4 Higher order Bernstein-type inequalities and their sharpness

P_n^(k)(e^ia)

≥(1−o(1))n^2kΩ(E_T, e^ia)^2k 2^kπ^2k

(2k−1)!!kP_nk_E

T. (49)

The quantityo(1) depends onE andk and tends to0 asn→ ∞.

Proof. We may assume thatnis even (because (n+ 1)²/n²= 1 +o(1)). We con-sider the trigonometric polynomial Tn/2(t) = e^−itn/2Pn e^it

. So, (48) follows now from applying Theorem 6 toTn/2.

Concerning (49), existence of such polynomials, in view of the remark above, follows from existence of trigonometric polynomialsT_n for which (29) holds.

4 Higher order Bernstein-type inequalities and their sharpness

LetE⊂(−π, π) be a compact subset, and fix a pointz₀=e^it0 which is in the one dimensional interior ofE_T. That is,{exp(it) :t₀−δ < t < t₀+δ} ⊂E_Tfor some smallδ >0. Denote by∂/∂n+and∂/∂n₋the outward and inward normal derivatives (w.r.t. the unit circle) correspondingly. Then (see [17], formulas (23) and (24) on p. 349)

1

2 1 + 2πωE_T e^it

= ∂g(e^it,∞)

∂n₊ = max

∂g(e^it,∞)

∂n₊ ,∂g(e^it,∞)

∂n₋

where g(z, w) =g_C\E

T(z, w) is Green’s function of C\E_T and ω_E

T(.) denotes the density of the equilibrium measure (w.r.t. arc length on the unit circle).

Now let us consider higher order Bernstein-type inequalities for trigonomet-ric polynomials.

Theorem 8. Let E⊂(−π, π)be a compact set andkbe a positive integer. Fix a closed intervalE0⊂IntE(subset of the one dimensional interior ofE). Then there existsC=C(E, E0, k)>0 such that for all trigonometric polynomialTn

with degreen, we have fort∈E0

T_n^(k)(t)

≤(1 +o(1))n^k 2πωE_T e^itk

kTnk_E. (50) whereo(1) is uniform in t∈E0 and uniform among all trigonometric polyno-mials having degree at mostn and tends to0 asn→ ∞.

Proof. We prove the theorem by induction onk, the casek= 1 was done in [13, Theorem 4].

Let

V(t) = 2πωE_T e^it .

Select a closed setE₀^∗⊃E0such thatE₀^∗ has no common endpoints either with E0or with E.

Consider anyδ >0 such that the intersection ofEwith theδ-neighborhood of E0 is still subset of of E₀^∗, and set fk,n,t₀(t) := Tn^(k)(t)Q(t), where Q(t) = Q_n1/3(t) is a fast decreasing trigonometric polynomial from Theorem 3 fort0∈ E0 (α⁰ andβ⁰ from Theorem 3 are chosen such a way that the interval [α⁰, β⁰] is in theδ-neighborhood ofE₀).

By (23) and (26), for thisf_k,n,t0 we have the upper bound O(n^2k) exp

−δ₁n^1/3

kT_nk_E=o(1)kT_nk_E

onE outside the δ-neighborhood oft0 withδ1>0 (uniform int0∈E0).

In the δ-neighborhood of any t0 ∈ E0, by kQkE ≤ 1 and by induction hypothesis applied toTn and toE₀^∗, we have

|fk,n,t₀(t)| ≤(1 +o(1))n^kkTnkEV(t)^k ≤(1 +o(1))n^k(1 +ε)^kkTnkEV(t0)^k, whereε→0 asδ→0. Here we used that by the continuity ofV(t), if t0∈E0

and|t−t0|< δ, thenV(t)≤(1 +ε)V(t0) with someεthat tends to 0 asδ→0.

Therefore,fk,n,t₀(t) is a trigonometric polynomial intof degree at mostn+n^1/3 for which

kfk,n,t0k ≤(1 +o(1))n^kkTnkEV(t₀)^k.

Upon applying Lukashov’s theorem from [13, Theorem 4] to the trigonometric polynomialf_k,n,t0(t) we obtain

|f_k,n,t⁰

0(t₀)| ≤(1 +o(1))n^k+1kT_nk_EV(t₀)^k+1. (51) Since (recall thatQ(t0) = 1)

f_k,n,t⁰ ₀(t0) =T_n^(k+1)(t0) +T_n^(k)(t0)(Q(t0))⁰,

and the second term on the right is at most O(n^k)O(n^2/3)kTnkE in modulus, by (26) and by the induction assumption, from (51) we get (50). It follows from the proof that the estimate is uniform int₀∈E₀.

