On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Lebesgue space with variable exponent

In this paper we investigate some properties of approximation polynomials in particular de la Vallee-Poussin means, Fejer 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.