On the error of the interpolation by bilinear splines

We study the problem of approximation of functions from the classes W^r,s H and W^r,s H ^,2by bilinear splines. For some values ofr ands, we obtain exact estimates of the error.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1554–1560, November, 1994.     

Minimal Morse functions on a pair of manifolds

The existence theorem for a minimal Morse function on a pair of manifolds (M ⁿ,N ^k), wheren-k 3,k 6, is proved.Translated from Ukrainskii Matematiccheskii Zhurnal, Vol. 45, No. 1, pp. 143–144, January, 1993.     

Computation of Feynman integrals of functionals that are functions of quadratic functionals

If the arguments of a function G: ^{d 1} are taken as quadratic functional defined on a space C of continuous functions, we obtain a functional G: C ¹.We give a formula for computing analytic Feynman integrals of such functionals. We also propose a method of approximate computation of sequential Feynman integrals based on replacing the kernel of an integral operator by a degenerate kernel.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 58–61.     

Generalized elements of best approximation

It is proved that generalized elements of best approximation (GEBA's) are approximated by minimizing nets in the(X^**, X^*) topology. Criteria for GEBA's are established. GEBA's are considered in ^* where is a 1-normalizing subspace of X^*.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 513–523, April, 1976.     

Pettis integrability of Stone transforms

Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x ^* fx^*B(X^*)}, whereB(X^*) is the unit ball inX ^*.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996.     

Interpolation in the class of generalized axisymmetric potential functions

We consider the interpolation of anx -harmonic function with an arbitrary real . We investigate the existence and uniqueness of thex -harmonic interpolation function and also convergence of the interpolation process for some particular node placements.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 30–39, 1988.     

On boundary properties of the components of polyharmonic functions

The following two classes of functions are introduced forp0: the classCU ^p (G) of uniformly continuous functions of orderp in a domainG , and the classM ^p (G) of functions of the boundedness of orderp inG. Criterions are established for ann-analytic function to belong to each of these classes.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 821–834, June, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-013766     

A note on Jackson's theorem for differentiable functions

From previously published results of the author on the exact upper bound of best approximations by trigonometric polynomials for classes of periodic differentiable functions are derived the values of the exact constants in Jackson's inequalities for 2-periodic functionsf C^r with modulus of continuity (f ^(r); t) for the r-th derivative which is convex upwards.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 517–522, November, 1972.     

Unique determination of the convolutions of measures in Rm,m⩾2, by their restriction to a set

Restrictions are indicated on a complex-valued measure in the spaceR ^m ,m 2., under which the n-fold convolution ^n*,n 2, is uniquely determined by its values on any semispacex ₁<r, rR.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 86–90, 1988.     

On the volume of certain manifolds with closed geodesics

It is proved that the ratio of the volume of a Riemannian manifold of cohomology typeCP ² orCP ³ for which all geodesics are closed and of length 2 to the volume of the standard sphere assumes a fixed value.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 102–106     

Limit distribution function of inhomogeneities in regions with random boundary. I

A study is made of the interaction of systems of charged particles with a membrane consisting of inhomogeneities randomly distributed in accordance with the same law in the neighborhoods of corresponding sites of a planar crystal lattice. A system of equations for the self-consistent potentialU ₁(x, ⁰,..., ^N ,...) and density of surface changes (x, ⁰,..., ^N ,...) is derived and solved.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 98–112, July, 1992.     

On homogenization of periodic parabolic systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in ^d and prove that the solutions converge in L ₂(^d) to the solution of the homogenized problem for each t > 0. For the L₂(^d)-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L ₂(^d)-norm of the initial value.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 86–90, 2004Original Russian Text Copyright © by T. A. SuslinaSupported by RFBR grant No. 02-01-00798.     

A remark on the essential self-adjointness of a Schrödinger operator with a singular potential

We examine the operators=–+v, v L_{2, loe} (R ⁿ ), where S satisfies a natural additional condition of a local nature. If a condition of Titchmarsh type is fulfilled at infinity, then S is essentially self-adjoint in L₂(Rⁿ).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 571–580, October, 1976.     

Gaussian approximations of Brownian motion in a stochastic integral

Given a Brownian motionB, Gaussian approximationsB , >0, of the form B _t = ₀ ^t K (u, s)dB _s du, t O , including polygonal and mollifier approximations, are considered. A limit theorem is proved for the integrals _T ⁰ X _t dB ^t as O. In particular, in the case of symmetric kernelsK the limit is the Fisk-Stratonovich integral ₀ ^t X _t odB _t.Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Vilnius Pedagogical University, Studentu 39, 2034 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 33, No. 4, pp. 508–526, October–December, 1993. Translated by V. Mackeviius     

Infinite p-groups containing exactly p2 solutions of the equation xP=1

We study arbitrary infinite 2-groups with three involutions and infinite locally finite p-groups (p2), containing p²–1 elements of order p. For odd p the groupG=a, where A is a direct product of two quasicyclic 3-groups ¦b¦=9, b³A, and subgroup A is generated by the elements of the commutator ladder of element b, is a unique infinite non-Abelian locally finite p-group whose equation xP=1 has p² solutions.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 11–18, July, 1976.     

Analytic continuation and superconvergence of series of homogeneous polynomials

LetD be a domain in ⁿ (n1) andx ⁰ D. We prove that a necessary and sufficient condition for the existence of a semicontinuous regular methodA such that the series expansion of any real-analytic functionf inD in homogeneous polynomials aroundx ⁰ is uniformly summed by this method tof(x) on compact subsets ofD is thatD be rectilinearly star-shaped with respect tox ⁰.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 708–714, November, 1996.In conclusion, I wish to my sincere gratitude to my scientific supervisor Prof. Dolzhenko for setting the problem and his attention to my work.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

On the spectrum of the C*-algebra of fourier multipliers in a cone

We describe the spectrum of the C^*-algebra generated by the Fourier multipliers (singular integral operators) on the space L₂ with a polynomial weight in an acute-angled convex cone of the space ⁿ. Bibliography: 12 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 61–82.     

Correctness of the Cauchy problem for trinomial higher-order operator differential equations

Criteria are established for the correctness of the Cauchy problem for the equationsy ⁽²ⁿ⁾+Ay ⁽ⁿ⁾++By=0,t [0, ), wheren > 1;A andB are arbitrary commuting self-adjoint operators in a Hilbert space. Forn=2, the criterion is illustrated by the example of the equation describing the dynamics of an exponentially stratified rotating compressible fluid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 704–714, May, 1993.     

Analytical description of strong extremum in a convex set

Let E, F be normed linear partially ordered spaces and E ⁺,F ⁺ the cones of positive elements in E and F. The mapping G: E F is given and X=G ¹ (F ⁺ ). Necessary and sufficient conditions are derived for x ₀ X to be a strong minimum (the least element). The conditions are applied to establish the existence of optimal correction in an improper linear programming problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 107–112, 1988.