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1.
Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then
moreover, for it is impossible to decrease the constants on . Here, are some explicitly constructed constants, is the modulus of continuity of order r for the function f, and are explicitly constructed linear operators with the values in the space of periodic splines of degree of minimal defect with 2n equidistant interpolation points. This assertion implies the sharp Jackson-type inequality
. Bibliography: 17 titles.  相似文献   

2.
The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system
(tan is the tangent component; is a solution) by the response operator ( is the normal). The parameters determine the velocity , the c-metric , and the time . It is shown that for any fixed , the operator determines and in uniquely. Bibliography: 15 titles.  相似文献   

3.
In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,
It is proved that coincides with the smallest eigenvalue of some matrix of order n+1. The main result of the paper states that, for every does not exceed and, for , is equal to the minimum of the quadratic functional
over the unit sphere of . Then it is calculated that Bibliography: 19 titles.  相似文献   

4.
This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])
for a solution to the Hamilton–Jacobi nonlinear partial differential equation
where the Cauchy data are now a function semicontinuous from below, is the usual norm in , , and is a positive evolution parameter. We proved that the Lax formula solves the Cauchy problem (2) at all points , fixed save for an exceptional set of points R of the F type, having zero Lebesgue measure. In addition, we formulate a similar Lax-type formula without proof for a solution to a new nonlinear equation of the Hamilton–Jacobi-type:
where is a diagonal positive-definite matrix, mentioned in Part I and having interesting applications in modern mathematical physics.  相似文献   

5.
We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let be a system of distinct points on and let be the family of all systems of nonoverlapping simply connected domains on such that . Let
where is the reduced module of the domain with respect to the point . At present, the problem concerning the value was solved completely for . In this work, we continue the previous author's investigations and consider the case . In addition, we consider the problem concerning the maximum of the sum
in the family introduced above, where , are arbitrary points of the circle , and is a positive number. We prove that if , then the maximum is attained only for systems of equidistant points of the circle . For , this result was obtained earlier by Dubinin who applied the method of symmetrization. It is shown that if , where is an even number, then equidistant points of the circle do not realize the indicated maximum. Bibliography: 11 titles.  相似文献   

6.
7.
Let be the class number of properly equivalent primitive binary quadratic forms of discriminant . The case of indefinite forms is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let be an arbitrarily slow monotonically increasing function such that . Then
(\log p)^{\alpha (p)} } \right\} = o(\pi (x)),$$ " align="middle" vspace="20%" border="0">
where . 2. Let F be an arbitrary sufficiently large positive constant. Then for x_F$$ " align="middle" border="0"> , the relation
F} \right\} \asymp \frac{{\pi (x)}}{F}$$ " align="middle" vspace="20%" border="0">
holds. 3. The relation
holds, where A is Artin's constant. Hence, for the majority of discriminants of the form , where , the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of for the majority of discriminants 0$$ " align="middle" border="0"> in the general case. Bibliography: 22 titles.  相似文献   

8.
A new numerical inequality for average power means is presented. Let and let be a sequence of positive numbers. Consider the operator . We denote by the superposition of these operators. The following assertion is proved: if . Bibliography: 2 titles.  相似文献   

9.
Let be the Hecke eigenbasis of the space of -cusp forms of weight 2. Let p be a prime. Let be the Hecke L-series of form . The following statements are proved:
and
We also give a correct proof of a previous author's theorem on automorphic L-functions. Bibliography: 12 titles.  相似文献   

10.
The paper deals with the system
where and are -matrix functions; is a boundary control; is the solution. The singularities of the fundamental solution corresponding to the controls ( is the Dirac -function) are under investigation. In the case of , the singularities of the fundamental solution are described in terms of the standard scale . In the presence of points an interesting effect occurs: singularities of intermediate (fractional) orders appear. Bibliography: 1 title.  相似文献   

11.
Let be the space of 2-periodic functions whose (r – 1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L p, S 2n,m is the space of 2-periodic splines of order m of minimal defect over the uniform partition . In this paper, we construct linear operators such that
where
To construct the operators X n,r,m, we use the same idea as in the polynomial case, i.e., the interpolation of Bernoulli kernels. As is proved, the operators X n,r,m converge to polynomial Akhiezer–Krein–Favard operators as . Bibliography: 10 titles.  相似文献   

12.
Let be a sequence of independent equidistributed random vectors with . Let , where and denotes the indicator function of the event in brackets. If, for example, are the gains and are the indicators of success in repetitions of a game of chance, then is the maximal gain along head runs (sequences of successes without interruptions) of length j. We investigate the asymptotic behavior of the values , , where is the length of the longest head run in . We show that the asymptotics of the values depend significantly on the growth rate of j and that these asymptotics vary from the strong noninvariance (as in the ErdsRéenyi law of large numbers) to the strong invariance (as in the CsöorgRévész strong approximation laws). We also consider the Shepp-type statistics. Bibliography: 17 titles.  相似文献   

13.
Let K be a compact space, let X be a closed subspace of C(K), and let be a positive measure on K. The triple is said to be regular if, for any positive function and for any , there exists a function such that on K and . The case where K is the unit sphere in and the subspace X is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for the regularity of a triple are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented. Bibliography: 16 titles.  相似文献   

14.
In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,
,
Theorem 1. Let . Then
For some values of and seminorms related to best approximations by trigonometric polynomials and splines in the uniform and integral metrics, the inequalities are sharp. Bibliography: 6 titles.  相似文献   

15.
Suppose that , , and are three discrete probability distributions related by the equation (E): , where denotes the k-fold convolution of In this paper, we investigate the relation between the asymptotic behaviors of and . It turns out that, for wide classes of sequences and , relation (E) implies that , where is the mean of . The main object of this paper is to discuss the rate of convergence in this result. In our main results, we obtain O-estimates and exact asymptotic estimates for the difference .  相似文献   

16.
Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis   总被引:1,自引:1,他引:0  
Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:
is summable on each finite interval, we establish some assertions similar to the following theorem: Let 0$$ " align="middle" border="0"> ,
Then for fV the series
uniformly converges with respect to and the following equality holds:
This theorem develops some results obtained by Zubov relative to the approximation of probability distributions. Bibliography: 4 titles.  相似文献   

17.
This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.  相似文献   

18.
A matrix is said to be accretive-dissipative if, in its Hermitian decomposition , both matrices B and C are positive definite. Further, if B= I n, then A is called a Buckley matrix. The following extension of the classical Fischer inequality for Hermitian positive-definite matrices is proved. Let be an accretive-dissipative matrix, k and l be the orders of A 11 and A 22, respectively, and let m = min{k,l}. Then For Buckley matrices, the stronger bound is obtained. Bibliography: 5 titles.  相似文献   

19.
20.
A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.  相似文献   

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