Algebraic polynomials least deviating from zero in measure on a segment
We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment $[–1, 1]$ with respect to a measure, or, more precisely, with respect to the functional $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. We also discuss an anal...
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| Date: | 2010 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508857581699072 |
|---|---|
| author | Arestov, V. V. Арестов, B. В. Арестов, B. В. |
| author_facet | Arestov, V. V. Арестов, B. В. Арестов, B. В. |
| author_sort | Arestov, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:19Z |
| description | We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment $[–1, 1]$ with respect to a measure, or, more precisely, with respect to the functional $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. We also discuss an analogous problem with respect to the integral functionals $∫_{–1}^1 φ (∣f (x)∣) dx$ for functions $φ$ that are defined, nonnegative, and nondecreasing on the semiaxis $[0, +∞)$. |
| first_indexed | 2026-03-24T02:31:52Z |
| format | Article |
| fulltext |
UDK 517.518.86
V. V. Arestov (Ural. un-t, Yn-t matematyky y mexanyky Ural. otd-nyq RAN, Rossyq)
ALHEBRAYÇESKYE MNOHOÇLENÁ,
NAYMENEE UKLONQGWYESQ OT NULQ
PO MERE NA OTREZKE
∗∗∗∗
We study the problem of algebraic polynomials with given leading coefficient that deviate least from
zero in measure on the segment [–,] 11 more ( precisely, with respect to the functional µ() f =
= mesxfx ∈≥ {}) [–,]:() 111. We discuss a similar problem with respect to the integral functionals
ϕfxdx () ()
−∫1
1
for functions ϕ which are defined, nonnegative, and nondecreasing on the half line
[,) 0+∞.
DoslidΩu[t\sq zadaça pro alhebra]çni mnohoçleny iz zadanym starßym koefici[ntom, wo naj-
menße vidxylqgt\sq vid nulq za mirog na vidrizku [–,] 11, a toçniße, vidnosno funkcionala
µ() f = mesxfx ∈≥ {}) [–,]:() 111. Obhovorg[t\sq analohiçna zadaça vidnosno intehral\nyx
funkcionaliv ϕfxdx () ()
−∫1
1
dlq funkcij ϕ , vyznaçenyx, nevid’[mnyx ta nespadnyx na piv-
osi [,) 0+∞.
1. Vvedenye. Pust\ Pm — mnoΩestvo alhebrayçeskyx mnohoçlenov
fx m() = axk
k
k
m
=
∑
0
(1.1)
porqdka m≥0 s kompleksn¥my koπffycyentamy. Pry m≥1 na mnoΩestve
Pm rassmotrym funkcyonal
µ() fm = mesxfx m ∈≥ {} [–,]:() 111,(1.2)
znaçenye kotoroho est\ mera Lebeha mnoΩestva toçek x∈[–,] 11, v kotor¥x
modul\ mnohoçlena fmm ∈P bol\ße lybo raven 1. Dlq fyksyrovannoho
y∈R vvedem velyçynu
σmy() = infµ() (): yxfxf mm
mmm 21
111
−
−−− −∈ {} P,(1.3)
kotorug moΩno ynterpretyrovat\ kak velyçynu nayluçßeho pryblyΩenyq
funkcyy yx mm 21− mnoΩestvom Pm−1 alhebrayçeskyx mnohoçlenov porqdka
m−1 otnosytel\no funkcyonala (1.2). Velyçynu (1.3) moΩno zapysat\ v
neskol\ko ynoj forme. Pust\ Pmy() — mnoΩestvo alhebrayçeskyx mnohoçle-
nov porqdka m, starßyj koπffycyent kotor¥x est\ ym21−, t. e. mnohoçlenov
vyda
∗
V¥polnena pry podderΩke Rossyjskoho fonda fundamental\n¥x yssledovanyj (proekt 08-
01-00213) y Prohramm¥ hosudarstvennoj podderΩky veduwyx nauçn¥x ßkol RF (proekt NÍ-
3208.2010.1).
© V. V. ARESTOV, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3291
292V. V. ARESTOV
fx m() = yxfx mm
m 21
1
−
− +(), fmm −− ∈ 11 P.
Tohda
σmy() = infµ():() ffy mmm ∈ {} P;(1.4)
πto uΩe est\ varyant zadaçy o mnohoçlenax, naymenee uklonqgwyxsq ot nulq.
VaΩnug rol\ v dal\nejßem budut yhrat\ mnohoçlen¥ Çeb¥ßeva pervoho
roda, kotor¥e dlq x∈[–,] 11 opredelen¥ formuloj Tx m() = cos() mx arccos.
