Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$
We study inequalities of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in the quasinorm of $L_0$ and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them.
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| Дата: | 2009 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3073 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509099814289408 |
|---|---|
| author | Adamov, A. N. Адамов, А. Н. Адамов, А. Н. |
| author_facet | Adamov, A. N. Адамов, А. Н. Адамов, А. Н. |
| author_sort | Adamov, A. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:40Z |
| description | We study inequalities of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in the quasinorm of $L_0$ and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them. |
| first_indexed | 2026-03-24T02:35:43Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.5
A. N. Adamov (Odes. nac. un-t, Yn-t matematyky, πkonomyky y mexanyky)
NERAVENSTVO TYPA TURANA
DLQ TRYHONOMETRYÇESKYX Y SOPRQÛENNÁX
TRYHONOMETRYÇESKYX POLYNOMOV V L0
We study inequalities of a Turan type for trigonometric and conjugate trigonometric polynomials in the
quasinorm L 0 and derivatives of any order. We present expressions for constants in these inequalities
and obtain their double-sided estimates.
Rozhlqnuto nerivnosti typu Turana dlq tryhonometryçnyx i sprqΩenyx tryhonometryçnyx po-
linomiv u kvazinormi L 0 ta poxidnyx bud\-qkoho porqdku. Navedeno vyrazy dlq konstant u cyx
nerivnostqx i otrymano ]x dvostoronni ocinky.
0. Vvedenye y formulyrovka osnovnoho rezul\tata. Pust\ Tn — mnoΩest-
vo tryhonometryçeskyx polynomov
T t a en k
ikt
k n
n
( ) =
=−
∑
porqdka n s koπffycyentamy ak yz polq C kompleksn¥x çysel, a ′Tn —
podmnoΩestvo polynomov yz Tn , u kotor¥x svobodn¥j çlen raven 0. Polynom
�T t i a e a en k
ikt
k
ikt
k
n
k
n
( ) = −
−
−
==
∑∑
11
naz¥vagt soprqΩenn¥m dlq Tn . Opredelym funkcyonal f p na otrezke
0 2, π[ ] pry 0 ≤ p ≤ ∞ sledugwym obrazom:
f p =
1
2 0
2 1
π
π
f t dtp
p
( )
/
∫
, 0 ≤ p < ∞,
f 0 = lim
p pf
→+0
= exp ln ( )
1
2 0
2
π
π
f t dt∫
, (0.1)
f ∞ = lim
p pf
→+∞
= f C = max ( ) : ,f t t ∈[ ]{ }0 2π .
Yzvestn¥ klassyçeskye neravenstva typa S. N. Bernßtejna y H. Sehe
Tn
r
p
( ) ≤ n Tr
n p , T Un n∈ , 0 ≤ p ≤ ∞, (0.2)
�Tn
r
p
( ) ≤ n Tr
n p , T Un n∈ , 1 ≤ p < ∞. (0.3)
© A. N. ADAMOV, 2009
986 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
NERAVENSTVO TYPA TURANA DLQ TRYHONOMETRYÇESKYX Y SOPRQÛENNÁX … 987
Snaçala b¥ly poluçen¥ neravenstva v sluçae ravnomernoj metryky ⋅ ∞
S.HN.HBernßtejnom [1] dlq tryhonometryçeskyx polynomov y H. Sehe [2] dlq so-
prqΩenn¥x tryhonometryçeskyx polynomov. Na norm¥ ⋅ p , 1 ≤ p < ∞, nera-
venstva (0.2) y (0.3) rasprostranyl A. Zyhmund [3, t. 2] (hl. X, (3.25)). Metod
Zyhmunda pry 0 < p < 1 (tryhonometryçeskaq ynterpolqcyq) yspol\zovaly
V.HY.HYvanov [4] y ∏. A. StoroΩenko, V. H. Krotov, P. Osval\d [5], no v nera-
venstvax typa (0.2) poqvylas\ dopolnytel\naq konstanta, zavysqwaq ot p . V
1981Hh. V. V. Arestov [6] predloΩyl nov¥j metod dokazatel\stva neravenstva
(0.2) dlq lgboho 0 ≤ p < 1, a takΩe eho analoha v klassax ϕ ( )L .
