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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 2 n 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 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
5.
We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,
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$\partial_t u=\Delta u+F(x,t,u) \quad{\rm in}
\;{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad{\rm in}
\;{\bf R}^N,\quad\quad ({\rm P})$\partial_t u=\Delta u+F(x,t,u) \quad{\rm in}
\;{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad{\rm in}
\;{\bf R}^N,\quad\quad ({\rm P}) 相似文献
6.
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
7.
We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem
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$\left\{{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\right.$\left\{\begin{array}{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\end{array}\right. 相似文献
8.
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
9.
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
10.
Let u and solve the problem where is an open set in
0\} ,n \geqslant 2,H = \Delta - \partial _t \hfill \\ \hfill \\ \end{gathered} $$
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is the heat operator,
denotes the characteristic function of ,
is the unit cylinder in n+1,
, and the first equation is satisfied in the sense of distributions. We obtain the optimal regularity of the function u, i.e., we show that
. Bibliography: 6 titles. 相似文献
11.
Let
be the Fejér kernel, C be the space of contiuous 2π-periodic functions f with the norm
, let
be the Jackson polynomials of the function f, and let
be the Fejér sums of f. The paper presents upper bounds for certain quantities like
which are exact in order for every function f ∈ C. Special attention is paid to the constants occurring in the inequalities obtained. Bibliography: 14 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 90–114. 相似文献
12.
An integral representation for the functional
is obtained. This problem is motivated by equilibria issues in micromagnetics.
相似文献
13.
In this work the authors study the conditions for the existence of diffusion equations in the cylinder Q = 3 D ×
+,
n
, satisfying the homogeneous Dirichlet or Neumann conditions on the side boundary of the cylinder Q and decreasing with respect to t as a power for t . 相似文献
14.
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
15.
In this paper, we prove the estimate , for every δ ∈ (0, ℓ N), where C = C( N) is a positive constant depending only on N and
. We show that the constant ℓ N in this estimate is optimal. We also present a class of maps from
into
, strictly larger than
, on which we can define the notion of degree and for which the previous inequality still holds. 相似文献
16.
We study the following system of Maxwell-Schrödinger equations $ \Delta u - u - \delta u \psi+ f(u)=0, \quad \Delta \psi + u^2 = 0 \mbox{in} {\mathbb R}^N , u, \;\psi > 0, \quad u, \;\psi \to 0 \ \mbox{as} \ |x| \to + \infty, $ where δ > 0, u, ψ : $\psi: {\mathbb R}^N \to {\mathbb R}We study the following system of Maxwell-Schr?dinger equations
where δ > 0, u, ψ :
, f :
, N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δK > 0 such that, for 0 < δ < δK, the system has a solution (uδ, ψδ) with the property that uδ has K spikes centered at the points
. Furthermore, setting
, then, as δ → 0,
approaches an optimal configuration for the following maximization problem:
Subject class: Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40 相似文献
17.
Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2, ℤ) and let denote the symmetric square L-function of f. A Voronoi type formula for and the relation are proved. Heuristic approaches to estimation of exponential sums arising in this connection are considered. Bibliography:
9 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 247–256. 相似文献
18.
In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation under which its solution satisfying the initial condition stabilizes to zero, i.e., there exists the limit uniform in x from every compact set K in ℝ N for any function u
0( x) belonging to a certain uniqueness class of the problem considered and growing not rapidly than
with a > 0 and b < 0 at infinity.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 79–97, 2006. 相似文献
19.
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
20.
We are interested in parabolic problems with L1 data of the type
with i, j=0, 1, ( i, j) (0, 0), 0 = 0 and 1 = 1. Here, is an open bounded subset of
with regular boundary and
is a Caratheodory function satisfying the classical Leray-Lions conditions and is a monotone graph in
with closed domain and such that
We study these evolution problems from the point of view of semi-group theory, then we identify the generalized solution of the associated Cauchy problem with the entropy solution of
in the usual sense introduced in [5]. 相似文献
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