Polynomials orthogonal with sign-sensitive weight

We introduce the notion of inner product with sign-sensitive weight and construct systems of nonsymmetrically orthonormalized polynomials. We also study some properties of such polynomials (for example, the properties of Fourier coefficients, quadrature formulas of Gauss type, etc.).Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 737–752, May, 1996.     

Rational approximation with sign-sensitive weight

For metrics of sign-sensitive weight, a generalization of the Jackson theorem and an analog of the Dolzhenko theorem on the estimate of the variation of a function in terms of its least rational deviations are obtained. In the same metric, two-sided estimates for rational deviations of the function signx are given.Translated fromMatematickeskie Zametki, Vol. 60, No. 5, pp. 715–725, November, 1996.     

Polynomial approximations of continuous functions with unbounded sign-sensitive weight

The conditions of arbitrarily exact polynomial approximation of functions continuous on a closed interval with respect to an unbounded sign-sensitive weight are obtained by using divided differences with multiple nodes. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Completeness in quasi-uniform spaces

We introduce a theory of completeness (the π-completeness) for quasi-uniform spaces which extends the theories of bicompleteness and half-completeness and prove that every quasi-uniform space has a π-completion. This theory is based on a new notion of a Cauchy pair of nets which makes use of couples of nets. We call them cuts of nets and our inspiration is due to the construction of the τ-cut on a quasi-uniform space (cf. [1], [20]). This new version of completeness coincides with bicompletion, half-completion and D-completion in extended subclasses of the class of quasi-uniform spaces.     

Completeness of systems of entire functions in spaces of holomorphic functions

Letf be an entire function in . For a broad class of distribution densities of the set Λ, a scale of sufficient conditions for the completeness of the system of functions {f(λ×z):λ∈Λ},z∈E, where , in the spaceH(E) of holomorphic functions onE with respect to the topology of uniform convergence on compact subsets is given in terms of the mutual indicator of the functionf and the setE. These conditions are new already forn=1 even ifE is a disk. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 603–616, October, 1999.     

Completeness of root vectors for a Douglis-Nirenberg elliptic problem with an indefinite weight in Lp spaces

We establish the completeness of root vectors for a non-selfadjoint Douglis-Nirenberg elliptic problem with an indefinite weight matrix in appropriate L_p Sobolev spaces.     

On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.     

Completeness in generalized ultrametric spaces

We analyze the relationship between four notions of completeness for Γ-ultrametric spaces. The notions we consider include Cauchy completeness, strong Cauchy completeness, spherical completeness and injectivity. In the process we show that the category of Γ-ultrametric spaces is equivalent to the category of flabby separated presheaves on Γ ^op .     

Completeness results in probabilistic metric spaces

In this paper, we consider complete probabilistic metrizable (topological) space and prove that any G_δ set in a complete probabilistic metric spaces is a topologically complete probabilistic metrizable space.     

Completeness of quasi-normed symmetric operator spaces

We show that (generalized) Calkin correspondence between quasi-normed symmetric sequence spaces and symmetrically quasi-normed ideals of compact operators on an infinite-dimensional Hilbert space preserves completeness. We also establish a semifinite version of this result.     

Completeness in quasi-metric spaces and Ekeland Variational Principle

In this paper we prove a quasi-metric version of Ekeland Variational Principle and study its connections with the completeness properties of the underlying quasi-metric space. The equivalence with Caristi-Kirk?s fixed point theorem and a proof of Clarke?s fixed point theorem for directional contractions within this framework are also considered.     

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ(2m 1, F) over a linear-able field F of characteristic 0 are shown to be complete groups, provided m > 3.     

Sparse exponential systems: Completeness with estimates

According to A. Beurling and H. Landau, if an exponential system {e ^iλt }_λ∈Λ is a frame in L ² on a set S of positive measure, then Λ must satisfy a strong density condition. We replace the frame concept by a weaker condition and prove that if S is a finite union of segments then the result holds. However, for “generic” S, very sparse sequences Λ are admitted. Supported in part by the Israel Science Foundation.     

Compactness in asymmetric normed spaces

A systematic study of precompact and compact subsets on asymmetric normed linear spaces is developed, centering our attention in the case of linear lattices with an asymmetric norm.