共查询到20条相似文献,搜索用时 62 毫秒
1.
D. Dryanov 《Constructive Approximation》2009,30(1):137-153
Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric
spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with
accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L
p
-metric proposed by A.I. Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L
p
-metric is ε
−1/2, ε→0+.
相似文献
2.
Sheng Fan ZHOU Qiu Li JIA Wei SHI 《数学学报(英文版)》2007,23(2):313-320
We obtain an estimate of the upper bound for Kolmogorov's ε-entropy for the bounded sets with small "tail" in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov's ε-entropy of the global attractor for lattice systems. 相似文献
3.
V. N. Belykh 《Siberian Mathematical Journal》2011,52(3):381-393
We calculate the principal term of the asymptotics of the Kolmogorov ɛ-entropy of some compact set of periodic infinitely differentiable functions that is continuously embedded into the space
of continuous periodic functions. 相似文献
4.
We address in this article the following two closely related problems. 1. How to represent functions with singularities (up
to a prescribed accuracy) in a compact way. 2. How to reconstruct such functions from a small number of measurements.
The stress is on a comparison of linear and non-linear approaches. As a model case, we use piecewise-constant functions on
[0,1], in particular, the Heaviside jump function ℋ
t
=χ
[0,t]. Considered as a curve in the Hilbert space L
2([0,1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact
in a context of step-functions in one or two variables.
Next, we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear
methods. Our main tools in this problem are Kolmogorov’s n-width and ε-entropy, as well as Temlyakov’s (N,m)-width.
On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We
discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis.
Finally, we informally discuss two problems in Computer Imaging which are parallel to problems 1 and 2 above: compression
of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer
Tomography), on the other.
This research was supported by the ISF, Grant No. 304/05, and by the Minerva Foundation. 相似文献
5.
Wolfgang Erb 《Journal of Fourier Analysis and Applications》2012,18(1):45-66
For the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space
localization of functions f having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value ε(f). Solving an optimization problem including the functional ε(f), we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these
optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the functional ε(f) to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an
example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained
optimal polynomials can be applied as filters in signal processing. 相似文献
6.
I. Ya. Tyrygin 《Ukrainian Mathematical Journal》1994,46(6):827-831
By the methods of differential pulse-code modulation and “generalized” polygonal lines, we obtain almost exact estimates for
the ɛ-entropy of classes simulating signals of various types. The complexity of coding and reconstruction of functions from
the classes under consideration is investigated. We present a numerical solution of the problem of minimization of constants
in the order-of-magnitude inequality for the ɛ-entropy of the classesKH
0
α
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46,
No. 6, pp. 760–764, June, 1994. 相似文献
7.
Summary We consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeε
γexp[-β/ε]. We estimateβ using the singularities of theε → 0+ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior
of the functions in the complex plane, are supported by our numerical calculations.
Deceased. 相似文献
8.
Fu-qi Yin Sheng-fan Zhou 《应用数学学报(英文版)》2006,22(3):469-486
In this paper, we establish the existence of a global attractor for a coupled κ-dimensional lattice dynamical system governed by a discrete version of the Klein-Gordon-SchrSdinger Equation. An estimate of the upper bound of the Kohnogorov ε-entropy of the global attractor is made by a method of element decomposition and the covering property of a polyhedron by balls of radii ε in a finite dimensional space. Finally, a scheme to approximate the global attractor by the global attractors of finite-dimensional ordinary differential systems is presented . 相似文献
9.
N. V. Parfinovych 《Ukrainian Mathematical Journal》2005,57(10):1652-1662
We determine the exact order of relative widths of classes W
1
r
of periodic functions in the space L
1 as n → ∞ under restrictions on higher derivatives of approximating functions.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1409–1417, October, 2005. 相似文献
10.
T. Q. Son J. J. Strodiot V. H. Nguyen 《Journal of Optimization Theory and Applications》2009,141(2):389-409
In this paper, ε-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective
function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce
the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming
any constraint qualification. Then, we prove that these conditions are also sufficient for ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate
a Wolfe-type dual problem which allows us to present ε-duality theorems and relationships between the KKT conditions and regular ε-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained
convex problem and the semidefinite programming problem. 相似文献
11.
12.
13.