V sluçae soprqΩenn¥x polynomov pry 0 ≤ p < 1 uΩe nel\zq perenesty ne-
ravenstvo (0.3) s toj Ωe konstantoj dlq lgboho r. V rabote [7] dokazano, çto
�Tn
r
p
( ) ≤ χ p n p
n r T( , ) , T Un n∈ , 0 ≤ p < 1, r ≥ 0,
hde
χ p n r( , ) ≤ χ0( , )n r ,
1
2
1
n
C n
n+ ≤ χ0 0( , )n ≤ 2 2
1C n
n+
.
Ocenky snyzu norm proyzvodn¥x polynoma çerez normu samoho polynoma,
t.He. neravenstva, protyvopoloΩn¥e (0.2), vperv¥e rassmotren¥ Turanom v rabo-
te [8]. S tex por poluçeno mnoho rezul\tatov v πtom napravlenyy, v mono-
hrafyy [9] ony systematyzyrovan¥. ∏ty neravenstva v¥polnqgtsq pry syl\n¥x
ohranyçenyqx na raspoloΩenye nulej funkcyj, naprymer v tryhonometryçes-
kom sluçae vse nuly dolΩn¥ naxodyt\sq na dejstvytel\noj osy. Poskol\ku v
dannoj rabote reç\ ydet o prostranstve L0 y na polynom¥ naklad¥vagtsq
ohranyçenyq druhoho vyda, okonçatel\n¥e ocenky sloΩno sravnyvat\.
Neravenstvo typa Turana dlq p = 0 vperv¥e b¥lo rassmotreno ∏. A. Storo-
Ωenko v rabote [10], a v rabote [11] poluçeno sledugwee neravenstvo dlq
tryhonometryçeskyx polynomov so svobodn¥m çlenom, ravn¥m 0:
Tn
r( )
0
≥
1
2 2
1 0C
T
n
n n+ , Tn n∈ ′T , r ≥ 1. (0.4)
Hlavnoj cel\g nastoqwej rabot¥ qvlqetsq utoçnenye konstant¥ v nera-
venstve (0.4) y poluçenye analoha dlq proyzvol\n¥x soprqΩenn¥x polynomov.
Teorema. Dlq lgboho n ≥ 1 ymegt mesto neravenstva
Tn
r( )
0
≥
1
0
0D
T
n r
n
,
, r ≥ 1,
(0.5)
�Tn
r( )
0
≥
1
0
0S
T
n r
n
,
, r ≥ 0, Tn n∈ ′T ,
hde
D zn r, ( ) =
C z
k n
n
k k
r
k k n
n
2
0
2
( ), −= ≠
∑ , S zn r, ( ) = sign ( )
( ),
k n
C z
k n
n
k k
r
k k n
n
−
−= ≠
∑ 2
0
2
(0.6)
y pry r ≥ 3
0 458 2
1, C n
n+ ≤ Dn r, 0
≤ 1 052 2
1, C n
n+
, (0.7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
988 A. N. ADAMOV
0 596 2
1, C n
n+ ≤ Sn r, 0
≤ 1 208 2
1, C n
n+
. (0.8)
1. Vspomohatel\n¥e rezul\tat¥. Pust\ Pn — mnoΩestvo alhebrayçes-
kyx mnohoçlenov Pn stepeny n s kompleksn¥my koπffycyentamy, kotor¥e
udobno zapys¥vat\ v vyde
P zn ( ) = C c zn
k
k
k
k
n
=
∑
0
. (1.1)
Mnohoçlenam
Λn z( ) = C zn
k
k
k
k
n
λ
=
∑
0
y (1.1), sleduq V. V. Arestovu, sopostavym mnohoçlen
Λn n n
k
k k
k
k
n
P z C c z( ) =
=
∑ λ
0
,
naz¥vaem¥j kompozycyej Sehe mnohoçlenov Pn y Λn (sm. [12], teorema II,
razdel V).
Ysxodq yz (0.1) opredelym pry 0 ≤ p ≤ ∞ funkcyonal ⋅
p
na edynyçnoj
okruΩnosty z = 1, poloΩyv f z
p
( ) = f eit
p
( ) .
Otmetym prostejßye svojstva kvazynorm¥ L0 , kotor¥e budem yspol\zo-
vat\ v dal\nejßem:
1) mul\typlykatyvnost\
∀ f, g L∈ 0 : f g f g⋅ =
0 0 0
, (1.2)
2) ocenka sverxu y snyzu
ess inf ( )
0 2≤ ≤t
f t
π
≤ f
0
≤ f ∞ = ess sup ( )
0 2≤ ≤t
f t
π
. (1.3)
Teorema A [13]. Dlq lgboho 0 ≤ p ≤ ∞ y lgb¥x dvux mnohoçlenov Λn ,
P zn n( ) ∈ P ymeet mesto neravenstvo
Λn n p
P ≤ Λn n p
P
0
.
