On the best convergence of trigonometric integrals

ОсНОВНОИ РЕжУльтАт Ё тОИ стАтьИ жАклУЧАЕт сь В слЕДУУЩЕМ. ЕслИ с ФИксИРОВАННыМr=2, 3,..., тО Дль тОгО, ЧтОБы ИМЕлО М ЕстО (*) гДЕ ДОстАтОЧНО ВыпОлНЕН ИЕ слЕДУУЩИх УслОВИИ: 1) ЕслИ $$1< \frac{{n_\nu + 1}}{{n_\nu }} = g_\nu \in N, \nu = 0,1,$$ тО Дль кАжДОгОΝ=0,1,... ИМЕ Ет МЕстО (**) 2) ЕслИ тО (***) $$n_{\nu + 1} \geqslant (2r - 1)n_\nu , \nu = 0,1,...$$ . ЕслИ И тО УслОВИь (**), (***) ьВльУтсь тАкжЕ НЕОБхОДИМыМИ Д ль (*).     

Ergodic properties of integral points on ellipsoids of genus

The ergodic and the intermixing theorems of Yu. V. Linnik (Mat. Sb.,43, No. 2, 257–276) are generalized to arbitrary positive quadratic forms of genus of , where Ω > 1 is an odd number, of invariants [Ω,1], defined by the character for all prime numbers ρ¦Ω. One obtains estimates for the remainder term. The method of proof is simplified.     

Peierls substitution and the Maslov operator method

We consider a periodic Schrödinger operator in a constant magnetic field with vector potential A(x). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: , where is the corresponding energy level of some auxiliary Schrödinger operator, assumed to be nondegenerate, and µ is a small parameter. In the present paper, we use V. P. Maslov’s operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation with the operator represented as a function depending only on the operators of kinetic momenta $ \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) $ .     

Some model nonstationary systems in the theory of non-Newtonian fluids. II

For the nonstationary quasilinear system one proves theorems of (local) existence and uniqueness of the generalized solution with a finite energy integral ; different variants of regularizing systems are constructed for which the generalized solution with a finite energy integral exists globally.     

For which p and is the equation true?

It is explained when the classes of p-absolutely summing and p-integral operators given on the space coincide. For a Banach space there is considered the following subset of the real line: . In the case when is an infinite-dimensional subspace of the space , it is proved that J_X=(1,2] if , and J_X?{2} if and χ is not isomorphic with a Hilbert space.     

Analytic continuation from a continuum to its neighborhood

Let be a positive number. A function f analytic in an open set ? ?? is called -analytic on the set E, E ??, if . THEOREM. Let K be a compact connected subset of the plane. For any >0 there exists an open neighborhood V of the set K such that any function -analytic on K coincides in some neighborhood of the set K with a function analytic in V. This theorem answers a question posed in the collection (RZhMat., 1979, 3B536, pp. 33–35 of the book).     

On form ideals and Artin-Rees condition

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring G_A(α). More precisely, when and and f_i?f'_iεα ^m m?0, ? i , we give a necessary and sufficient condition to have , involving the first syzygies modules both of (f₁,...,f_n) and (f'₁,...,f'_n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f₁,...,f_n is regular and for an integer q, then f₁?f'_i ? α^q+1 ? i implies .     

Bahadur asymptotic efficiency of ω2 -type tests in the many-sample case

Consider independent samples of sizen ₁,n ₂,... ,n ₂ and the empirical distribtion constructed using these samples. We test the hypothesis that all the samples are drawn from a population with the same continuous distribution function F (x). The test statistic is the weight function) is defined as generalizes Kiefer's well-known statistic originally proposed for the case K=2 and q,= 1. A rough asymptotics is obtained for the probability of large deviations of the statistic , which allows to give explicit expressions for the Bahadur local exact slopes and to compare the statistics for various K and q in the sense of Bahadur efficiency. In conclusion, the question posed by Renyi is considered: to what extent is it advisable to use statistics of the type instead of pooling all the samples and using a one-sample test?     

On pointwise estimate for Gamma operators

For linear combinations of Gamma operators, if 0＜a＜2r, 1/2-1/2r≤λ≤l, or 0≤λ＜1/2-1/2r(r≥2),0＜a＜r+10＜a<(r+1)/1-λ, we obtain an equivalent theorem with ωuλρ(f ,t) instead of ωrλφ(f,t), where ωuφ(f,t) is theDitzian-Totik moduli of smoothness.     

Metrische Räume mit Dreiseitverbindbaren

This paper presents a system of axioms for n-dimensional metric geometry. For every group satisfying the axioms there exist a group-space and an embedding of into a projective-metric space Ω. We construct an isomorphism of onto a subgroup of a special orthogonal group O _n+1 ^* (K,f). This group belongs to a metric vector space (V,f) over a field K of characteristic ≠ 2 where dim rad V≦1. The (full) groups o _n+1 ^* (K,f) are models of the system of axioms.     

Modulus of boundary values of analytic functions of class

Let ω be a modulus of continuity, be the class of all functions analytic in the unit disk of the complex plane and such that . A condition is given (depending essentially on ω), necessary for a nonnegative function defined on the unit circle to coincide with the modulus of some function of class .     

Riemann—Lebesgue theorem

Пусть?(x) — ограниченн ая функция на отрезке [0,1] и ее функция распределен ияΦ(t) удовлетворяет услов ию $$\Phi \left( t \right) + \Phi \left( { - t} \right) = 1.$$ Еслиf(x) — конечная поч ти всюду функция, то дл яF _n (t) — функции распределе ния произведенияf(x)?(nx) — вы полнены соотношения и В частности, еслиf(x) — и нтегрируемая функци я, то из (1) следует, что $$\mathop {\lim }\limits_{n \to \infty } \mathop \smallint \limits_0^1 f\left( x \right)\varphi \left( {nx} \right)dx = 0 $$      

Completely- matrices

A matrix is a real square matrixM such that for everyq the linear complementarity problem: Findw andz satisfyingw = q + Mz, w ≥ 0, z ≥ 0, w ^T z = 0, has a solution. We characterize the class of completely- matrices, defined here as the class of -matrices all of whose nonempty principal submatrices are also -matrices.     

Singular strictly increasing functions and a problem on partitions of closed intervals

We establish that the problem of constructing a strictly increasing singular function is equivalent to the problem of constructing subsets and of a closed interval [a; b] ? ? such that (1) ∩ = ø; (2) ∪ = [a; b]; (3) the Lebesgue measures of the intersections of and with an arbitrary interval J ? [a; b] are positive.     

Interpolation by analytic functions smooth along the boundary. II

In this paper one considers the following classes of functions, analytic in the circle of the complex plane: , . One establishes that the well-known Carleson condition is necessary and sufficient for the solvability of the interpolation problems, intrinsic for the given class, on sets lying along a ray. In this connection, the class of sequences to be interpolated is described in terms of divided differences of order S. See also author's paper, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,30, 167–169 (1972).