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We study some problems of imbedding of the sets of ψ-integrals of the functions f $$ \epsilon $$ L and determine the orders of approximations of functions from these sets by Fourier's sums. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the inverse problem of determination of the coefficient of a minor term of a semilinear parabolic equation in the presence of a nonlinear boundary condition. The additional condition is given in the nonlocal integral form. A uniqueness theorem and a 'conditional' stability theorem are proved. [ABSTRACT FROM AUTHOR]

compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function. [ABSTRACT FROM AUTHOR]

We obtain asymptotic equalities for upper bounds of deviations of the Poisson integrals on the class of continuous functions $ C_{\beta, \infty }^\psi $ in the metric of the space C. [ABSTRACT FROM AUTHOR]

We determine the exact values of upper bounds of approximations by biharmonic Poisson integrals on classes of conjugate differentiable functions in uniform and integral metrics. [ABSTRACT FROM AUTHOR]

We prove the existence of nontrivial functions in ℝn, n ≥ 2, with vanishing integrals over balls of fixed radius and given majorant of growth. [ABSTRACT FROM AUTHOR]

Under certain conditions on continuous functions μ, λ, a, and f, we prove the inequality and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence. [ABSTRACT FROM AUTHOR]

Let G ∈ H (ℂ+), where H (ℂ+) is the class of functions analytic in the half plane ℂ+ = { z: Re z > 0} and such that . In the case where a singular boundary function G is identically constant and G( z) ≠ 0 for all z ∈, ℂ+, we establish conditions equivalent to the condition $$G(z)\exp \left\{ {\frac{{2\sigma }}{\pi }zlnz - cz} \right\} \notin H^p (\mathbb{C}_ + )$$ , where H p (ℂ+) is the Hardy space, in terms of the behavior of G on the real semiaxis and on the imaginary axis. [ABSTRACT FROM AUTHOR]

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions. [ABSTRACT FROM AUTHOR]

For sequences of functions from a Sobolev space satisfying special integral estimates, we, in one case, establish a lemma on the choice of pointwise convergent subsequences and, in a different case, prove a theorem on convergence of the corresponding sequences of generalized derivatives in measure. These results are applied to the problem of existence of the entropy solutions of nonlinear equations with degenerate coercivity and L 1-data. [ABSTRACT FROM AUTHOR]

We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of $${\bar \psi }$$ -integrals of functions of many variables. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the nth-order derivatives of a singular Cauchy integral and for the boundary values of the nth-order derivatives of a Cauchy-type integral are obtained. [ABSTRACT FROM AUTHOR]

We study the problem of the Baire classification of integrals g ( y) = ( If)( y) = ∫ X f( x, y) dμ( x), where y is a parameter that belongs to a topological space Y and f are separately continuous functions or functions similar to them. For a given function g, we consider the inverse problem of constructing a function f such that g = If. In particular, for compact spaces X and Y and a finite Borel measure μ on X, we prove the following result: In order that there exist a separately continuous function f : X × Y → ℝ such that g = If, it is necessary and sufficient that all restrictions g| Y n of the function g: Y → ℝ be continuous for some closed covering { Y n : n ∈ ℕ} of the space Y. [ABSTRACT FROM AUTHOR]

We find asymptotic formulas for the least upper bounds of the deviations of Fourier operators on classes of functions locally summable on the entire real axis and defined by-integrals. On these classes, we also obtain asymptotic equalities for the upper bounds of functionals that characterize the simultaneous approximation of several functions. [ABSTRACT FROM AUTHOR]

We present a criterion for the weak compactness of continuous measure-valued processes in terms of the weak compactness of families of certain space integrals of these processes. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas for the deviations of Fourier operators on the classes of continuous functionsandin the uniform metric. We also establish asymptotic laws of decrease of functionals characterizing the problem of the simultaneous approximation of-integrals of continuous functions by Fourier operators in the uniform metric. [ABSTRACT FROM AUTHOR]

We consider two-parameter fractional integrals and Weyl, Liouville, and Marchaut derivatives and substantiate some of their properties. We introduce the notion of generalized two-parameter Lebesgue-Stieltjes integral and present its properties and computational formulas for the case of differentiable functions. The main properties of two-parameter fractional integrals and derivatives of Hölder functions are considered. As a separate case, we study generalized two-parameter Lebesgue-Stieltjes integrals for an integrator of bounded variation. We prove that, for Hölder functions, the integrals indicated can be calculated as the limits of integral sums. As an example, generalized two-parameter integrals of fractional Brownian fields are considered. [ABSTRACT FROM AUTHOR]

We establish upper bounds for a group of ?*-deviations of Faber sums on the classes of ?-integrals in a complex plane introduced by Stepanets. [ABSTRACT FROM AUTHOR]

On the classes of Poisson integrals of functions belonging to unit balls in the spaces C and L, we obtain asymptotic equalities for the upper bounds of approximations by de la Vallée Poussin sums in the uniform metric and the integral metric, respectively. [ABSTRACT FROM AUTHOR]

Some comparison results are formulated for systems of differential inequalities with different initial points. [ABSTRACT FROM AUTHOR]