The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.
Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. These operators have been proven to be bounded, and an exponent of boundedness in the space of integrable functions has been found for the case when hypersurfaces are given by parametric equations.
The article considers the behavior of the sum of the Dirichlet series (F(s) = sumlimits_n {kern 1pt} {{a}_{n}}{{e}^{{{{lambda }_{n}}s}}},) (0 < {{lambda }_{n}} uparrow infty ,) which converges absolutely in the left half-plane ({{Pi }_{0}}), on a curve arbitrarily approaching the imaginary axis—the boundary of this half-plane. We have obtained a solution to the following problem: under what additional conditions on (gamma ) will the strengthened asymptotic relation the type of Pólya for the sum F(s) of the Dirichlet series be valid in the case when the argument (s) tends to the imaginary axis along (gamma ) over a sufficiently massive set.
The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular, the Laplace–Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary noncompact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.
We present a solution of the Blaschke problem much simpler than in [1]: find all regular curvilinear three-webs formed by the pencils of circles.
The solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates, is studied. The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in a Sobolev space, with the solvability of this equation being established using the contraction mapping principle.
In this paper we consider a (2 times 2) operator matrix (H). We construct an analog of the well-known Faddeev equation for the eigenvectors of (H) and study some important properties of this equation, related with the number of eigenvalues. In particular, the Birman–Schwinger principle for (H) is proven.
In this work, in an unbounded domain, we prove the correctness of the problem with combined Tricomi and Frankl conditions on one boundary characteristic for one class of mixed-type equations.
Two closely related problems are discussed, viz., solving the Riemann boundary value problem for analytic functions and some of their generalizations in the domains of the complex plane with nonrectifiable boundaries and constructing a generalization of the curvilinear integral onto nonrectifiable curves that preserves the properties important for the complex analysis. This review reflects the current state of the topic, with many of the results being quite recent. At the end of the work, a number of unsolved problems are given, each of which can serve as a starting point for scientific research.
This article is concerned with the study of the inverse problem of determining the difference kernel in a Volterra type integral term function in the third-order Moore–Gibson–Thompson (MGT) equation. First, the initial-boundary value problem is reduced to an equivalent problem. Using the Fourier spectral method, the equivalent problem is reduced to a system of integral equations. The existence and uniqueness of the solution to the integral equations are proved. The obtained solution to the integral equations of Volterra-type is also the unique solution to the equivalent problem. Based on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original inverse problem is proved.
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