The application of the area principle to the Bieberbach — Eilenberg functions

A series of inequalities are obtained for the logarithmic areas of functions of the form and univalent the circle ¦z¦<1, and also for the Taylor coefficients of the Bieberbach-Eilenberg functions.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 609–618, June, 1972.     

Conditions for the self-adjointness of a quasi-elliptic operator

We prove the following theorem for the operator considered in L₂(Rⁿ) (the m_k are natural numbers): If for sufficiently large |x|, then L is a self-adjoint operator.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 709–716, November, 1976.In conclusion I wish to thank R. S. Ismagilov for useful advice.     

A bound for the coefficient c4 for one-sheeted functions in terms of ¦c2¦

In the class S of functions which are regular and single-sheeted in the circle ¦z¦ < 1, the bound for ¦c₄¦ in terms of ¦c₂¦, obtained by Al'fors, is improved. The crudest bound ¦c₄¦ –< 4/15 (11 + ¦c₂¦) is better than that of Al'fors: ¦c₄¦ –< Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 127–130, August, 1972.     

Limits of indeterminacy of sequences obtained from a given sequence using a regular transformation

The problem considered is how there can be a set of weak accumulation points of the subsequences of a sequence obtained from a given sequence by using a regular transformation of the class T(C, C) when the terms of the sequences are elements of a reflexive Banach space. T(C, C) denotes the class of complex regular matrices (c_mn) (c_mn=a _mn+ib_mn, wherea _mn and b_mn are real numbers) satisfying the conditions and Translated from Matematicheskie Zametki, Vol. 16, No. 6, pp. 887–897, December, 1974.In conclusion the author thanks D. E. Men'shov for his help and interest, and S. B. Stechkin for valuable advice.     

Kernels of sequences of complex numbers and their regular transformations

It is proved that , where U(a, r) is the ball of radius r with center at the pointa, is the smallest closed convex set containing the kernel of any sequence {y_n} obtained from the sequence {x_n} by means of a regular transformation (c_nk) satisfying the condition , where x, x_n, c_nk (n, k=1, 2,...) are complex numbers.Traslated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 815–823, December, 1977.     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

On estimates for the Green's function for a multipoint boundary problem

We obtain new estimates for the Green's function G(t, s) for a boundary problem of the Vallée-Poussin type: under certain hypotheses we prove the existence of non-negative functions g(t), h(t), u(t) such that g(t) h(s)¦G(t, s)¦g(t) and ¦G(t, s)¦u(t) ¦G(, s)¦, where h(t) and u(t) are positive on sets of positive measure. These estimates allow us to apply effectively the methods of the theory of cones to investigate non-linear boundary problems.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 533–540, November, 1968.The author profoundly thanks M. A. Krasnosel'skii and A. Yu. Levin for their valuable advice and interest in this work.     

Noneffectiveness of a class of regular matrices

We show that if a sequence {_j} is such that ₁>_{2 3}..., then for any bounded sequence {S_n} the equation implies the equation . This theorem generalizes a theorem of N. A. Davydov [2].Translated from Matematicheskie Zametki, Vol. 16, No. 3, pp. 361–364, September, 1974.In conclusion the author thanks N. A. Davydov for useful advice in the writing of this paper.     

Коэффициенты рядов Ф урье и класс Липшица

We generalize and sharpen certain results concerning Fourier series from the Lipschitz class. In particular, for sinnx we prove the following: Let ¦b_n¦n^–2L(n) where L(x) is a continuous and slowly oscillating function. Then

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

Optimal error estimates of a locally one-dimensional method for the multidimensional heat equation

For the multidimensional heat equation in a parallelepiped, optimal error estimates inL ₂(Q) are derived. The error is of the order of +¦h¦² for any right-hand sidef L ₂(Q) and any initial function ; for appropriate classes of less regularf andu ₀, the error is of the order of ((+¦h¦² ), 1/2<1.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 185–197, August, 1996.     

(i)-Convergence and its application to a sequence of functions

Let , where A is a directed set containing cofinal chains — a generalized sequence in a complete chain. It is established that every such sequence contains a monotonic cofinal sub-sequence. For a monotonically increasing (decreasing) bounded sequence , by definition, we put . For an arbitrary sequence the (i)-limit is defined as the common (i)-limit of its monotonic cofinal sub-sequences. The properties of (i)-convergence and some of its applications to generalized sequences of mappings are discussed.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 809–820, December, 1973.     

Coefficients of univalent functions which assume no pair of values W and −W

In this paper we study the behavior of the coefficients of functions univalent in the disk ¦z¦< 1 and assuming there are no pair of values Wand –W. In particular, we establish the asymptotic behavior of b_n (n); for the coefficients we obtain the estimate ¦b_n¦ < 2.34 exp {l/4n} (n = 2,3, ...) and for each function of the class indicated we prove, subject to a certain condition, the relationship b_n+1¦–¦b_n=O(n^–1/2).Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 3–14, January, 1972.     

The kernel of a sequence obtained from a given sequence with the aid of a regular transformation

The smallest set is found that contains the kernel of a sequence obtained from a sequence of elements {x_n} of a Banach space with the aid of a regular transformation of the class T(C, C). Here T(C, C) is the set of complex matrices (c_nk) (a_nk+ib_nk) satisfying the conditions .Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 707–716, May, 1976.In conclusion the author would like to express his thanks to A. A. Melentsov for his assistance and attention to the work.     

On a nonlinear stochastic integral equation with application to control systems

The aim of the present paper is to study a nonlinear stochastic integral equation of the form

A class of lacunary trigonometric series

It is shown that there exists a sequence of natural numbers {n_k} which does not belong to the class B₂ and which cannot be decomposed into a finite number of lacunary sequences such that: a) if the series converges on a set of positive measure, then the series consisting of the squares of the coefficients converges; b) for each set E of positive measure we can remove from the system a finite number of terms with the result that what is left is a Bessel system in L²(E); and c) if the series converges to zero on a set of positive measure, then each coefficient is zero.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 781–788, December, 1973.In conclusion the author wishes to thank V. F. Emel'yanov for posing the problem and for helping to solve it.     

Convergence of orthogonal series to K+∞

For any sequence {b_n} such that. =, a uniformly bounded system _n(x)} orthonormal on [0, 1], is constructed such that the series diverges to + on some set E [0, 1], 0<mes E<1, for any order of the terms.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 129–136, August, 1970.     

Entropy of stochastic processes homogeneous with respect to a commutative group of transformations

In order to define the entropy of a stochastic field homogeneous with respect to a countable commutative group of transformations G, one fixes a sequence {A_n} of finite subsets of the group G and considers the upper limit of the sequence of mean entropies of the iterates of the decomposition P. i.e., , where ¦A_n¦ is the number of elements in A_n. It is proved that for a fixed stochastic field and all sequences {A_n} satisfying the Folner condition, the limit of the means exists and is unique. If the sequence {A_n} is such that for all stochastic fields invariant under G, the entropy calculated in terms of it is the same as that calculated for a Folner-sequence, then {A_n} satisfies the Folner condition. In the case when G is a -dimensional lattice Z, the Folner condidition coincides with the Van Hove condition.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 447–462, March, 1978.The author thanks D. V. Anosov for valuable advice, and also B. M. Gurevich and A. M. Stepin for helpful discussions concerning this note.     

On linear summation methods of Fourier series

Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ _k и их разностейΔλ _k=λ _k?λ _k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS _m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1.