Bernstein-Szegõ inequality for trigonometric polynomials in the space L0

Abstract

Inequalities of the form kfn (α) cos θ + fn (α) sin θkp ≤ Bn(α, θ)pkfnkp for classical derivatives of order α ∈ N and Weyl derivatives of real order α ≥ 0 of trigonometric polynomials fn of order n ≥ 1 and their conjugates for real θ and 0 ≤ p ≤ ∞ are called Bernstein-Szegõ inequalities. They are generalizations of the classical Bernstein inequality (α = 1, θ = 0, p = ∞). Such inequalities have been studied for more than 90 years. The problem of studying the Bernstein-Szegõ inequality consists in analyzing the properties of the best (the smallest) constant Bn(α, θ)p, its exact value, and extremal polynomials for which this inequality turns into an equality. G. Szegõ (1928), A. Zygmund (1933), and A. I. Kozko (1998) showed that, in the case p ≥ 1 for real α ≥ 1 and any real θ, the best constant Bn(α, θ)p is nα. For p = 0, Bernstein-Szegõ inequalities are of interest at least because the constant Bn(α, θ)p is the largest for p = 0 over 0 ≤ p ≤ ∞. In 1981, V. V. Arestov proved that, for r ∈ N and θ = 0, the Bernstein inequality is true with the constant nr in the spaces Lp, 0 ≤ p < 1; i.e., Bn(r, 0)p = nr. In 1994, he proved that, for p = 0 and the derivative of the conjugate polynomial of order r ∈ N ∪ {0}, i.e., for θ = π/2, the exact constant grows exponentially in n; more precisely, Bn(r, π/2)0 = 4n+o(n). In two recent papers of the author (2018), a similar result was obtained for Weyl derivatives of positive noninteger order for any real θ. In the present paper, we prove that the formula Bn(α, θ)0 = 4n+o(n) holds for derivatives of nonnegative integer orders α and any real θ 6= πk, k ∈ Z. © 2019 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.