Abstract. We investigate the approximation of a conjugate function by the Fejér sums of the Fourier series of the conjugate function and obtain the estimate between the derivatives of the conjugate functions and the derivatives of the conjugate trigonometric polynomials in the weighted Orlicz spaces with Muckenhoupt weights. We prove inverse theorem of approximation theory for the derivatives conjugate functions in the weighted Orlicz spaces.Mathematics subject classification (2010): 41A10, 42A10, 41A25, 46E30.

Let be a quasi-smooth curve in the complex plane C. In this study, a direct theorem of approximation theory in certain subclasses of the functions which have continuous derivatives through order r on a closed curve is proved. Mathematics Subject Classification Definitions, Some Auxiliary Results and Main ResultLet be an arbitrary restricted Jordan curve with two-component complements = C = 1 ∪ 2 , (0 ∈ 1 , ∞ ∈ 2 ). Let us consider the functions w = φ i (z), (i = 1, 2), that conformally and univalently map, respectively, i onto i , 1 = {w : |w| < 1}, 2 = {w : |w| > 1} , with norm φ 1 (0) =0, φ 1 (0) > 0, φ 2 (∞) = ∞, lim z→∞ 1 z φ 2 (z) > 0. Let us extend each φ i (z), (i = 1, 2) continuously up to the bound = ∂ 1 = ∂ 2 (generally speaking φ 1 (z) = φ 2 (z) for z ∈ ). We preserve the notation φ i , (i = 1, 2) for the extension. Let z = i (w) be the inverse mapping of w = φ i (z), (i = 1, 2) . LetFor arbitrary natural number n we set1+ (−1) i n (z) := inf ζ 1+ (−1) i n |ζ − z| , ρ1 n (z) := min ρ 1+ (−1) i n (z) , i = 1, 2

In the present work, we investigate estimates of the deviations of the periodic functions from the linear operators constructed on the basis of its Fourier series in reflexive weighted Orlicz spaces with Muckenhoupt weights. In particular, the orders of approximation of Zygmund and Abel-Poisson means of Fourier trigonometric series were estimated by the k − th modulus of smoothness in reflexive weighted Orlicz spaces with Muckenhoupt weights.

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In this work the approximation problems of the functions by matrix transforms in weighted Orlicz spaces with Muckenhoupt weights are studied. We obtain the degree of approximation of functions belonging to Lipschitz class Lip(α, M, ω) through matrix transforms T (A) n (x, f) , and Nörlund means Nn (x, f) of their trigonometric Fourier series.