DOI: 10.2478/s12175-011-0031-4 Math. Slovaca 61 (2011), No. 4, 601–618 APPROXIMATION IN WEIGHTED ORLICZ SPACES Ramazan Akg¨ un* — Daniyal M. Israfilov** (Communicated by J´ an Bors´ ık ) ABSTRACT. We prove some direct and converse theorems of trigonometric ap- proximation in weighted Orlicz spaces with weights satisfying so called Mucken- houpt’s A p condition. c 2011 Mathematical Institute Slovak Academy of Sciences 1. Introduction A function Φ is called Young function if Φ is even, continuous, nonnegative in R, increasing on (0, ∞) such that Φ (0) = 0, lim x→∞ Φ(x)= ∞. A Young function Φ said to satisfy ∆ 2 condition (Φ ∈ ∆ 2 ) if there is a constant c 1 > 0 such that Φ (2x) ≤ c 1 Φ(x) for all x ∈ R. Two Young functions Φ and Φ 1 are said to be equivalent (we shall write Φ ∼ Φ 1 ) if there are c 2 ,c 3 > 0 such that Φ 1 (c 2 x) ≤ Φ(x) ≤ Φ 1 (c 3 x) , for all x> 0. A nonnegative function M : [0, ∞) → [0, ∞) is said to be quasiconvex if there exist a convex Young function Φ and a constant c 4 ≥ 1 such that Φ(x) ≤ M (x) ≤ Φ(c 4 x) , for all x ≥ 0 holds. Let T := [-π,π]. A function ω : T → [0, ∞] will be called weight if ω is measurable and almost everywhere (a.e.) positive. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 41A25, 41A27; Secondary 42A10, 46E30. K e y w o r d s: direct theorem, inverse theorem, Orlicz space, Muckenhoupt weight, modulus of smoothness.