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The sharp values of some known -widths of classes of functions analytic in the unit disk in the Bergman space given by characteristic of smoothness and majorant satisfying certain restrictions were found. [ABSTRACT FROM AUTHOR]

The problem of finding the supremums of the best simultaneous polynomial approximations of some classes of functions analytic in the unit disk and belonging to the Bergman space is considered. The indicated function classes are defined by the averaged values of the th-order moduli of continuity of the highest derivative bounded from above by some majorant . [ABSTRACT FROM AUTHOR]

Exact constants are found in inequalities of the Jackson–Stechkin type for smoothness characteristics determined by averaging the norm in of the mth order finite differences of the functions f. For classes of functions defined by the smoothness characteristic , and the majorants satisfying a certain condition, the exact values of different n-widths are calculated. [ABSTRACT FROM AUTHOR]

In the metric L2, we establish exact inequalities, which relate the best approximations by trigonometric "angles" for the functions f(x, y) differentiable and 2𝜋-periodic in each variable to the integrals containing the moduli of continuity of higher order for mixed derivatives of these functions. For some classes of functions defined by the moduli of continuity, we determine the Kolmogorov and linear quasiwidths. [ABSTRACT FROM AUTHOR]

We consider the problem of the mean-square approximation of complex functions regular in a domain D ⊂ C by Fourier serieswith respect to an orthogonal (inD) systemof functions {ϕk(z)}, k = 0, 1, 2,.... For the case inwhich D = {z ∈ C: |z| < 1}, we obtain sharp estimates for the rate of convergence of the Fourier series in the orthogonal system {zk}, k = 0, 1, 2,..., for classes of functions defined by a special mth-order modulus of continuity. Exact values of the series of n-widths for these classes of functions are calculated. [ABSTRACT FROM AUTHOR]

For some classes of functions of two variables defined by their moduli of continuity, sharp upper bounds for the approximation of functions by interpolation bilinear splines are obtained. [ABSTRACT FROM AUTHOR]

We compute the exact values of widths for various widths for the classes W(Φ, μ), μ ≥ 1, of analytic functions in the disk belonging to the Hardy space H, q ≥ 1, whose averaged moduli of continuity of the boundary values of the derivatives with respect to the argument f, r ∈ N, are dominated by a given function Φ. For calculating the linear and Gelfand n-widths, we use best linear approximation for these functions. [ABSTRACT FROM AUTHOR]

We consider some extremal problems of approximation theory of functions on the whole real axis ℝ by entire functions of the exponential type. In particular, we find the exact values of the mean ν-widths of classes of functions, defined by the modules of continuity of the mth order ω and majorants ψ satisfying the special type of restriction. [ABSTRACT FROM AUTHOR]