approximation des fonctions continues sur tout l’axe reel at l’une de
ses applications // Bull. Math. France. – 1924. – 52. – P. 399 – 410.
2. Branges L. The Bernstein problem // Proc. Amer. Math. Soc. – 1959. – 10. – P. 825 – 832.
3. Axyezer N., Bernßtejn S. Obobwenye teorem¥ o vesov¥x funkcyqx y prymenenye k prob-
leme momentov // Dokl. AN SSSR. � 1953. � 92. � S. 1109 � 1112.
4. Pollard H. Solution of Bernstein’s approximation problem // Proc. Amer. Math. Soc. – 1959. – 4. –
P. 869 – 875.
5. Merhelqn S. N. Vesov¥e pryblyΩenyq mnohoçlenamy // Uspexy mat. nauk. � 1956. � 11. �
S. 107 � 152.
6. Sodin M., Yuditskii P. Another approach to de Branges’ theorem on wighted polynomial approxi-
mation // Proc. Ashkelon Workshop Complex Function Theory (Isr. Math. Conf. Proc., May 1996).
– Providence, RI: Amer. Math. Soc., 1997. – 11. – P. 221 – 227.
7. Borichev A., Sodin M. The Hamburger moment problem and weighted polynomial approximation
on discrete subsets of the real line // J. Anal. Math. – 1998. – 71. – P. 219 – 264.
8. Bakan A. G. Polynomial density in L R dp( ),1 µ and representation of all measures which generate
a determinate Hamburger moment problem // Approxim., Optimiz. and Math. Economics. –
Heidelberg; New York: Physica, 2001. – P. 37 – 46.
9. Akilov G. P., Kantorovich L. V. Functional analysis in normed spaces. – New York: Macmillan,
1964. – 773 p.
10. Riesz M. Sur le problème des moments et le theoreme de Parseval correspondant // Acta Litt. Acad.
Sci. Szeged. – 1923. – 1. – P. 209 – 225.
11. Berg Ch., Thill M. Rotation invariant moment problem // Acta math. – 1991. – 167. – P. 207 – 227.
12. Widder D. W. The Laplas transform. – Princeton Univ. Press, 1941. – Vol. 1. – 406 p.
13. Bateman H., Erdely A. Higher transcendental functions. – New York: McGraw-Hill, 1953. –
Vol. 1.
14. Abramowitz M., Stegun I. Handbook of mathematical functions // Nat. Bur. Stand. Appl. Math. Ser.
– 1964. – 55.
15. Bakan A., Ruscheweyh St. Representation of measures with simultaneous polynomial denseness in
L dp ( , )R µ , 1 ≤ p < ∞ // Ark. mat. – 2005. – 43, # 2. – P. 221 – 249.
16. Berg Ch., Christensen J. P. R. Exposants critiques dans le problème des moments // C. r. Acad. sci.
Paris. – 1983. – 296. – P. 661 – 663.
Poluçeno 24.06.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
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| id | umjimathkievua-article-3020 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:42Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/50/6a5982f9d3e7512586d415212ee33150.pdf |
| spelling | umjimathkievua-article-30202020-03-18T19:43:35Z On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ О полноте алгебраических полиномов в пространствах $L_p (ℝ, dμ)$ Bakan, A. G. Бакан, А. Г. Бакан, А. Г. We prove that the theorem on the incompleteness of polynomials in the space $C^0_w$ established by de Branges in 1959 is not true for the space $L_p (ℝ, dμ)$) if the support of the measure μ is sufficiently dense Доведено, що встановлена Луї де Бранжем у 1959 році теорема про поліноміальну неповноту у просторі $C^0_w$ не є правильною для простору $L_p (ℝ, dμ)$, якщо носій міри д є достатньо щільним. Institute of Mathematics, NAS of Ukraine 2009-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3020 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 3 (2009); 291-301 Український математичний журнал; Том 61 № 3 (2009); 291-301 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3020/2789 https://umj.imath.kiev.ua/index.php/umj/article/view/3020/2790 Copyright (c) 2009 Bakan A. G. |
| spellingShingle | Bakan, A. G. Бакан, А. Г. Бакан, А. Г. On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ |
| title | On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ |
| title_alt | О полноте алгебраических полиномов в пространствах $L_p (ℝ, dμ)$ |
| title_full | On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ |
| title_fullStr | On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ |
| title_full_unstemmed | On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ |
| title_short | On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$ |
| title_sort | on the completeness of algebraic polynomials in the spaces $l_p (ℝ, dμ)$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3020 |
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