In the Orlicz type spaces S M , we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre K-functionals in the spaces S M . 1 2 STANISLAV CHAICHENKO, ANDRII SHIDLICH AND FAHREDDIN ABDULLAYEV where f (k) := [f ] (k) = 12π 2π 0 f (t)e −ikt dt, k ∈ Z, are the Fourier coefficients of the function f , and investigated some approximation characteristics of these spaces, including in the context of direct and inverse theorems. Stepanets and Serdyuk [25] introduced the notion of kth modulus of smoothness in S p and established direct and inverse theorems on approximation in terms of these moduli of smoothness and the best approximations of functions. Also this topic was investigated actively in [26], [30], [24], [29, Ch. 3] and others.In the papers [19], [20] some results for the spaces S p were extended to the Orlicz sequence spaces. In particular, in [20] the authors found the exact values of the best n-term approximations and Kolmogorov widths of certain sets of images of the diagonal operators in the Orlicz spaces. The purpose of this paper is to combine the above mentioned studies and prove direct and inverse theorems in the Orlicz type spaces S M in terms of best approximations of functions and moduli of smoothness of fractional order. PreliminariesAn Orlicz function M (t) is a non-decreasing convex function defined for t ≥ 0 such that M (0) = 0 and M (t) → ∞ as t → ∞. Let S M be the space of all functions f ∈ L such that the following quantity (which is also called the Luxemburg norm of f ) is finite:(2.1) Functions f ∈ L and g ∈ L are equivalent in the space S M , when f − g M = 0. The spaces S M defined in this way are Banach spaces. In case M (t) = t p , p ≥ 1, they coincide with the above-defined spaces S p .Let T n , n = 0, 1, . . ., be the set of all trigonometric polynomials τ n (x):= |k|≤n c k e ikx of the order n, where c k are arbitrary complex numbers. For any function f ∈ S M , we denote by E n (f ) M := inf τn−1∈Tn−1 f − τ n−1 M (2.2) where S n−1 (f, x) = |k|≤n−1 f (k)e ikx is the Fourier sum of the function f . According to (3.1) and (2.4), we have E n (f ) M = inf a > 0 : |k|≥n M (|ψ k f ψ (k)|/a) ≤ 1

In weighted Orlicz type spaces S p, µ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is in a certain sense the best. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre K-functionals is shown in the spaces S p, µ .

In Musilak-Orlicz type spaces S M , direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

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