Pry m≥1 starßyj koπffycyent πtoho mnohoçlena raven 21 m− (sm., napry-
mer, [1], hl. 3). Poπtomu zadaça (1.3), (1.4) netryvyal\na lyß\ pry y>1; qs-
no, çto dostatoçno ohranyçyt\sq sluçaem y>1.
Otnosytel\no zadaçy (1.4) spravedlyvo sledugwee utverΩdenye.
Teorema 1.1. Pry m≥1, y>1 ymeet mesto ravenstvo
σmy() = 211 − () −ym/.(1.5)
Bolee toho, mnohoçlen
fx m
∗() = Tyx m
m 1/ ()(1.6)
y eho sdvyhy
fxh m
∗− (), h ≤ 11 −−ym/,(1.7)
prynadleΩat mnoΩestvu Pmy() y qvlqgtsq πkstremal\n¥my mnohoçlenamy v
zadaçe (1.4) (mnohoçlenamy, naymenee uklonqgwymysq ot nulq), y tol\ko
takye mnohoçlen¥ reßagt zadaçu (1.4).
∏kstremal\n¥e mnohoçlen¥ v rassmotrenn¥x v dannoj rabote zadaçax okaza-
lys\ vewestvenn¥my, poπtomu rezul\tat¥ rabot¥ (teorem¥ 1.1, 3.1 y 4.1) osta-
gtsq spravedlyv¥my dlq mnohoçlenov (1.1) s vewestvenn¥my koπffycyentamy.
Dlq tryhonometryçeskyx polynomov zadaçy, podobn¥e tem, kotor¥e obsuΩ-
dagtsq v dannoj rabote, y rodstvenn¥e ym, yssledovan¥ ranee sovmestno avto-
rom y A. S. Mendelev¥m [2].
2. Rezul\tat¥ P. L. Çeb¥ßeva y H. Poja otnosytel\no alhebrayçeskyx
mnohoçlenov, naymenee uklonqgwyxsq ot nulq. Zadaça (1.4) svqzana s ne-
skol\kymy druhymy πkstremal\n¥my zadaçamy dlq mnohoçlenov, v çastnosty s
zadaçej o mnohoçlenax s fyksyrovann¥m starßym koπffycyentom, naymenee
uklonqgwymysq ot nulq otnosytel\no ravnomernoj norm¥ na kompaktax.
Mnohoçlen¥ Çeb¥ßeva (naymenee uklonqgwyesq ot nulq s edynyçn¥m
starßym koπffycyentom) na kompaktax kompleksnoj ploskosty yzuçalys\
mnohymy matematykamy y ymegt mnohoçyslenn¥e pryloΩenyq (sm., naprymer,
[3]). Opyßem bolee podrobno rezul\tat¥ P. L. Çeb¥ßeva y H. Poja
otnosytel\no alhebrayçeskyx mnohoçlenov, naymenee uklonqgwyxsq ot nulq.
Pust\ �m = �m() R, m≥1, est\ mnoΩestvo alhebrayçeskyx mnohoçlenov
Px m() = xcx m
k
k
k
m
+
=
−
∑
0
1
s edynyçn¥m starßym y proyzvol\n¥my (kompleksn¥my) ostal\n¥my koπffy-
cyentamy. P. L. Çeb¥ßev [4] naßel naymen\ßee uklonenye ot nulq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
ALHEBRAYÇESKYE MNOHOÇLENÁ, NAYMENEE UKLONQGWYESQ OT NULQ … 293
Em([–,]) 11 = infPP mCmm [–,]
:
11
∈ {} �(2.1)
na otrezke [–,] 11 mnohoçlenov yz klassa �m. A ymenno, on pokazal, çto
Em([–,]) 11 =
1
21 m−
, m≥1,(2.2)
y πkstremal\n¥m qvlqetsq mnohoçlen
Px m
∗() =
1
21 mmTx
−
(), Tx m() = cos() mx arccos.