Lemma. Esly koπffycyent¥ ak mnohoçlenov
Q z a z
a
z zn
n k
k
n
n k n k( ) ( )= + +
=
+ −∑0
1 2
y
�Q z ia z
a
z zn
n k
k
n
n k n k( ) ( )= + −
=
+ −∑0
1 2
vewestvenn¥, neotrycatel\n¥ y dlq nekotoroho 0 ≤ k < n udovletvorqgt
uslovyqm ak ≥ … ≥ an y a0 = … = ak−1 = 0, to
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
NERAVENSTVO TYPA TURANA DLQ TRYHONOMETRYÇESKYX Y SOPRQÛENNÁX … 989
Qn 0
≤ sup
0≤ ≤j n
ja , �Qn 0
≤ sup
0≤ ≤j n
ja . (1.4)
Dlq
�Q zn ( ) y a0 = 0 lemma dokazana V. V. Arestov¥m [7] (lemma 3), v ob-
wem sluçae dokazatel\stvo analohyçno.
Napomnym opredelenye dzeta-funkcyy Rymana ζ( )x =
1
1 nxn=
∞∑ y smewen-
noj dzeta-funkcyy Rymana ζ( , )x a =
1
1 ( )n a xn +=
∞∑ . Lehko vydet\, çto funk-
cyy opredelen¥ na x > 1 y monotonno ub¥vagt na πtom mnoΩestve. Yspol\-
zovav πty oboznaçenyq, pryvedem summu nekotor¥x rqdov:
1
2 1
1 2
0 ( )
( ) ( )
k
r
r
k
r
+
= −
=
∞
−∑ ζ , r > 1,
1
2 1
2 1
2
21 1
−
−
+
−
=
∞
=
∞
∑ ∑( )
( ) ( )
k
k
k
kr
k
r
k
= 2 1 2 1( ) ( ) ( )− − −−r r rζ ζ , r > 2, (1.5)
1
2 1
2
1
4 2
4
1
4 30 0 0( ) ( ) ( )k k kr
k
r
k
r
k+
−
+
−
+=
∞
=
∞
=
∞
∑ ∑ ∑ =
= ( )( ) ( ) ,1 2 1 2 2
3
4
1 2 2− − −
− − −r r rr rζ ζ , r > 1.
2. Dokazatel\stvo osnovnoj teorem¥. Sopostavym tryhonometryçeskomu
polynomu Tn n∈ T alhebrayçeskyj mnohoçlen
P T a zn n n k
n k
n
k n
n
2 2= = ∈+
=−
∑τ ( ) P .
Lehko vydet\, çto esly Tn n∈ ′T , to
T t e T en n n
it( ) ( ) ( )int= − τ , T t i e D T en
r
n r n n
r it( ) ( )int
,
( )= ( )−
2 τ
y
T t i e S T en
r
n r n n
r it( ) ( )int
,
( )= ( )+ −1
2 τ � .
Prymenqq teoremu A, poluçaem (0.5) y (0.6).
Perejdem k osnovnoj çasty dokazatel\stva — ocenke velyçyn konstant
Dn r, 0
y Sn r, 0
. Lehko zametyt\, çto mnohoçlen¥ Dn r, y Sn r, udovletvo-
rqgt uslovyqm lemm¥, yz çeho sleduet, çto Dn r, 0
≤ 2 2
1C n
n+
, r ≥ 1, y
Sn r, 0
≤ 2 2
1C n
n+
, r ≥ 0. Vtoroe neravenstvo rasprostranqet rezul\tat ∏.HA.HSto-
roΩenko yz [11] na sluçaj soprqΩenn¥x polynomov. Perejdem k uluçßenyg
πtyx ocenok. Rassmotrym neçetn¥e r ≥ 3. Preobrazuem Dn r, sledugwym ob-
razom:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
990 A. N. ADAMOV
D zn r, ( ) = z z
z
C
k
zn n
n k
r
k
n
j k
j
k
−
+
=
− +
=
−
∑ ∑1 2
1
2 1
0
1
=
= z z
z
C
k
n n
n k
r
k
n
−
+
+ +
=
−
∑1
2 1
2
2 1
0
1 2
( )
( )/
+ +
+
−
=
− + +
=
∑ ( )
( )
z z
C
j
k k
k
n
n
n j
r
j1
1
2
2 1
2 100
1 2( )/n k− −
∑
=
= ( ) ( ),z Q zn r
2 1− .
