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. | Turk J Math (2018) 42: 2916 – 2925 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Linear methods of summing Fourier series and approximation in weighted Orlicz spaces Sadulla Z. JAFAROV∗, Department of Mathematics and Science Education, Faculty of Education, Muş Alparslan University, Muş, Turkey Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan Received: • Accepted/Published Online: • Final Version: Abstract: 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. Key words: Boyd indices, weighted Orlicz space, Muckenhoupt weight, modulus of smoothness, Zygmund mean, AbelPoisson mean 1. Introduction Let M (u) be a continuous increasing convex function on [0, ∞) such that M (u)/u → 0 if u → 0, and M (u)/u → ∞ if u → ∞ . We denote by N the complementary of M in Young’s sense, . N (u) = max {uv − M (v) : v ≥ 0} if u ≥ 0. We will say that M satisfies the ∆2 −condition if M (2u) ≤ cM (u) for any u ≥ u0 ≥ 0 with some constant c, independent of u. Let T denote the interval [−π, π] , C the complex plane, and Lp (T), 1 ≤ p ≤ ∞ , the Lebesgue space of measurable complex-valued functions on T. e M (T) denote the set of all Lebesgue measurable functions f : T → C For a given Young function M , let L for which ∫ M (|f (x)|) dx 0, M −1 (ty) first considered by Matuszewska and Orlicz [24], are called the Boyd indices of the Orlicz spaces LM (T) . It is known that the indices αM and βM satisfy 0 ≤ αM ≤ βM ≤ 1, αN + βM = 1 , αM +
