We investigate the problem of approximation of functions f holomorphic in the unit disk by means A f r ρ, ( ) as ρ → 1 -. In terms of the error of approximation by these means, a constructive characteristic of classes of holomorphic functions H p r Lip α is given. The problem of the saturation of A f r ρ, ( ) in the Hardy space H p is solved.

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 .

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