PokaΩem, çto Q zn r, ( ) udovletvorqet uslovyg lemm¥. Dejstvytel\no, ub¥va-
nye koπffycyentov, naçynaq so vtoroho, oçevydno, a sravnyvaq perv¥j y vto-
roj koπffycyent¥, ymeem
2
2
2
2
1
2 C
k
n
n k
r
k
n +
=
[ ]
∑
( )
/
≤
C
k
Cn
n
k
n
n2
1
2
1
2
2
1
2
1
12
+
=
∞
+∑ =
π
≤ C n
n
2
1+ =
C
k
n
n k
r
k
n
2
2 1
0
1 2
2 1
+ +
=
−[ ]
+
∑
( )
( )/
.
Na osnovanyy lemm¥ y (1.5)
Dn r, 0
≤
C
k
n
n k
r
k
n
2
2 1
0
1 2
2 1
+ +
=
−[ ]
+
∑
( )
( )/
≤ C
k
r Cn
n
r
k
r
n
n
2
1
0
2
11
2 1
1 2+
=
∞
− +
+
= −∑
( )
( ) ( )ζ .
Ocenka snyzu s pomow\g (1.3) y (1.4) pryvodyt k neravenstvu
Dn r, 0
= Qn r, 0
≥ min ( ),z n rQ z
=1
≥ C k
C
k
n
n n
n k
r
k
n
2
1 2
2 1
1
1 2
2 1
2 1
+
+ +
=
−[ ]
− −
+
∑ ( )
( )
( )/
–
– 2
2
2
2
1
2
k
C
k
n
n k
r
k
n +
=
[ ]
∑
( )
/
≥ C
k
k
k
k
n
n
r r
kk
2
1
11
1
2 1
2 1
2
2
+
=
∞
=
∞
−
−
+
−
∑∑
( ) ( )
=
= 2 1 2 1 2
1( ) ( ) ( )− − −
− +r
n
nr r Cζ ζ .
Oçevydno, çto konstant¥ pered C n
n
2
1+
monotonno stremqtsq k 1 pry uvely-
çenyy r. Pry r = 3 poluçaem konstant¥ v (0.7).
Perejdem k ocenke sverxu y snyzu velyçyn¥ Sn r, 0
(r ≥ 3 neçetn¥e). Ana-
lohyçno preobrazovanyqm Dn r, budem v¥delqt\ velyçynu z
z
+
1
, pryçem,
tak kak delymosty nacelo v obwem sluçae net, kaΩdoe slahaemoe s çetn¥my
stepenqmy predstavym tak:
z zk k2 2+ − = ( ) ( ) ( )z z z zj k
j
k
j j k+ − +( ) + −− + +
=
−
+ − −∑1 1
0
1
2 1 2 11 1 .
Menqq porqdok summyrovanyq v dvojnoj summe, moΩno zapysat\
S z z z
z
W z V zn r n r n r
n
2
11
, , ,( ) ( )= +
−
− ,
hde
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
NERAVENSTVO TYPA TURANA DLQ TRYHONOMETRYÇESKYX Y SOPRQÛENNÁX … 991
V
C
j
n r
j n
n j
r
j
n
,
/
( )
( )
= − +
+
=
[ ]
∑2 1
2
1 2
2
1
2
,
W zn r, ( ) = z
C
j
n j n
n j
r
j
n
−
+ +
=
−
−
+
∑1 2
2 1
0
1 2
1
2 1
( )
( )
( )/
+
+ ( ) ( )
( )
( )
z z
C
j k
k k j n
n j k
r
j
n k
+ −
+ +
−
+ + +
=
− −
1
2 1
2
2 1
0
1 //2
1
1 [ ]
=
−
∑∑
k
n
.
