Periodica Mathematica Hungarica https://doi.org/10.1007/s10998-019-00288-z On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Lebesgue space with variable exponent Ahmet Testici 1 © Akadémiai Kiadó, Budapest, Hungary 2019 Abstract In this paper we investigate some properties of approximation polynomials in particular de la Vallée-Poussin means, Fejér means and partial sums of Fourier series in weighted Lebesgue spaces with variable exponent. In addition to these we prove a simultaneous type theorem and some theorems on the equivalence of modulus of smoothness and the K -functional in weighted Lebesgue space with variable exponent. Keywords Trigonometric polynomial · Lipschitz class · de la Vallée-Poussin mean · Fourier series · Muckenhoupt weight · Best approximation Mathematics Subject Classification 41A10 · 42A10 · 41A30 · 41A17 1 Introduction and main results Let T := [0, 2π ] and let p (·) : T →[1, ∞) be a Lebesgue measurable 2π periodic func- tion. The variable exponent Lebesgue space L p(·) (T) is defined as the set of all Lebesgue measurable 2π -periodic functions f such that ρ p(·) ( f ) :=  2π 0 | f (x )| p(x ) dx < ∞. We suppose that the considered exponent functions p (·) satisfy the conditions 1 < p − := essinf x ∈T p (x ) ≤ esssup x ∈T p (x ) := p + < ∞. In addition to this requirement if there exist a positive constant c such that | p (x ) − p ( y )| ln (1/ |x − y |) ≤ c, x , y ∈ T, 0 < |x − y |≤ 1/2, (1.1) then we say that p (·) ∈ P 0 (T). B Ahmet Testici testiciahmet@hotmail.com 1 Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey 123