S pomow\g lynejnoj zamen¥ lehko proverqetsq, çto dlq lgboho otrezka I =
= [,] ab dlyn¥ I = ba − = 2ρ, ρ>0, velyçyna
EI m() = infPP mCImm ()
:∈ {} �
ymeet znaçenye
EI m() = 2
2
ρ
m
;
sootvetstvenno peresçyt¥vaetsq πkstremal\n¥j mnohoçlen. K prymeru, dlq
otrezka [–,] ρρ takov¥m qvlqetsq mnohoçlen
gx m
∗() = 2
2
ρ
ρ
m
mT
x
.(2.3)
Dlq kompaktnoho podmnoΩestva Q⊂R poloΩym
EQ m() = infPP mCQmm ()
:∈ {} �.(2.4)
H. Pojq yzuçal naymen\ßee znaçenye
Em() 2ρ = inf():() EQQ m∈ {} Q2ρ(2.5)
velyçyn¥ (2.4) po semejstvu Q = Q() 2ρ vsex kompaktn¥x podmnoΩestv Q⊂R
çyslovoj osy, mera kotor¥x ravna fyksyrovannomu çyslu 2ρ, ρ>0, y doka-
zal sledugwee utverΩdenye (sm., naprymer, [5, c. 23]).
Teorema 2.1. Pry lgbom ρ>0 dlq lgboho mnoΩestva Q∈Q() 2ρ v¥pol-
nqetsq neravenstvo
EQ m() ≥ 2
2
ρ
m
,
pryçem znak ravenstva dostyhaetsq lyß\ v tom sluçae, kohda Q est\ otrezok
(dlyn¥ 2ρ). Kak sledstvye,
Em() 2ρ = 2
2
ρ
m
.(2.6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
294V. V. ARESTOV
3. Dokazatel\stvo teorem¥ 1.1. Svqz\ zadaç.
Lemma 3.1. Pry lgb¥x m≥1, y>1 v zadaçe (1.4) suwestvuet πkstre-
mal\n¥j mnohoçlen; vse nuly πkstremal\noho mnohoçlena vewestvenn¥ y leΩat
na otrezke [–,] 11.
Dokazatel\stvo. Mnohoçlen¥ fy mm ∈P() moΩno zapysat\ v vyde
fx m() = yxz m
k
k
m
21
1
−
=
− ∏(),(3.1)
hde {} zkk
m
=1 — (neobqzatel\no vewestvenn¥e) korny mnohoçlena fm. Pry kaΩ-
dom k , 1 ≤ k ≤ m , oboznaçym çerez xk toçku otrezka I = [–,] 11, blyΩaj-
ßug k toçke zk v kompleksnoj ploskosty C . PoloΩym
"fx m() = yxx m
k
k
m
21
1
−
=
− ∏().
∏to est\ mnohoçlen yz Pmy(), vse korny kotoroho leΩat na I . Pry πtom v¥-
polnqetsq neravenstvo
"fx m() ≤ fx m(), x∈I. Esly mnohoçlen fm ymeet
xotq b¥ odyn koren\ vne I , to vsgdu, krome kornej mnohoçlena fm, prynad-
leΩawyx I , πto neravenstvo budet strohym y, kak sledstvye, ymeet mesto
strohoe neravenstvo µ() "fm < µ() fm. Sledovatel\no, dejstvytel\no, v zadaçe
(1.4) moΩno ohranyçyt\sq mnohoçlenamy, vse m kornej kotor¥x prynadleΩat
otrezku I .
Funkcyonal µ() fm, fy mm ∈P(), qvlqetsq neprer¥vnoj funkcyej kornej
mnohoçlena fm. Na kompakte Im πta funkcyq dostyhaet svoeho naymen\ßeho
znaçenyq.
Lemma dokazana.
Dokazatel\stvo teorem¥ 1.1. Pust\ fy mm ∈P() est\ πkstremal\n¥j
mnohoçlen zadaçy (1.4). Eho moΩno predstavyt\ v vyde
fx m() = ygx m
m 21−(), gx m() = () xxj
j
m
−
=
∏
1
;(3.2)
mnohoçlen gm prynadleΩyt mnoΩestvu �m (y, v sylu lemm¥ 3.1, vse eho
korny leΩat na [–,] 11). Rassmotrym dva (kompaktn¥x) mnoΩestva
Hm = xfx m ∈≥ {} [–,]:() 111,
Gm = xfx m ∈≤ {} [–,]:() 111 = xgxy m
m ∈≤ {} −− [–,]:()() 11211;
ymeem HG mm + = 2 y Hm = σmy(). PoloΩym ρ = 12 −σmy()/, tohda
2ρ = 2−σmy() = Gm.(3.3)
V sylu opredelenyq (2.5) y utverΩdenyq (2.6) spravedlyva cepoçka sootno-
ßenyj
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
ALHEBRAYÇESKYE MNOHOÇLENÁ, NAYMENEE UKLONQGWYESQ OT NULQ … 295
2
2
ρ
m
= Em() 2ρ ≤ EG mm () ≤ gmCGm ()
=
1
21 ym−
.(3.4)
Otsgda sleduet, çto
2ρ ≤ 21 ym −/.(3.5)
Sootnoßenyq (3.3) y (3.5) dagt dlq velyçyn¥ (1.4) sledugwug ocenku snyzu:
σmy() = 22 −ρ ≥ 211 − () −ym/.(3.6)
Mnohoçlen (1.6) prynadleΩyt mnoΩestvu Pmy() y daet ocenku, obratnug
(3.6). Takym obrazom, ravenstvo (1.5) provereno. Odnovremenno dokazano, çto
mnohoçlen (1.6) qvlqetsq πkstremal\n¥m v zadaçe (1.4). Oçevydno, çto tohda y
vse mnohoçlen¥ (1.7) budut πkstremal\n¥my. Ubedymsq, çto druhyx πkstre-
mal\n¥x mnohoçlenov net.