Perexodq k norme L0 , yz (1.2) ymeem
Sn r, 0
= W z
z
V z
Q z
n r
n r
n
n r
,
,
, ( )0
1
0
1
+
−
−
. (2.1)
PokaΩem, çto W zn r, ( ) udovletvorqet uslovyqm lemm¥. Dejstvytel\no, vse
koπffycyent¥ poloΩytel\n¥. Proverym neravenstva meΩdu koπffycyenta-
my. Sootnoßenye meΩdu perv¥m y vtor¥m koπffycyentamy oçevydno pry r ≥
≥ 3, dlq dokazatel\stva monotonnoho ub¥vanyq ostal\n¥x koπffycyentov
rassmotrym raznost\ sosednyx:
a ak k− +1 =
=
C
j k
C
j k
Cn
n j k
r
n
n j k
r
2
4 1
2
4 2
4 1 4 2
+ + + + + +
+ +
−
+ +
−
( ) ( )
22
4 3
2
4 4
4 3 4 4
n
n j k
r
n
n j k
rj k
C
j k
+ + + + + +
+ +
+
+ +
( ) ( )
=
− −[ ]
∑
j
n k
0
1 4( )/
,
hde dlq udobstva zapysy C n
l
2 0= pry l > 2n.
PokaΩem, çto kaΩdoe slahaemoj neotrycatel\no. Dejstvytel\no, oçevydno,
çto C n
n k
2
+ – 2 2
1C n
n k+ + + C n
n k
2
2+ + ≥ 0 pry k ≥ 0 y
1
k r
–
2
1( )k r+
+
1
2( )k r+
≥ 0
pry k ≥ 0 y r ≥ 0. Yz toΩdestv
a b a bk k k k+ + +2 2 =
( ) ( )a a b bk k k k− −+ +2 2
2
+
+H
( ) ( )a a b bk k k k+ ++ +2 2
2
,
( ) ( )a a b bk k k k+ ++ +2 2
2
– 2 1 1a bk k+ + =
=
1
2
2 21 2 1 2( ) ( )a a a b b bk k k k k k− + − ++ + + + +
+H ( ) ( )a a a b a b b bk k k k k k k k− + + − ++ + + + + +2 21 2 1 1 1 2
sleduet, çto esly posledovatel\nosty ak y bk neotrycatel\n¥, monotonno
ub¥vagt y v¥polnqgtsq neravenstva ak – 2 1ak+ + ak+2 ≥ 0, bk – 2 1bk+ +
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
992 A. N. ADAMOV
+ bk+2 ≥ 0 dlq lgboho k, to y dlq posledovatel\nosty a bk k ony toΩe budut
v¥polnqt\sq. Poπtomu
C
k
n
n k
r
2
+
– 2
1
2
1C
k
n
n k
r
+ +
+( )
+
C
k
n
n k
r
2
2
2
+ +
+( )
≥ 0,
y, sledovatel\no,
C
k
C
k
C
k
n
n k
r
n
n k
r
n
n k
r
2 2
1
2
2
2
1 2
+ + + + +
−
+
+
+
( ) ( )
+
C
k
C
k
C
k
n
n k
r
n
n k
r
n
n k
2
1
2
2
2
3
1
2
2 3
+ + + + + +
+
−
+
+
+( ) ( ) ( ))r
≥ 0,
otkuda poluçaem neobxodymoe sootnoßenye.
Takym obrazom, ocenka sverxu prynymaet vyd
Wn r, 0
≤ ( )
( )
( )/
−
+
+ +
=
−[ ]
∑ 1
2 1
2
2 1
0
1 2
j n
n j
r
j
n C
j
≤ C n
n
2
1+
.
Po formulam (1.3) y (1.5) ymeem
Wn r, 0
≥ min ( ),
z
n rW z
=1 0
≥
≥
C
k
C
k
n
n k
r
k
n
n
n k
2
2 1
0
1 2
2
4 2
2 1
2
4
+ +
=
−[ ] + +
+
−∑
( ) (
( )/
++=
−[ ]
∑
20
2 4
)
( )/
r
k
n
– 4
4 3
2
4 3
0
3 4 C
k
n
n k
r
k
n + +
=
−[ ]
+
∑
( )
( )/
≥
≥ C
k k k
n
n
r
k
r
k
2
1
0 0
1
2 1
2
1
4 2
4
1
4 3
+
=
∞
=
∞
+
−
+
−
+
∑ ∑
( ) ( ) ( )rr
k=
∞
∑
0
≥
≥ ( ) ( ) ( ) ,1 2 1 2 2
3
4
1 2 2
2− − −
− − −r r r
nr r Cζ ζ nn+1 ≥ 0 66457 2
1, C n
n+ .