Robert Samuel Simon 《Israel Journal of Mathematics》2006,156(1):285-309
A stochastic game isvalued if for every playerk there is a functionr
k:S→R from the state spaceS to the real numbers such that for every ε>0 there is an ε equilibrium such that with probability at least 1−ε no states is reached where the future expected payoff for any playerk differs fromr
k(s) by more than ε. We call a stochastic gamenormal if the state space is at most countable, there are finitely many players, at every state every player has only finitely many
actions, and the payoffs are uniformly bounded and Borel measurable as functions on the histories of play. We demonstrate
an example of a recursive two-person non-zero-sum normal stochastic game with only three non-absorbing states and limit average
payoffs that is not valued (but does have ε equilibria for every positive ε). In this respect two-person non-zero-sum stochastic
games are very different from their zero-sum varieties. N. Vieille proved that all such non-zero-sum games with finitely many
states have an ε equilibrium for every positive ε, and our example shows that any proof of this result must be qualitatively
different from the existence proofs for zero-sum games. To show that our example is not valued we need that the existence
of ε equilibria for all positive ε implies a “perfection” property. Should there exist a normal stochastic game without an
ε equilibrium for some ε>0, this perfection property may be useful for demonstrating this fact. Furthermore, our example sews
some doubt concerning the existence of ε equilibria for two-person non-zero-sum recursive normal stochastic games with countably
many states.
This research was supported financially by the German Science Foundation (Deutsche Forschungsgemeinschaft) and the Center
for High Performance Computing (Technical University, Dresden). The author thanks Ulrich Krengel and Heinrich Hering for their
support of his habilitation at the University of Goettingen, of which this paper is a part. 相似文献
14.
V. Maiorov 《Advances in Computational Mathematics》2006,25(4):435-450
We consider the manifolds H
n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝd. For example, neural networks and radial basis functions are the manifolds of type H
n(φ). We obtain estimates for pseudo-dimension of the manifold H
n(φ) for wide collection of the generator function φ. The estimates have the order O(d
2
n) in degree scale, that is the order is proportional to number of parameters of the manifold H
n(φ). Moreover the estimates for ɛ-entropy of the manifold H
n(φ) are obtained.
Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10
V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel. 相似文献
15.
We study the existence and asymptotic convergence when t→+∞ for the trajectories generated by
where is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t)→ 0 when t→+∞ . This method is obtained from the following variational characterization of Newton's method:
where H is a real Hilbert space. We find conditions on the approximating family and the parametrization to ensure the norm convergence of the solution trajectories u(t) toward a particular minimizer of f . The asymptotic estimates obtained allow us to study the rate of convergence as well. The results are illustrated through
some applications to barrier and penalty methods for linear programming, and to viscosity methods for an abstract noncoercive
variational problem. Comparisons with the steepest descent method are also provided.
Accepted 5 December 1996 相似文献
16.
A. S. Romanyuk 《Ukrainian Mathematical Journal》1998,50(8):1242-1252
We obtain estimates exact in order for the trigonometric widths of the Besov classes B
p,θr of periodic functions of many variables in the space L
q for 1 ≤ p ≤ 2 < q < p/(p - 1).
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1089–1097, August, 1998. 相似文献
17.
We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different
nonlinear Robin boundary conditions σε(u
ε) + εκ
m
(u
ε) = εg
(m)
ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without
using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained.
Bibliography: 60 titles. Illustrations: 1 figure. 相似文献
18.
Kersten Schmidt Sébastien Tordeux 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,42(1):603-626
We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we
consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets
the models are well defined for any order and have optimal convergence meaning that the H
1-modelling error for an expansion with N terms is bounded by O(ε
N+1) in the exterior of the sheet and by O(ε
N+1/2) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying
curvature validate the theoretical results. 相似文献
19.
The diametral dimension of a nuclear Fréchet spaceE, which satisfies (DN) and (Ω), is related to power series spaces Λ1(ε) and Λ∞(ε) for some exponent sequence ε. It is proved thatE contains a complemented copy of Λ∞(ε) provided the diametral dimensions ofE and Λ∞(ε) are equal and ε is stable. Assuming Λ1(ε) is nuclear, any subspace of Λ1(ε) which satisfies (DN), can be imbedded intoE. Applications of these results to spaces of analytic functions are given.
Support of Turkish Scientific and Technical Research Council is gratefully acknowledged. 相似文献
20.
O. V. Fedunyk 《Ukrainian Mathematical Journal》2006,58(1):103-117
We obtain exact order estimates for the linear widths of the classes B
p,θ
Ω
of periodic functions of many variables in the space L
q
for certain values of the parameters p and q.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 93–104, January, 2006. 相似文献