Linear methods for approximation of some classes of holomorphic functions from the Bergman space

We construct a linear method {ie910-01} for the approximation (in the unit disk) of classes of holomorphic functions {ie910-02} that are the Hadamard convolutions of the unit balls of the Bergman space A _p with reproducing kernels {ie910-03}. We give conditions for ψ under which the method {ie910-04} approximates the class {ie910-05} in the metrics of the Hardy space H _s and the Bergman space A _s , 1 ≤ s ≤ p, with an error that coincides in order with the value of the best approximation by algebraic polynomials. Translated from in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 783–795, June, 2008.     

A point source ofSH wave in a medium with vertical boundary in the case where the velocity of wave propagation is a smoothly increasing function of depth

By the method of separation of variables, the solution of the equation that describes propagation of the SH waves in the case of a point source is constructed in the half-plane y≤0, The coefficients of this equation are piecewise constant functions. The asymptotics of the solution as w→+∞ is found. Bibliography: 6 titles. Translated fromapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 44–55. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by T. N. Surkova.     

Solvability of nonlinear equations in a cone of a banach space

The solvability conditions for the equation Tu+F(u)=0 are found in the case where the operator [T+F′(u)]⁻¹ exists only for u∈K, where K is a cone in a Banach space X. An application concerning the solvability of boundary-value problems for systems of second-order differential equations is provided. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 225–230. Translated by L. Yu. Kolotilina.     

On the coincidence of the canonical embeddings of a metric space into a Banach space

Recall the two classical canonical isometric embeddings of a finite metric space X into a Banach space. That is, the Hausdorff–Kuratowsky embedding x → ρ(x, ⋅) into the space of continuous functions on X with the max-norm, and the Kantorovich–Rubinshtein embedding x → δ _x (where δ _x , is the δ-measure concentrated at x) with the transportation norm. We prove that these embeddings are not equivalent if |X| > 4. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 153–161.     

Inversion of a theorem on action of analytic functions and multiplicative properties of some subclasses of the hardy space H∞

In this paper, function spaces V∩l _A ^p (w) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values of a function in a disk (for example, C_A,Lip _Aα). We denote by l _A ^p (w) the space of power series such that their Taylor coefficients are p-summable with weight w. For an analytic function Φ acting in a space of this type, we prove the following alternative: either Φ″(z)≡0, or the space is a Banach algebra with respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embeddingmult(V∩l _A ^p (w))↪multl _A ^p . An estimate for the l^p-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by Salem-Zygmund. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 50–72. Translated by S. Shimorin.     

Convergence of a sequence of weakly regular set functions

The present paper is devoted to generalizations of the Dieudonné theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonné theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary σ-topological space, and take values in a uniform space. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 103–110, July, 1997. Translated by O. V. Sipacheva     

Nonsingular transformations of a class of tempered Lévy processes

Let ξ(t), t ∈ [0, 1], be a tempered stable process. By {ie4447-01} we denote the law of ξ in the Skorokhod space {ie4447-02}. For the measure {ie4447-03}, we construct the group of nonsingular transformations of {ie4447-04}. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 38–53.     

Intrinsic normalizations of a hyperplane distribution on the Grassmann manifold. I

We consider intrinsic normalizations of a hyperplane distribution on the Grassmann manifold of ann-dimensional projective space. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 257–273, April–June, 1999. Translated by V. Mackevičius     

On a multidimensional version of the tomographic problem

The main result: If the dimension of a Euclidean space with measure is high enough, only the first observation from a long series of observations with random independent choice of projection gives essential information on the measure. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 295–301. Translated by V. Sudakov.     

Extensions of dynamical systems and the martingale approximation method

Let T be a measure-preserving transformation of a probability space (X, F, μ) and let A be the generator of a μ-symmetric Markov process with state space X. Under the assumption that A is an “eigenvector” for T an extension of T is constructed in terms of A. By means of this extension a version of the central limit theorem is proved via approximation by martingales. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 10–19. Translated by V. Sudakov.     

Wavelet decompositions on a manifold

A general method for constructing chains of embedded spline spaces on a smooth (not necessarily compact) manifold is suggested. A wavelet decomposition is obtained for the case of an arbitrary vector space. The results are illustrated by constructing the wavelet decompositon of a chain of embedded spaces of the B _φ-splines of zero order on a smooth manifold. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 26–38.     

The nonstationary heat-conduction problem for a heat-sensitive space with a spherical cavity

We propose a method of finding closed-form solutions of nonstationary heat-conduction problems for heat-sensitive spaces with simple nonlinearity. The approach is illustrated using the example of a space with a spherical cavity. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 100–104.     

Singularity and absolute continuity of measures associated with the rotation of a circle

The problem of whether the infinite convolution of certain discrete distributions naturally associated with the rotation of a circle through an irrational angle of α is singular or absolutely continuous for different values of α is studied. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 323–336. This paper is partially supported by the Russian Foundation for Basic Research, grant 94-01-00921. Translated by N. A. Sidorov.     

Local description of homogeneous real hypersurfaces of the two-dimensional complex space in terms of their normal equations

In the paper, a classification of real hypersurfaces of the space ℂ² that admit transitive actions of local Lie groups of holomorphic transformations is constructed. Any nonspherical Levi nondegenerate homogeneous surface is determined by the triple of real coefficientsN ₅₂₀ ² ,N ₄₄₀, ImN ₄₂₁ of a Moser normal equation. All such surfaces are described by several quadratic curves in the space of above coefficcients. This work was partially supported by RFBR grant 96-01-01002. Voronezh State Academy of Architecture and Civil Engineering. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 33–42, April–June, 2000. Translated by A. V. Loboda     

Lattice of subalgebras of the ring of continuous functions and Hewitt spaces

The latticeA(X) of all possible subalgebras of the ring of all continuous ℝ-valued functions defined on an ℝ-separated spaceX is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line ℝ. The main achievement of the paper is the proof of the fact that any Hewitt spaceX is determined by the latticeA(X). An original technique of minimal and maximal subalgebras is applied. It is shown that the latticeA(X) is regular if and only ifX contains at most two points. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 687–693, November, 1997. Translated by A. I. Shtern     

The asymptotic expansion of the solution to the Cauchy problem for a parabolic equation in a perforated space

We consider the Cauchy problem for a second-order parabolic equation with rapidly oscillating coefficients of the forma(ɛ ⁻ x,ɛ ^−2k t) in a perforated space ℝ _ɛ ⁿ . We construct a complete asymptotic expansion for the solution of the said problem and obtain an estimate for the remainder term in this expansion. Bibliography: 8 titles. To my dear Teacher, Olga Arsenievna Oleinik This research was supported in part by Grant MIE000 from the International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 19, pp. 000-000. 0000.     

Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L ²-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝⁿ, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.