PredpoloΩym, çto mnohoçlen fy mm ∈P() qvlqetsq πkstremal\n¥m v zada-
çe (1.4) y gmm ∈� est\ mnohoçlen, svqzann¥j s fm sootnoßenyem (3.2). Na
πtyx mnohoçlenax vse neravenstva (3.4), y, kak sledstvye, (3.5), obratqtsq v ra-
venstvo. V sylu teorem¥ 2.1 ravenstvo Em() 2ρ = EG mm () oznaçaet, çto mno-
Ωestvo Gm qvlqetsq otrezkom [,] hh −+ ρρ dlyn¥ 2ρ, ρ = ym −1/. ∏tot ot-
rezok prynadleΩyt [–,] 11. Sledovatel\no, Gm = [,] hh −+ ρρ , h ≤ 1−ρ .
Ravenstvo EG mm () = gmCGm ()
vleçet teper\, çto gm qvlqetsq sootvetst-
vugwym sdvyhom mnohoçlena (2.3). Kak sledstvye, fx m() = Tyxh m
m 1/() − ().
Teorema 1.1 dokazana.
Yz dokazatel\stva teorem¥ 1.1 (a, vproçem, formal\no yz formulyrovok te-
orem 1.1 y 2.1) vydno, çto spravedlyvo sledugwee utverΩdenye.
Teorema 3.1. Zadaçy (1.4) y (2.5) svqzan¥ sootnoßenyqmy
Em() 2ρ =
1
21 ym−
, 2 = 2ρσ +my(), ρ = ym −1/, y>1.(3.7)
4. Mnohoçlen¥, naymenee uklonqgwyesq ot nulq otnosytel\no yn-
tehral\n¥x funkcyonalov. Zadaça o mnohoçlenax s fyksyrovann¥m starßym
koπffycyentom, naymenee uklonqgwyxsq ot nulq, yzuçena ne tol\ko otnosy-
tel\no ravnomernoj norm¥. Ona xoroßo yssledovana (sm., naprymer, [6], § 15) v
prostranstvax L
p
(,) −11, 1≤<∞ p, nadelenn¥x normoj
f
L
p
(,) −11
=
1
2
1
11
fxdx
p
p
()
/
−
∫
.(4.1)
Vproçem, lyß\ pry p = 1 y p = 2 mnohoçlen¥ naymen\ßeho uklonenyq v¥py-
san¥ qvno; a ymenno, takov¥my qvlqgtsq (pry sootvetstvugwej normyrovke)
mnohoçlen¥ Çeb¥ßeva vtoroho roda () p=1 y mnohoçlen¥ LeΩandra () p=2
(sm., naprymer, [7], § 2.9). Mnohoçlen¥, naymenee uklonqgwyesq ot nulq, yz-
vestn¥ [8] ewe v prostranstve L011 (,) −, a toçnee, otnosytel\no funkcyonala
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
296V. V. ARESTOV
f
L011 (,) − = expln()
1
2
1
1
fxdx
−
∫
.(4.2)
Pust\ teper\ Φ — mnoΩestvo funkcyj ϕ , opredelenn¥x, neotrycatel\n¥x
y neub¥vagwyx na poluosy [,) 0+∞. V dannom punkte budet obsuΩdat\sq za-
daça o mnohoçlenax, naymenee uklonqgwyxsq ot nulq otnosytel\no yntehral\-
n¥x funkcyonalov
ϕfxdx () ()
−
∫
1
1
, ϕ∈Φ.(4.3)
Vo mnoΩestve Pm alhebrayçeskyx mnohoçlenov porqdka m rassmotrym
lynejn¥j operator Am, kotor¥j mnohoçlenu (1.1) sopostavlqet eho starßug
komponentu, normyrovannug mnoΩytelem 21 −− () m:
()() Aft mm = 21 −− () m
m
m ax, fmm ∈P.(4.4)
Netrudno ponqt\, çto zadaça o mnohoçlene, naymenee uklonqgwemsq ot nulq
otnosytel\no (proyzvol\noj) norm¥ na prostranstve Pm, πkvyvalentna prob-
leme v¥çyslenyq norm¥ operatora (4.4). V çastnosty, rezul\tat P. L. Çeb¥ße-
va (2.2) oznaçaet, çto norma operatora (4.4) na mnoΩestve Pm s ravnomernoj
normoj ravna edynyce.
