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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]

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 find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of mth order in the space L as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L. [ABSTRACT FROM AUTHOR]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L2[0, 2π] satisfying the constraint , where 0 < h < ∞, 1/ r < p ≤ 2, r ∈ ℕ, and ω m( f( r); t) is the modulus of continuity of mth order of the derivative f( r)( x) ∈ L2[0, 2 π]. [ABSTRACT FROM AUTHOR]