Weighted Remez- and Nikolskii-Type Inequalities on a Quasismooth Curve
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Weighted Remez- and Nikolskii-Type Inequalities on a Quasismooth Curve Vladimir Andrievskii1
Received: 20 July 2017 / Revised: 17 December 2017 / Accepted: 26 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We establish sharp L p , 1 ≤ p < ∞, weighted Remez- and Nikolskii-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) curve in the complex plane. Keywords Polynomial · Quasismooth curve · Remez inequality · Nikolskii inequality Mathematics Subject Classification 30A10 · 30C10 · 30C62
1 Introduction From the numerous generalizations of the classical Remez inequality (see, for example, [5,8,10,20]), we mention three results which are the starting point of our analysis. Let |S| be the linear measure (length) of a Borel set S in the complex plane C. By Pn we denote the set of all complex polynomials of degree at most n ∈ N := {1, 2, . . .}. The first result is due to Erdélyi [7]. Assume that for pn ∈ Pn and T := {z : |z| = 1}, we have π (1.1) |{z ∈ T : | pn (z)| > 1}| ≤ s, 0 < s ≤ . 2 Then, | pn (eit )|2 is a trigonometric polynomial of degree at most n and, by the Remeztype inequality on the size of trigonometric polynomials (cf. [7, Thm. 2] or [5, p. 230]), we obtain
Communicated by Doron Lubinsky.
B 1
Vladimir Andrievskii [email protected] Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
123
V. Andrievskii
|| pn ||C(T) ≤ e2sn , 0 < s ≤
π . 2
(1.2)
Here, || · ||C(S) means the uniform norm over S ⊂ C. The second result is due to Mastroianni and Totik [17]. Let Tn be a trigonometric polynomial of degree n ∈ N, 1 ≤ p < ∞, and W : [0, 2π ] → {x ≥ 0} be an A∞ weight function. Then, according to [17, (5.2) and Thm. 5.2], there are positive constants c1 and c2 depending only on the A∞ constant of W and p, such that for a measurable set E ⊂ [0, 2π ] with |E| ≤ s, 0 < s ≤ 1, we have
|Tn | W ≤ c1 exp(c2 sn) p
[0,2π ]
[0,2π ]\E
|Tn | p W.
(1.3)
The third result, which is due to Andrievskii and Ruscheweyh [4], extends (1.2) valid for all pn ∈ Pn satisfying (1.1) to the case of algebraic polynomials considered on a Jordan curve ⊂ C instead of the unit circle T. In the present paper, we always assume that is quasismooth (in the sense of Lavrentiev), see [19], i.e., for every z 1 , z 2 ∈ , (1.4) |(z 1 , z 2 )| ≤ |z 1 − z 2 |, where (z 1 , z 2 ) is the shorter arc of between z 1 and z 2 (including the endpoints) and ≥ 1 is a constant. Let be the unbounded component of C\, where C := C ∪ {∞}. Denote by the conformal mapping of onto D∗ := {z : |z| > 1} with the normalization (z) > 0. z→∞ z
(∞) = ∞, (∞) := lim For δ > 0 and A, B ⊂ C, we set d(A, B) = dist(A, B) :=
inf
z∈A,ζ ∈B
|z − ζ |,
δ := {ζ ∈ : |(ζ )| = 1 + δ}. Let the function δ(t) = δ(t, ), t > 0, be defined by the equation d(, δ(t) ) = t and let diam S be the diameter of a set S ⊂ C. According to [4, Thm. 2], if for pn ∈ Pn , |{z ∈ : | pn (z)| > 1}| ≤ s
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