Dlq funkcyy ϕ∈Φ oboznaçym çerez cm() ϕ naymen\ßug konstantu v ne-
ravenstve
ϕ21
1
1
−−
−
() ∫() m
m
m axdx ≤ cfxdx mm ()() ϕϕ()
−
∫
1
1
, fmm ∈P.(4.5)
PoloΩym
cm
∗ = sup(): cmϕϕ∈ {} Φ.(4.6)
NyΩe v teoreme 4.1 budet ukazano toçnoe znaçenye πtoj velyçyn¥.
Klassu Φ prynadleΩyt, v çastnosty, funkcyq ϕ∗, opredelennaq sootno-
ßenyqmy
ϕ∗() u =
001
11
,[,),
,[,).
u
u
∈
∈+∞
(4.7)
Dlq πtoj funkcyy funkcyonal¥ (4.3) y (1.2) sovpadagt:
ϕ∗
−
() ∫fxdx m()
1
1
= µ() fm.(4.8)
Poπtomu velyçyna αm = cm() ϕ∗ qvlqetsq naymen\ßej konstantoj v nera-
venstve
µ() () 21 −− m
m
m ax ≤ αµ mmf (), fmm ∈P.(4.9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
ALHEBRAYÇESKYE MNOHOÇLENÁ, NAYMENEE UKLONQGWYESQ OT NULQ … 297
Teorema 4.1. Pry lgbom m≥1 ymegt mesto ravenstva
cm
∗ = αm = 1.(4.10)
Teorema 4.1 v¥tekaet yz pryvedenn¥x nyΩe lemm 4.1 y 4.2.
Rassmotrym neskol\ko bolee obwug sytuacyg. Pust\ Λm — proyzvol\n¥j
lynejn¥j operator v Pm. Dlq funkcyy ϕ∈Φ oboznaçym çerez cm() ϕ =
= cmm (,) Λϕ nayluçßug (naymen\ßug vozmoΩnug) konstantu v neravenstve
ϕ()() Λmmftdt ()
−
∫
1
1
≤ cftdt mmm (,)() Λϕϕ()
−
∫
1
1
, fmm ∈P.(4.11)
Otmetym, çto, naprymer, dlq funkcyy ϕpu() = up konstanta () (,)/ cmmp
p Λϕ1
qvlqetsq normoj operatora Λm na podprostranstve Pm, nadelennom normoj
prostranstva Lp. Predstavlqet ynteres naybol\ßaq yz konstant cmm (,) Λϕ,
t. e. velyçyna
cmm () Λ = sup(,): cmm ΛΦ ϕϕ∈ {}.(4.12)
V sylu (4.8) velyçyna αmm () Λ = cmm (,) Λϕ∗ qvlqetsq naymen\ßej kon-
stantoj v neravenstve
µ() Λmmf ≤ αµ mmmf ()() Λ, fmm ∈P.(4.13)
Na mnoΩestve tryhonometryçeskyx polynomov neravenstva typa (4.13) yzuçal
A. H. Babenko [9] v 1992 hodu. On poluçyl dvustoronnye ocenky konstant¥ v ta-
kom neravenstve dlq dovol\no ßyrokoho klassa operatorov y, v çastnosty, dlq
klassyçeskyx operatorov vzqtyq starßej harmonyky y dyfferencyrovanyq.
Esly Λm/≡0, to
αmm () Λ ≥ 1.(4.14)
Dejstvytel\no, predpoloΩym, çto mnohoçlen fmm ∈P obladaet svojstvom
Λmmf/≡0. Rassmotrym semejstvo mnohoçlenov {} , ufu m>0. Ymeem
µ() ufm = mestftu m ∈≥ {} [–,]:()/ 111 → 2, u→+∞.