Corollary 9. Let E ⊂ (−π, π) be again a compact set and k be a positive integer. Fix a closed intervalE0⊂IntE. Then there existsC=C(E, E0, k)>0 such that for all algebraic polynomial Pn with degree n, we have for z = e^it, t∈E0

P_n^(k)(z)

≤(1 +o(1))n^k

2^k (1 + 2πω_E

T(z))^kkPnk_E

T (52)

whereo(1) is uniform inz=e^it,t∈E₀ and independent ofP_n, but it tends to 0as n→ ∞.

Proof. As in the proof of Corollary 7, we may assume that nis even (because

It, together with Fa`a di Bruno’s formula (1) and Theorem 8 yields that

Corollary 9 extends Theorem 1 of the paper [17] to higher derivatives of algebraic polynomials and the proof of sharpness is similar to the proof of [17], Theorem 2.

Theorem 10. Under assumption of Corollary 9, inequality (52) is sharp, that is, there is a sequence of polynomialsPn6≡0,n= 1,2, . . ., such that

Proof. We enclose E_T into a setGwith the following properties:

• Gis a finite union of disjointC²smooth Jordan domains: there are finitely many disjointC²Jordan curvesS1, . . . , Smsuch that ifGj is the bounded connected components ofC\Sj, thenG=∪^m_j=1Gj,

• E_T is a boundary arc of the boundary∂G,

• the component ofGthat containszlies in the closed unit disk,

• every point ofGis of distance≤η from a point ofE_T, whereη is a given positive number.

Then the boundary Γ =∂G=∪^m_j=1S_j is a family of disjoint Jordan curves.

Furthermore, let n+ = z be the normal at z to Γ pointed to the interior of Ω =C\G.

Ifε >0 is given, then for sufficiently smallη we have (see e.g. [15], pp. 350-351

∂g_Ω(z,∞)

∂n+

≥(1−ε)∂g_C\E

T(z,∞)

∂n+

. (53)

By the sharp form of the Hilbert lemniscate theorem [15], Theorem 1.2, there is a Jordan curveσsuch that

• σ contains Γ in its interior except for the pointz, where the two curves touch each other,

• σis a lemniscate, i.e. σ={ζ: |VN(ζ)|= 1}for some algebraic polynomial VN of degreeN, and

•

∂g_C\σ(z,∞)

∂n₊ ≥(1−ε)∂gΩ(z,∞)

∂n₊ . (54)

We may assume thatV_N⁰(z)>0. The Green’s function of the outer domain ofσis _N¹ log|VN(.)|, and its normal derivative is

∂g_C\σ(z,∞)

∂n+

= 1

N|V_N⁰(z)|= 1 NV_N⁰(z).

Consider now, for all largen, the polynomialsP_n(.) =V_N(.)^[n/N]. This is a polynomial of degree at mostn, its supremum norm onσ is 1, and by Fa`a di Bruno formula (1), it can be shown that (see also [8], subsection 10.2)

P_n^(k)(z)

=n^k ∂g_C\σ(z,∞)

∂n+

!^k

+O(n^k−1).

Thus, in view of (53) and (54), we may continue

P_n^(k)(z)

≥(1−ε)^2kn^k ∂g_C\E

T(z,∞)

∂n+

!^k

+O(n^k−1).

Note also thatkPnkE_T≤ kPnkσ= 1 by the maximum principle.

Corollary 11. Under assumption of Theorem 8, inequality (50) is sharp, for there is a sequence of trigonometric polynomialsT_n6≡0,n= 1,2, . . ., such that

T_n^(k)(t)

≥(1−o(1))n^k 2πω_E

T e^itk

kTnk_E. whereo(1) depends onE andk and tends to0 asn→ ∞.

Proof. Existence of such trigonometric polynomialsTnfollows immediately from the existence of corresponding (in the sense of the proof of Corollary 9) algebraic polynomialsP2n from Corollary 9.

Acknowledgement

The research of the first author has been supported by Russian Science Founda-tion under project 14-11-00022 (in the part concerning polynomial inequalities).

The second author was supported by the J´anos Bolyai Scholarship of Hun-garian Academy of Sciences.

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Sergei Kalmykov

School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240, P.R. China

Far Eastern Federal University, 8 Sukhanova Street, Vladivostok, 690950, Rus-sia

email address: sergeykalmykov@inbox.ru B´ela Nagy

MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, Uni-versity of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary

email address: nbela@math.u-szeged.hu

In document Higher Markov and Bernstein inequalities and fast decreasing polynomials with prescribed zeros (Pldal 23-29)