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1.
The following classes of functions analytic in the unit disk are considered: and where
is the Nevanlinna characteristic and
is a properly changing positive function on (0,1]. Necessary and sufficient conditions on
are established under which the classes
and
are invariant under the operators of differentiation and integration. Bibliography: 7 titles. 相似文献
2.
Let
, and let
be Epstein's zeta-function of the form Q. It is proved that for |t| > C > 0 one has the estimate Bibliography: 9 titles. 相似文献
3.
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
4.
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
5.
Let
be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [ a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators
generated by a multiplier matrix
defined by the following relations: and In the case || = || = 1/2, we prove the following statements for the Jacobi polynomials (these statements are similar to known results for the trigonometrical system). Consider the cases
and Under some conditions on a function , the values
and
equal and In addition, we show that for the Fourier–Legendre summation methods ( = = 0) generated by the multiplier function
, the limit and supremum of the sequence of Lebesgue constants may differ. Bibliography: 11 titles. 相似文献
6.
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
7.
In this note we prove the following theorem: Let u be a
harmonic function in the unit ball
and
. Then there is a
constant C =
C( p,
n) such that
. 相似文献
9.
We prove the existence and the uniqueness of a weak solution to the mixed boundary problem for the elliptic-parabolic equation |
1, \hfill \\ \end{gathered} $$
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with a monotone nondecreasing continuous function b. Such equations arise in the theory of non-Newtonian filtration as well as in the mathematical glaciology. Bibliography: 16 titles. 相似文献
10.
We prove certain local bifurcation results for the mean curvature problem. . This is achieved by applying standard local bifurcation theory. The use of certain equivalent weighted and homogeneous Sobolev spaces was proved to be crucial. 相似文献
11.
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
12.
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
13.
We prove new pointwise inequalities involving the gradient of a function
, the modulus of continuity
of the gradient
, and a certain maximal function
and show that these inequalities are sharp. A simple particular case corresponding to
and
is the Landau type inequality
, where the constant 8/3 is best possible and . 相似文献
14.
Let an n × n Hermitian matrix A be presented in block 2 × 2 form as
, where A 12 ≠ 0, and assume that the diagonal blocks A 11 and A 22 are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds where
and ‖ ⋅ ‖ is the spectral norm. Further, in the positive-definite case, several equivalent conditions necessary and sufficient for both of the above bounds to be attained are provided. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 172–194. 相似文献
15.
This paper discusses the Cauchy problem of the equation
with initial datum a measure. Under the assumption of the parameters, one proves the existence and non-existence of the non-negative
generalized solution.
Member in Partner Group of MPI for Math. in Leipzig at AMSS.
Supported by the Research Fund for Returned Overseas Chinese Scholars 20010107 相似文献
16.
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$$
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, 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.
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 2 n equidistant interpolation points. This assertion implies the sharp Jackson-type inequality . Bibliography: 17 titles. 相似文献
18.
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)),$$
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|
where
. 2. Let F be an arbitrary sufficiently large positive constant. Then for
x_F$$
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, the relation |
F} \right\} \asymp \frac{{\pi (x)}}{F}$$
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holds. 3. The relation holds, where A is Artin's constant. Hence, for the majority of discriminants of the form
, where
, the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of
for the majority of discriminants
0$$
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in the general case. Bibliography: 22 titles. 相似文献
19.
A power series
with radius of convergence equal 1 is called a (p,A)-lacunary one if n k ≥ Ak p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition , where , for some ε > 0, then f ≡ 0. We construct a (p,A)-lacunary series f
0 such that with a constant C 0 = C 0(p,A) > 0. Bibliography: 4 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2003, pp. 135–149. 相似文献
20.
In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form
where n 3 is odd, > 0, > 0.This research was supported by the grants 1/8055/01 and 1/0026/03 of Scientific Grant Agency of Ministry of Education of Slovak Republic and Slovak Academy of Sciences. 相似文献
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