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 the paper, exact constants in direct and inverse approximation theorems for functions of several variables are found in the spaces S p. The equivalence between moduli of smoothness and some K-functionals is also shown in the spaces S p .

In the spaces S p of functions of several variables, 2π-periodic in each variable, we study the approximative properties of operators A △ ̺,r and P △ ̺,s , which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes S p H ω .Keywords and phrases: space S p , classes H ω , linear methods.Mathematics subject classification: 42B05, 26B30, 26B35.

3 . We obtain direct and inverse approximation theorems of 2π-periodic functions by Taylor-Abel-Poisson operators in the integral metric. Keywords: direct and inverse approximation theorems; K-functional; Taylor-Abel-Poisson means 2000 MSC: 42B05, 26B30, 26B35 UDC: 517.5It is well-known that any function f ∈ L p , f ≡ const, can be approximated by its Abel-Poisson means f (̺, ·) with a precision not better than 1 − ̺. It relates to the so-called saturation property of this approximation method. From this property, it follows that for any f ∈ L p , the relation f − f (̺, ·) p = o(1 − ̺), ̺ → 1−, holds only in the trivial case where f ≡ const. Therefore, any additional restrictions on the smoothness of functions don't give us the order of approximation better than 1 − ̺. In this connection, a natural question is to find a linear operator, constructed similarly to the Poisson operator, which takes into account the smoothness properties of functions and at the same time, for a given functional class, is the best in a certain sense. In [17], for classes of convolutions, whose kernels were generated by some moment sequences, the authors proposed the general method of construction of similar operators that take into account properties of such kernels and hence, the smoothness of functions from corresponding classes. One example of such operators are the operators A ̺,r , which are the main subject of study in this paper.The operators A ̺,r were first studied in [14], where in the terms of these operators, the author gave the structural characteristic of Hardy-Lipschitz classes H r p Lip α of functions of one variable, holomorphic on the unit circle of the complex plane. In [15], in terms of approximation estimates of such operators in some spaces S p of Sobolev type, the authors give a constructive description of classes of functions of several variables, whose generalized derivatives belong to the classes S p H ω .

Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces BS p of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.