Dlq ocenky vtoroho v¥raΩenyq yssleduem povedenye velyçyn¥ f t( ) =
=
V e
W e
n r
it n
n r
it
,
( )
, ( )
−1
. Na 0 ≤ t ≤ 2π f neprer¥vno dyfferencyruema y prynymaet
tol\ko dejstvytel\n¥e znaçenyq. PoloΩym
C f t f t
t t
= +
≤ ≤ ≤ ≤
1
2 0 2 0 2
inf ( ) sup ( )
π π
,
(2.2)
ε
π π
= −
≤ ≤ ≤ ≤
1
2 0 2 0 2
sup ( ) inf ( )
t t
f t f t .
Pry r = 3 velyçyn¥ v (2.2) udovletvorqgt neravenstvam 0 < C ≤ 0,28317, 0 <
< ε ≤ 0,09302, pry r > 3 πty velyçyn¥ budut ub¥vat\.
Teper\, oboznaçaq t
C
0 2
= arccos , dlq
H z z
z
V z
W z
n r
n r
n
n r
,
,
,
( )
( )
= +
−
−1 1
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
NERAVENSTVO TYPA TURANA DLQ TRYHONOMETRYÇESKYX Y SOPRQÛENNÁX … 993
poluçaem
H en r
it
, ( ) ≤ 2 cos t C− + ε , 0 ≤ t ≤ t0, 2π – t0 ≤ t ≤ 2π,
H en r
it
, ( ) ≤ 2 cos t C− − ε , t0 ≤ t ≤ 2π – t0
,
y
Hn r, 0
≤ exp ln cos ln cos
1
2 2
0
0
0π
ε ε
π
t C dt t C dt
t
t
− + + − −
∫∫
≤
≤ 1,20726. (2.3)
Snyzu analohyçno, oboznaçaq t1 = arccos
C + 2
2
ε
y t2 = arccos
C − 2
2
ε
, ymeem
H en r
it
, ( ) ≥ 2 cos t C− − ε , 0 ≤ t ≤ t1,
H en r
it
, ( ) ≥ 2 cos t C− + ε , t2 ≤ t ≤ π.
V sylu neprer¥vnosty na otrezke t t1 2,[ ] est\ nul\ H e tn r
it
, ( ) � , y, tak kak
′f t( ) ≤ 0,2 pry r ≥ 3,
1
2
1
2
π
δln cos t e dtit
t
t
− ( )∫ ≥
1
1 73
1
2
π
ln , ( )t t dt
t
t
−∫ � ≥ − +
ε
ε
0 226 0 658
1
, , ln .
Takym obrazom, analohyçno (2.3) ymeem Hn r, 0 ≥ 0,89735. Teper\ yz (2.1) po-
luçaem konstant¥ v (0.8). V sluçae çetn¥x r ≥ 4 rassuΩdenyq dlq Sn r, y
Dn r, menqgtsq mestamy, y ocenky (0.7) y (0.8) ostagtsq spravedlyv¥my.
Teorema dokazana.
Zameçanye. Pry vozrastanyy r konstant¥ v (0.7) y (0.8) moΩno uluçßat\.
Pry r = 2 metod dokazatel\stva pozvolqet poluçyt\ ewe odnu ocenku:
Dn, 2 0
≤ 1,2337 C n
n
2
1+
.
TakΩe yz rezul\tatov V. V. Arestova [2] sleduet, çto ocenku (0.5) moΩno
rasprostranyt\ dlq lgboho 0 < p ≤ ∞ v takom Ωe vyde
Tn
r
p
( ) ≥
1
0
D
T
n r
n p
,
, r ≥ 1,
y
�Tn
r
p
( ) ≥
1
0
S
T
n r
n p
,
, r ≥ 0, Tn n∈ ′T ,
no toçnost\ konstant¥ pry πtom ne harantyruetsq.
Avtor v¥raΩaet blahodarnost\ ∏. A. StoroΩenko za obsuΩdenye poluçen-
n¥x rezul\tatov.
1. Bernßtejn S. N. Sobranye soçynenyj. – M.: Yzd-vo AN SSSR, 1952 – 1959. – T. 1, 2.
2. Szegö G. Über einen satz des Hern Serge Bernstein // Schriftenr. Königsberg. Gelehrten Ges. –
1928. – 5, # 4. – S. 59 – 70.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
994 A. N. ADAMOV
3. Zyhmund A. Tryhonometryçeskye rqd¥: V 2 t. – M.: Myr, 1965. – T. 1, 2.
4. Yvanov V. Y. Prqm¥e y obratn¥e teorem¥ teoryy pryblyΩenyq v metryke L p dlq 0 < p <
< 1 // Mat. zametky. – 1975. – 18. – S. 641 – 658.