Takoe Ωe svojstvo ymeet y velyçyna µ(()) Λmm uf = µ(()) uf mm Λ. Podstavlqq
funkcyy {} , ufu m>0 v (4.13), poluçaem neravenstvo (4.14).
Funkcyq (4.7) prynadleΩyt klassu Φ, poπtomu ymeet mesto neravenstvo
αmm () Λ ≤ cmm () Λ. Na samom Ωe dele dve poslednye velyçyn¥ sovpadagt.
∏tot fakt, v obwem-to, qsen. V klasse tryhonometryçeskyx polynomov on do-
kazan v [10]; dlq polnot¥ yzloΩenyq pryvedem eho dokazatel\stvo v Pm.
Lemma 4.1. Pry lgbom m≥1 dlq lgboho odnorodnoho (v çastnosty, ly-
nejnoho) operatora Λm v Pm ymeet mesto ravenstvo
supcmm (,): ΛΦ ϕϕ∈ {} = αmm () Λ.(4.15)
Dokazatel\stvo. Esly Λm≡0, to obe velyçyn¥ v (4.15) ravn¥ nulg.
Budem sçytat\, çto Λm/≡0. Na dannom πtape ymeem 1≤≤ αmmmm c ()() ΛΛ.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
298V. V. ARESTOV
Dlq obosnovanyq lemm¥ ostalos\ dokazat\, çto dlq lgboj funkcyy ϕ∈Φ
v¥polnqetsq neravenstvo
cm() ϕ = cmm (,) Λϕ ≤ αmm () Λ;
pry πtom moΩno sçytat\, çto αmm () Λ<+∞.
Oboznaçym çerez Φc mnoΩestvo funkcyj ϕ , neprer¥vn¥x, neotrycatel\-
n¥x y neub¥vagwyx na poluosy [,) 0∞ so svojstvom ϕ() 00 =. PredpoloΩym
vnaçale, çto ϕ∈Φc. Pust\ f — yzmerymaq, ohranyçennaq funkcyq na [–,] 11,
k prymeru, ffmm =∈P. Rassmotrym xarakterystyçeskug funkcyg
λ() u = λ(;) uf = mestftu ∈≥ {} [–,]:() 11, u∈∞ [,) 0,(4.16)
funkcyy f. Kak yzvestno (sm., naprymer, [11], §10.12, vproçem, πto lehko pro-
veryt\ samostoqtel\no), ymeet mesto formula
ϕftdt () ()
−
∫
1
1
= – ϕλ ()() udu
0
∞
∫.
Vzqv poslednyj yntehral po çastqm, poluçym predstavlenye
ϕftdt () ()
−
∫
1
1
= λϕ ()() udu
0
∞
∫.(4.17)
Pry lgbom u>0 ymeem (sm. (4.16))
λ(;) uf = λ1;
f
u
= µ
f
u
.
Poπtomu dlq lgboho mnohoçlena fm pry lgbom u>0 v¥polnqetsq neraven-
stvo
λ() ; uf mm Λ ≤ αλ mmm uf (); () Λ.(4.18)
Poskol\ku αmm () Λ≥1, πto neravenstvo ymeet mesto y pry u=0. V sylu
(4.17) y (4.18) ymeem
ϕΛmmftdt () ()
−
∫
1
1
= λϕ () ;() ufdu mm Λ
0
∞
∫ ≤
≤ αλϕ mmm ufdu ();() () Λ
0
∞
∫ = αϕ mmmftdt ()() Λ()
−
∫
1
1
.
Sledovatel\no, esly ϕ∈Φc, to cm() ϕ ≤ αmm () Λ.
Rassmotrym proyzvol\nug funkcyg ϕ∈Φ. Ona ymeet ne bolee çem sçet-
noe mnoΩestvo toçek razr¥va; oboznaçym πto mnoΩestvo çerez UU=() ϕ.
MoΩno sçytat\, çto v kaΩdoj toçke razr¥va u>0 funkcyq ϕ neprer¥vna
sprava. Funkcyg ϕ moΩno predstavyt\ v vyde summ¥ ϕϕϕ =+ cs neprer¥v-
noj funkcyy ϕcc ∈Φ y funkcyy skaçkov ϕs. Dlq funkcyy ϕc (po dokazan-
nomu) ymeem cmc () ϕ ≤ αmm () Λ. Ubedymsq, çto takoe Ωe neravenstvo ymeet
mesto y dlq funkcyy ϕs. Neravenstvo (4.18) moΩno ynterpretyrovat\ kak ne-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
ALHEBRAYÇESKYE MNOHOÇLENÁ, NAYMENEE UKLONQGWYESQ OT NULQ … 299
ravenstvo cmu () δ ≤ αmm () Λ dlq funkcyy skaçka δu v toçke u>0, oprede-
lennoj formulamy
δut() =
00
1
,,
,.
