Sharp integral inequalities for fractional derivatives of trigonometric polynomials

Abstract

We study sharp estimates of integral functionals for operators on the set T n of real trigonometric polynomials f n of degree n1 in terms of the uniform norm

f n

C 2π of the polynomials and similar questions for algebraic polynomials on the unit circle of the complex plane. P.Erdös, A.P.Calderon, G.Klein, L.V.Taikov, and others investigated such inequalities. In this paper, we, in particular, show that the sharp inequality

D αf n

q≤n α

cos t

q

f n

∞ holds on the set T n for the Weyl fractional derivatives Dα f n of order α 1 for 0 ≤ q < ∞. For q = ∞ (α1), this fact was proved by Lizorkin (1965) [12]. For 1 ≤ q < ∞ and positive integer α, the inequality was proved by Taikov (1965) [23]; however, in this case, the inequality follows from results of an earlier paper by Calderon and Klein (1951) [6]. © 2012 Elsevier Inc.