Transformation of measures generated by jump Lévy processes

Let ξ(t), t ∈ [0, 1], be a jump Lévy process. We denote by the law of ξ in the Skorokhod space [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of -preserving transformations of the space [0, 1]. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 175–188.     

On generalized Beurling algebras

We consider the space, where Ω is a family of weights. We give a sufficient condition, on Ω, for to be locally convex algebra with continuous multiplication. Examples of such weights are also provided.     

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.     

On a uniqueness theorem for functions with a sparse spectrum

We present an example of a set { } satisfying the following two conditions: (1) there exists a nonzero positive singular measure on the unit circle { } with spectrum in Λ; (2) if the spectrum of f∈L¹ { } is contained in Λ and f vanishes on a set of positive measure, then f=0. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 7–14. Translated by A. B. Aleksandrov.     

Invariants of classCk for finite coxeter groups and their representations in terms of anisotropic spaces

This article is devoted to the study of representations of C^k { } functions f invariant with respect to finite Coxeter groups W in the form f=F op, where p is a base in the algebra of W-invariant polynomials. We examine the lowering of smoothness of F as compared with f and conclude that this lowering has an anisotropic nature and that, more precisely, at each point Po it is described by a vector { }. We examine the cases W=A_n, B_n, D_n, { }; in each case the greatest component { } of { } is equal to the Coxeter number of the stabilizer W_yo of the point yo, where po=p(yo). Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 46–70. Translated by S. Yu. Pilyugin.     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

Lacunary series and pseudocontinuations

The main goal of this paper is to prove the following statement. Let be a function holomorphic and of bounded characteristic in the unit disk , where E is a Λ(1)-subset of ℤ₊. Assume that f has a pseudocontinuation of bounded characteristic to the annulus {zɛℂ:1<|z|<R}. Then f admits analytic continuation to the disk R . In particular, f is a polynomial if R=+∞. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 16–32. Translated by A. B. Aleksandrov.     

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

Protuberance effect in the generalized functional Strassen-Révész law

The set of increments of the Wiener process

Isometric embeddings of coinvariant subspaces of the shift operator

Let θ be an inner function. The main aim of this paper is to describe all positive measures on the unit circle such that for all continuous functions f∈H²⊖θH². Bibliography: 8 titles. Translated fromZapiski Nauchnykh Serninarov POMI, Vol. 232, 1996, pp. 5‐15. Translated by V. V. Kapustin.     

Rational operators of the space of formal series

The main result of this paper is the following theorem: the group ring of the universal covering of the group SL(2, ℝ) is embeddable in a skew field with valuation in the sense of Mathiak and the valuation ring is an exceptional chain order in the skew field , i.e., there exists a prime ideal that is not completely prime. In this ring, every divisorial right fractional ideal is principal, and the linearly ordered set of all divisorial fractional right ideals is isomorphic to the real line. This theorem is a consequence of the fact that the universal covering group satisfies sufficient conditions for the embeddability of the group ring of a left ordered group in a skew field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 9–53, 2006.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

The second main theorem for moving targets

The defect relation for holomorphic maps in respect to slowly moving target hyperplanes in is proved. The sharp defect boundn+1 is obtained. Communicated by Yoram Sagher     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

The theory of multi-dimensional polynomial approximation

We consider the problem of polynomial approximation to a real valued functionf defined on a compact set . An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional Jackson type theorem which is stronger than previously known results even for the interval [−1, 1]. A strong multidimensional Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including for 0<q≤∞. We have found that the space of polynomials ℙ on a compact setX induces a semimetric which encapsulates the local structure of ℙ. Any semimetric ρ equivalent to suffices for the rough theory presented here. Many examples of sets and their metrics are presented.     

On the numerical construction of an attractor for the Navier-Stokes system

In this paper the problem of the numerical approximation of the minimal global B-attractor for a semiflow generated by the Navier-Stokes equations in a two-dimensional bounded domain Ω is considered. The method suggested here is based on the formula , where G^N is a sequence of compact subsets of L₂(Ω), . The procedure of constructing G^N is finite and includes the numerical solution of the Navier-Stokes equations by means of the Galerkin method, together with an explicit finite-difference discretization in time. Bibliography: 5 titles. To dear teacher Olga A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 91–97. Translated by V. I. Ochkur.