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We study extremal problems related to the best joint polynomial approximation of functions analytic in the unit disk and belonging to the Bergman space . The problem of joint approximation of periodic functions and their derivatives by trigonometric polynomials was considered by Garkavy [1] in 1960. Then, in the same year, Timan [2] considered this problem for classes of entire functions defined on the entire axis. The problem of joint approximation of functions and their derivatives is considered in more detail in Malozemov's monograph [3], where some classical theorems of the theory of approximation of functions are presented and generalized. In the present paper, a number of exact theorems are obtained and sharp upper bounds for the best joint approximations of a function and its successive derivatives by polynomials and their respective derivatives on some classes of complex functions belonging to the Bergman space are calculated. [ABSTRACT FROM AUTHOR]

Exact constants in Jackson–Stechkin type inequalities are found for the smoothness characteristics , , determined by averaging the norm of finite differences of th order of functions . A solution is given of the extremal problem of finding the supremum for best joint polynomial approximations of functions and their successive derivatives on some classes of functions from whose averaged norms of finite differences are bounded above by . [ABSTRACT FROM AUTHOR]

In Jackson–Stechkin type inequalities for the smoothness characteristic , , we find exact constants determined by averaging the norms of finite differences of th order of a function . We solve the problem of best joint approximation for a certain class of functions from , whose smoothness characteristic averaged with a given weight is bounded above by the majorant . The exact values of -widths of some classes of functions are also 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]

Considering Banach Hardy spaces and weighted Bergman spaces, we find the sharp values of the Bernstein, Kolmogorov, Gelfand, and linear n-widths for the classes of analytic functions on the unit disk whose moduli of continuity of the rth derivatives averaged with weight are majorized by a given function satisfying some constraints. [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]

The article presents mathematical analysis of the problem of the approximate evaluation of curvilinear integrals of the first kind for some classes of functions and classes of space curves.

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived. [ABSTRACT FROM AUTHOR]

The article focuses on the use of Jackson-Stechkin type inequalities for the study of periodic functions. It says that computing Jackson-Stechkin sharp constants is an external problem of function approximation theory. It also mentions several mathematical methods related to periodic functions including Minkowski's inequality, Rolle's theorem, and Fourier coefficients.

The article discusses the best linear methods and width values for some analytic function classes in the Bergman weight space. It presents results associated to the exact value calculation in analytical function classes on unit disk and the best linear method construction. It presents the interpretation of the results of the problems provided as assertions on the functions' coding and optimal reconstruction. It also explains the calculation of Taylor coefficient absolute values' least upper bounds.

We consider the problem of finding exact inequalities for the best approximations of periodic differentiable functions by trigonometric polynomials and the m-order moduli of continuity in the space L and present their applications. For some classes of functions defined by the indicated moduli of continuity, we calculate the exact values of n-widths in the space L. [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 find the exact values of the n-widths for the classes of periodic differentiable functions in L[0, 2 π] satisfying the constraint where h > 0, m ∈ ℕ, r ∈ ℤ, $\tilde \Omega _m $( f; t) is the generalized mth order continuity modulus of the derivative f ∈ L[0, 2 π], while Φ( t) is an arbitrary increasing function such that Φ(0) = 0. [ABSTRACT FROM AUTHOR]

The article discusses the exact inequalities relating the best approximations of periodic differentiable functions by trigonometric polynomials and order moduli of continuity in linear spaces. It provides the function of the space of the Lebesque measurable periodic real functions having finite norms. It also denotes the function of the subspace of all trigonometric polynomials at a degree to be associated with the functions of Fourier expansion. The value of the approximation by elements of the subspace is given as illustrated in a formula. A formula is also given for solving the extremal problems of the theory of approximation of differentiable periodic functions of trigonometric polynomials on in spaces within the Jackson-Stechkin-type inequalities.

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]

We determine the exact values of Kolmogorov and linear quasiwidths for some classes of differentiable periodic functions of two variables in the Hilbert space L2( Q). [ABSTRACT FROM AUTHOR]

The article provides information on the applicability of the analytic function classes for weight Banach spaces' disk. It mentions the move of the authors in finding the exact formulation as well as the equation to satisfy the disk's analytic function. It cites how the author's find the appropriate calculation in determining the mentioned disk, along with the formulas. Meanwhile, it denotes the similarities between the Banach space of complex valued functions and Hardy Banach space, in terms of ascertaining the finite form. Furthermore, the theorems as well as the computing methods are also highlighted.