≤<
≥
tu
tu
V sylu neravenstva (4.14) funkcyq skaçka v toçke 0
δ0()t =
00
10
,,
,,
t
t
=
>
obladaet πtym Ωe svojstvom. Funkcyq ϕs predstavyma v vyde
ϕs = quu
uU
()δ
∈
∑,
hde qu()≥0 est\ velyçyna skaçka v toçke razr¥va u funkcyy ϕ . Otsgda sle-
duet, çto dlq funkcyy skaçkov ϕs dejstvytel\no v¥polnqetsq neravenstvo
cms () ϕ ≤ αmm () Λ. Okonçatel\no ymeem cm() ϕ = maxcc mcms (),() ϕϕ {} ≤
≤ αmm () Λ.
Lemma dokazana.
Lemma 4.2. Pry lgbom m≥1 dlq nayluçßej konstant¥ αm v neraven-
stve (4.9) ymeet mesto formula
αm = 1.(4.19)
Dokazatel\stvo. Dlq konstant¥ αm spravedlyva formula
αm =
sup
()
:
() () µ
µ
21 −−
∗ ∈
m
m
m
m
m
ax
f
fm P;(4.20)
zdes\ verxnqq hran\ beretsq po mnoΩestvu Pm
∗
mnohoçlenov fmm ∈P, dlq koto-
r¥x µ() fm>0 yly, çto to Ωe samoe, fmC[–,] 11
= max():[–,] fxx m∈ {} 11 >
> 1 . Predstavym starßyj koπffycyent mnohoçlena fmm ∈∗ P v vyde am =
= ym21−. Tohda, ysxodq yz predstavlenyq (4.20), moΩem zapysat\
αm = sup
()
()
y
m
m
yx
y >1
µ
σ
.
Pry y>1 ymeem
µ() yxm = 211 − () −ym/.
Prymenqq teoremu 1.1, poluçaem
αm = sup
()
() /
y
m
m
yx
y >− − 1
1 21
µ
= 1;
tem sam¥m ravenstvo (4.19) provereno.
Lemma dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
300V. V. ARESTOV
Teorema 4.1 v sylu lemm¥ 4.2 qvlqetsq çastn¥m sluçaem lemm¥ 4.1.
Sledstvye. Dlq lgboj funkcyy ϕ∈Φ vo mnoΩestve Pm pry m≥1
ymeet mesto neravenstvo
ϕfxdx m() ()
−
∫
1
1
≥ ϕ21
1
1
−−
−
() ∫() m
m
m axdx, fmm ∈P.(4.21)
Pry lgbom m≥1 na klasse vsex funkcyj ϕ∈Φ (a ymenno, na funkcyy (4.7))
neravenstvo (4.21) neuluçßaemo.
Dlq sravnenyq otmetym sledugwyj fakt. V prostranstve L111 (–,) nayme-
nee uklonqgwymsq ot nulq (so starßym koπffycyentom, ravn¥m 1) qvlqetsq
normyrovann¥j mnohoçlen Çeb¥ßeva vtoroho roda (sm., naprymer, [7], p. 2.9.31)
Ux m() =
1
2
1
12 m
mx
x
sin() +
−
arccos
y pry πtom
Uxdx m()
−
∫
1
1
=
1
21 m−
.
Sledovatel\no, dlq funkcyy ϕ1() u = u nayluçßej v neravenstve (4.5) qvlq-
etsq konstanta
cm() ϕ1 =
2
1 m+
.
1.Suetyn P. K. Klassyçeskye ortohonal\n¥e mnohoçlen¥. – M.: Nauka, 1979.
2.Arestov V. V., Mendelev A. S. O tryhonometryçeskyx polynomax, naymenee uklonqgwyxsq
ot nulq // Dokl. AN. – 2009. – 425, # 6. – S. 733 – 736.
3.Smyrnov V. Y., Lebedev N. A. Konstruktyvnaq teoryq funkcyj kompleksnoho peremennoho.
– M.; L.: Nauka, 1964.
