Ukrainian Mathematical Journal, Vol. 69, No. 2, July, 2017 (Ukrainian Original Vol. 69, No. 2, February, 2017) BERNSTEIN–NIKOL’SKII-TYPE INEQUALITIES FOR TRIGONOMETRIC POLYNOMIALS WITH AN ARBITRARY CHOICE OF HARMONICS H. M. Vlasyk UDC 517.5 We obtain the Bernstein–Nikol’skii-type inequalities for trigonometric polynomials with an arbitrary choice of harmonics. Introduction Prior to the statement of the problem and presentation of the history of its investigation, we give necessary notation and definitions. Let L q be a space of functions f 2π -periodic and summable to a power q, 1  q< 1, (resp., essentially bounded for q = 1) on the segment [−π, π]. The norm in this space is defined as follows: kf k Lq = kf k q = 8 > > > > > > < > > > > > > : 0 @ 1 2π π Z −π |f (x)| q dx 1 A 1 q , 1  q< 1, ess sup x2[−π,π] |f (x)|, q = 1. For a function f 2 L 1 , we consider its Fourier series X k2Z ˆ f (k)e ikx , where ˆ f (k)= 1 2π π Z −π f (x)e −ikx dx are the Fourier coefficients of the function f. In what follows, we always assume that the function f 2 L 1 satisfies the condition π Z −π f (x) dx =0. Further, let ψ(τ ) 6=0, τ 2 N, be an arbitrary function of natural argument and let β be an arbitrary fixed real number. If a series X k2Z\{0} ˆ f (k) ψ(|k|) e i(kx+β π 2 sign k) Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 2, pp. 147–156, February, 2017. Original article submitted July 6, 2016. 0041-5995/17/6902–0173 c  2017 Springer Science+Business Media New York 173 DOI 10.1007/s11253-017-1355-1