5. StoroΩenko ∏. A., Krotov V. H., Osval\d P. Prqm¥e y obratn¥e teorem¥ typa DΩeksona v
prostranstvax L p , 0 < p < 1 // Mat. sb. – 1975. – 98, # 3. – S. 395 – 415.
6. Arestov V. V. Ob yntehral\n¥x neravenstvax dlq tryhonometryçeskyx polynomov y yx
proyzvodn¥x // Yzv. AN SSSR. Ser. mat. – 1981. – 45. – S. 3 – 22.
7. Arestov V. V. Neravenstvo Sehe dlq soprqΩennoho tryhonometryçeskoho polynoma v
L0 H// Mat. zametky. – 1994. – 56, v¥p.H6. – S. 10 – 26.
8. Turan P. Über die Ableitung von polynomen // Compos. math. – 1939. – 7. – P. 89 – 95.
9. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov.
– Kyev: Nauk. dumka, 1992.
10. StoroΩenko ∏. A. K odnoj zadaçe Malera o nulqx polynoma y eho proyzvodnoj // Mat. sb.
— 1996. – 187, # 5. – S. 111 – 120.
11. StoroΩenko ∏. A. Neravenstvo typa Turana dlq kompleksn¥x polynomov v L0 -metryke //
Yzv. vuzov. Matematyka. – 2008. – # 5. – S. 1 – 6.
12. Polya H., Sehe H. Zadaçy y teorem¥ yz analyza. – M.: Nauka, 1978. – T. 1, 2.
13. Arestov V. V. Yntehral\n¥e neravenstva dlq alhebrayçeskyx mnohoçlenov na edynyçnoj
okruΩnosty // Mat. zametky. – 1990. – 48, v¥p.H4. – S. 7 – 18.
Poluçeno 15.07.08,
posle dorabotky — 10.02.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, #7
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| id | umjimathkievua-article-3073 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:43Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6b/5499bc436872a593dbcd8b6275ae9c6b.pdf |
| spelling | umjimathkievua-article-30732020-03-18T19:44:40Z Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ Неравенство типа Турана для тригонометрических и сопряженных тригонометрических полиномов в $L_0$ Adamov, A. N. Адамов, А. Н. Адамов, А. Н. We study inequalities of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in the quasinorm of $L_0$ and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them. Розглянуто нерівності типу Турана для тригонометричних i спряжених тригонометричних поліномів у квазінормі $L_0$ та похідних будь-якого порядку. Наведено вирази для констант у цих нерівностях і отримано їх двосторонні оцінки. Institute of Mathematics, NAS of Ukraine 2009-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3073 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 7 (2009); 986-995 Український математичний журнал; Том 61 № 7 (2009); 986-995 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3073/2895 https://umj.imath.kiev.ua/index.php/umj/article/view/3073/2896 Copyright (c) 2009 Adamov A. N. |
| spellingShingle | Adamov, A. N. Адамов, А. Н. Адамов, А. Н. Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ |
| title | Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ |
| title_alt | Неравенство типа Турана для тригонометрических и сопряженных
тригонометрических полиномов в $L_0$ |
| title_full | Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ |
| title_fullStr | Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ |
| title_full_unstemmed | Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ |
| title_short | Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in $L_0$ |
| title_sort | inequality of the turan type for trigonometric polynomials and conjugate trigonometric polynomials in $l_0$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3073 |
| work_keys_str_mv | AT adamovan inequalityoftheturantypefortrigonometricpolynomialsandconjugatetrigonometricpolynomialsinl0 AT adamovan inequalityoftheturantypefortrigonometricpolynomialsandconjugatetrigonometricpolynomialsinl0 AT adamovan inequalityoftheturantypefortrigonometricpolynomialsandconjugatetrigonometricpolynomialsinl0 AT adamovan neravenstvotipaturanadlâtrigonometričeskihisoprâžennyhtrigonometričeskihpolinomovvl0 AT adamovan neravenstvotipaturanadlâtrigonometričeskihisoprâžennyhtrigonometričeskihpolinomovvl0 AT adamovan neravenstvotipaturanadlâtrigonometričeskihisoprâžennyhtrigonometričeskihpolinomovvl0 |