On approximating periodic functions using linear approximation methods

In the paper, a generalization of a known theorem by Hardy and Young is obtained; a formula interrelating the integral of a 2π-periodic function over the period with the integral over the entire axis is established; new approximation characteristics for functions belonging to saturation classes of continuity modules of different orders for the spaces L_p of periodic functions are provided, and some issues concerning approximation, in the uniform metric, of continuous periodic functions even with respect to each of their variables and having nonnegative Fourier coefficients are considered. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 134–164.     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

Strongly regular growth of entire functions of order zero

For entire functions of order zero we introduce a new concept of regularity of growth, which is shown to possess properties similar to those which characterize the concept of totally regular growth of entire functions of finite order in the sense of Levin-Pflüger. Translated fromMatematicheskie Zameiki, Vol. 63, No. 2, pp. 196–208, February, 1998. This research was partially supported by the International Science Foundation under grant No. UCR000.     

On superpositions of continuous functions defined on the baire space

For continuous functions specified on the Baire space, conditions for the representability of a function of several variables as a superposition of functions of a smaller number of variables are considered. With the use of linear functions of the form (1+α)t, a boundary value of the modulus of continuity separating the positive from the negative solution of the problem is found. For the case in which the problem has a negative solution, a constructive method for obtaining (n+1)-variable continuous functions with modulus of continuity ϕ(t) that are not representable as superposition ofn-variable continuous functions with the same modulus of continuity ϕ(t) is suggested. Translated fromMatermaticheskie Zametki, Vol. 66, No. 5, pp. 696–705, November, 1999.     

Entire functions of bernstein's class that are not fourier-stieltjes transforms

We consider certain subclasses of the class of entire functions of exponential type bounded on the real axis. We construct functions that belong to these subclasses but are not Fourier-Stieltjes transforms. Particular attention is given to the distribution of zeros of such functions. The results obtained allow us to study the stability of completeness of systems of exponentials inC andL ^p under small perturbations of the exponents. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 367–380, March, 1997. Translated by M. A. Shishkova     

Approximation of(\bar \psi ,\bar \beta )-differentiable periodic functions of many variables

Best-approximation estimates are obtained in the integral and uniform metric on classes of periodic functions of many variables, which are defined by restrictions imposed on the mixed generalized derivative introduced by Stepanets. In this case, theharmonic of trigonometric polynomials, which are used for approximation of the classes of functions under consideration, are taken from the so-called hyperbolic crosses.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 367–377, March, 1993.     

Growth and approximation of generalized biaxially symmetric potentials in several complex variables

The paper deals with the aspect of the growth of entire functions of several complex variables. The growth of the entire functions with respect to each of the variables separately has been studied by defining partial order and partial type. Finally, we have studied the growth and polynomial approximation of entire function generalized biaxially symmetric potential with respect to each of the variables separately.     

On stability of interconnected finite difference systems

We give a method of construction of Lyapunov functions in the form of a linear form with respect to moduli of variables, for which there exist Krasovskii constants in the case of asymptotic stability, for linear systems with constant coefficients and some types of nonlinear systems of finite-difference equations. An application of the above functions as components of a vector Lyapunov function allowed us to obtain conditions on asymptotic stability for interrelated finite-difference systems.Translated from Dinamicheskie Sistemy, No. 8, pp. 68–71, 1989.     

A boundary-value problem in an unbounded strip

Using the generalized scheme of the method of separation of variables we propose a new approach to the study of a boundary-value problem in an unbounded strip for hyperbolic partial differential equations of even order with respect to time. In the space of entire functions of exponential type (with respect to the spatial variable) we single out a unique solution class for the problem and exhibit an algorithm for constructing a solution of it. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 16–21.     

Intermediate Growth of Solvable Lie Superalgebras

Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponential. Recently the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F(A ^q , k) with a finite number of generators k, and which are free with respect to a fixed solubility length q. Later, an application of generating functions allowed us to obtain more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras; we announce that main results and describe the methods. Our goal is to compute the growth for F(A ^q , m, k), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is obtained in the generality of free polynilpotent Lie superalgebras. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Lower bound for the best approximations of periodic summable functions of two variables and their conjugates in terms of Fourier coefficients

In terms of Fourier coefficients, we establish lower bounds for the sum of norms and the sum of the best approximations by trigonometric polynomials for functions from the space L(Q ²) and functions conjugate to them with respect to each variable and with respect to both variables, provided that these functions are summable. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1042–1050, August, 2008.     

Interphase Hamiltonian and first-order phase transitions: A generalization of the Lee-Yang theorem

We generalize the Pirogov-Sinai theory and prove the results applicable to first-order phase transitions in the case of both bulk and surface phase lattice models. The region of first-order phase transitions is extended with respect to the chemical activities to the entire complex space ℂ^Ф, where Φ is the set of phases in the model. We prove a generalization of the Lee-Yang theorem: as functions of the activities, the partition functions with a stable boundary condition have no zeros in ℂ^Ф. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 98–123, October, 2007.     

Necessary conditions for an extremum in the calculus of variations with nonsmooth phase variables

Necessary conditions for an extremum in a general isoperimetric problem not involving partial derivatives with respect to phase variables are obtained. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 922–930, June, 1997. Translated by N. K. Kulman     

Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros

It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 101–112.     

An estimate of a Dirichlet series with real coefficients on a ray

Best possible conditions are found under which the logarithm of the absolute value of an entire Dirichlet series with real coefficients rarely changing sign increases on the positive ray in the same way as in the whole plane. This result essentially strengthens a similar result due to Pavlov concerning the growth of Taylor series on the positive ray. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 810–823, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00259.     

Rate of growth of hypercyclic entire functions

Let

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be a weighted shift operator on the space of entire functions. Suppose that |a_a| → ∞ monotonically, a particular such operator being the differentiation operator Df = f′.l It is known that T possesses hypercyclic functions, that is, functions f for which is dense in . In this paper a sharp result on the permissible rates of growth to T-hypercyclic entire functions is obtained, and it is shown that for every permissible growth rate there is a dense subspace of T-hypercyclic functions satisfying this growth condition. These results generalise earlier results by the author and by Armitage on the differentiation operator.     

Isometry of functional spaces with different number of variables

We construct spaces of real functions of n + k variables that are isometric to spaces of real functions given on an n-dimensional Euclidean space. We present certain properties and examples of delta-like kernels used for the construction of isometric spaces of functions with different number of variables. We prove certain assertions that enable one to construct delta-like kernels with many variables by using delta-like kernels with smaller number of variables. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1027–1045, August, 1998.     

Condition for the constancy of regression of a polynomial statistic on a sample mean

Conditions are obtained, under which from the constancy of the regression of a polynomial statistic of independent random variables on a sample mean there follows that the characteristic functions of the considered random variables are entire and of finite order. There are given also conditions on a polynomial statistic that are sufficient for the characterization of a normal distribution by the property of the constancy of the regression of this statistic on a sample mean.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 106–114, 1990.     

Polynomiality criterion for entire functions of several complex variables

The “radial” polynomiality criterion for entire functions of several complex variables is proved. Translated fromMatermaticheskie Zametki, Vol. 66, No. 4, pp. 500–502, October, 1999.     

The almost sure behavior of a random sequence

The almost sure behavior of a sequence of random variables is studied. The results obtained are stated in terms of characteristic functions of these variables. No moment conditions are used. Bibliography:5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 252–255.