M athematical I nequalities & A pplications Volume 14, Number 2 (2011), 359–371 SIMULTANEOUS AND CONVERSE APPROXIMATION THEOREMS IN WEIGHTED LEBESGUE SPACES YUNUS E. YILDIRIR AND DANIYAL M. I SRAFILOV (Communicated by J. Marshall Ash) Abstract. In this paper we deal with the simultaneous and converse approximation by trigono- metric polynomials of the functions in the Lebesgue spaces with weights satisfying so called Muckenhoupt’s A p condition. 1. Introduction and the main results Let T :=[-π , π ]. A positive almost everywhere (a.e.), integrable function w : T → [0, ∞] is called as a weight function. With any given weight w we associate the w - weighted Lebesgue space L p w (T) consisting of all measurable functions f on T such that ‖ f ‖ L p w (T) = ‖ fw‖ L p (T) < ∞. Let 1 < p < ∞ and 1/ p + 1/q = 1. A weight function w belongs to the Mucken- houpt class A p (T) if ⎛ ⎝ 1 |I |  I w p (x)dx ⎞ ⎠ 1/ p ⎛ ⎝ 1 |I |  I w -q (x)dx ⎞ ⎠ 1/q  c with a finite constant c independent of I , where I is any subinterval of T and |I | denotes the length of I . For formulation of the new results we will begin with some required informations. Let f (x) ∼ ∞ ∑ k=-∞ c k e ikx = a 0 2 + ∞ ∑ k=1 (a k cos kx + b k sin kx) (1) and ˜ f (x) ∼ ∞ ∑ k=1 (a k sin kx - b k cos kx) Mathematics subject classification (2010): 41A10, 42A10. Keywords and phrases: Best approximation,weighted Lebesgue space, mean modulus of smoothness, fractional derivative. c  , Zagreb Paper MIA-14-29 359