Sharp integral inequalities for fractional derivatives of trigonometric polynomials

Abstract

We study sharp estimates of integral functionals for operators on the set Tn of real trigonometric polynomials fn of degree n≥1 in terms of the uniform norm ‖fn‖C2π 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 ‖Dαfn‖q≤nα‖cost‖q‖fn‖∞ holds on the set Tn for the Weyl fractional derivatives Dαfn of order α≥1 for 0≤q<∞. For q=∞ (α≥1), 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].