4.Çeb¥ßev P. L. Teoryq mexanyzmov, yzvestn¥x pod nazvanyem parallelohrammov. Polnoe
sobranye soçynenyj P. L. Çeb¥ßeva: V 5 t. T. 2. Matematyçeskyj analyz. – M.; L.: Yzd-vo
AN SSSR, 1947. – S. 23 – 51.
5.Bernßtejn S. N. ∏kstremal\n¥e svojstva polynomov. – M.: ONTY, 1937.
6.Nykol\skyj S. M. Kvadraturn¥e formul¥. – M.: Nauka, 1979.
7.Tyman A. F. Teoryq pryblyΩenyq funkcyj dejstvytel\noho peremennoho. – M.: Fyzmat-
hyz, 1960.
8.Hlaz¥ryna P. G. Neravenstvo brat\ev Markov¥x v prostranstve L0 na otrezke // Mat. za-
metky. – 2005. – 78, # 1. – S. 59 – 65.
9.Babenko A. H. Neravenstva slaboho typa dlq tryhonometryçeskyx polynomov // Tr. Yn-ta
matematyky y mexanyky Ural. otd-nyq RAN. – 1992. – 2. – S. 34 – 41.
10.Arestov V. V. Nekotor¥e πkstremal\n¥e zadaçy dlq tryhonometryçeskyx polynomov otno-
sytel\no funkcyonalov typa ϕ - norm¥ // Tr. MeΩdunar. let. mat. ßkol¥ S. B. Steçkyna po
teoryy funkcyj. – Tula: Yzd-vo Tul. un-ta, 2007. – S. 18 – 21.
11.Xardy H., Lyttl\vud DΩ., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948.
Poluçeno 19.10.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 3
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| id | umjimathkievua-article-2868 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:31:52Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f2/6af2b584aab89c86605adf597e8f0cf2.pdf |
| spelling | umjimathkievua-article-28682020-03-18T19:39:19Z Algebraic polynomials least deviating from zero in measure on a segment Алгебраические многочлены, наименее уклоняющиеся от нуля по мере на отрезке Arestov, V. V. Арестов, B. В. Арестов, B. В. We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment $[–1, 1]$ with respect to a measure, or, more precisely, with respect to the functional $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. We also discuss an analogous problem with respect to the integral functionals $∫_{–1}^1 φ (∣f (x)∣) dx$ for functions $φ$ that are defined, nonnegative, and nondecreasing on the semiaxis $[0, +∞)$. Досліджується задача про алгебраїчні многочлени із заданим старшим коефіцієнтом, що найменше відхиляються від нуля за мірою на відрізку $[–1, 1]$, а точніше, відносно функціонала $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. Обговорюється аналогічна задача відносно інтегральних функціоналів $∫_{–1}^1 φ (∣f (x)∣) dx$ для функцій $φ$, визначених, невід'ємних та неспадних на півосі $[0, +∞)$. Institute of Mathematics, NAS of Ukraine 2010-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2868 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 3 (2010); 291–300 Український математичний журнал; Том 62 № 3 (2010); 291–300 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2868/2487 https://umj.imath.kiev.ua/index.php/umj/article/view/2868/2488 Copyright (c) 2010 Arestov V. V. |
| spellingShingle | Arestov, V. V. Арестов, B. В. Арестов, B. В. Algebraic polynomials least deviating from zero in measure on a segment |
| title | Algebraic polynomials least deviating from zero in measure on a segment |
| title_alt | Алгебраические многочлены, наименее уклоняющиеся от нуля по мере на отрезке |
| title_full | Algebraic polynomials least deviating from zero in measure on a segment |
| title_fullStr | Algebraic polynomials least deviating from zero in measure on a segment |
| title_full_unstemmed | Algebraic polynomials least deviating from zero in measure on a segment |
| title_short | Algebraic polynomials least deviating from zero in measure on a segment |
| title_sort | algebraic polynomials least deviating from zero in measure on a segment |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2868 |
| work_keys_str_mv | AT arestovvv algebraicpolynomialsleastdeviatingfromzeroinmeasureonasegment AT arestovbv algebraicpolynomialsleastdeviatingfromzeroinmeasureonasegment AT arestovbv algebraicpolynomialsleastdeviatingfromzeroinmeasureonasegment AT arestovvv algebraičeskiemnogočlenynaimeneeuklonâûŝiesâotnulâpomerenaotrezke AT arestovbv algebraičeskiemnogočlenynaimeneeuklonâûŝiesâotnulâpomerenaotrezke AT arestovbv algebraičeskiemnogočlenynaimeneeuklonâûŝiesâotnulâpomerenaotrezke |