### $\mathcal{O}$-regularly varying functions in approximation theory.

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A systematic method for the calculus of Bernstein's polynomial is described. It consists of reducing the problem to a homogeneous linear system of equations that may be constructed by fixed rules. Several problems about its computer implementation are discussed.

This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author's contributions to the subject.

We investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

The direct and inverse problems of approximation theory in the subspace of weighted generalized grand Lebesgue spaces of 2π-periodic functions with the weights satisfying Muckenhoupt's condition are investigated. Appropriate direct and inverse theorems are proved. As a corollary some results on constructive characterization problems in generalized Lipschitz classes are presented.

The present paper does not introduce a new approximation but it modifies a certain known method. This method for obtaining a periodic approximation of a periodic solution of a linear nonhomogeneous differential equation with periodic coefficients and periodic right-hand side is used in technical practice. However, the conditions ensuring the existence of a periodic solution may be violated and therefore the purpose of this paper is to modify the method in order that these conditions remain valid....