共查询到20条相似文献,搜索用时 15 毫秒
1.
Peter M. Gruber 《Monatshefte für Mathematik》2002,135(4):279-304
We estimate the error of asymptotic formulae for volume approximation of sufficiently differentiable convex bodies by circumscribed
convex polytopes as the number of facets tends to infinity. Similar estimates hold for approximation with inscribed and general
polytopes and for vertices instead of facets. Our result is then applied to estimate the minimum isoperimetric quotient of
convex polytopes as the number of facets tends to infinity.
Received 16 July 2001 相似文献
2.
D. Braess 《Constructive Approximation》2001,17(1):147-151
Although Newman's trick has been mainly applied to the approximation of univariate functions, it is also appropriate for
the approximation of multivariate functions that are encountered in connection with Green's functions for elliptic differential
equations. The asymptotics of the real-valued function on a ball in 2-space coincides with that for an approximation problem
in the complex plane. The note contains an open problem.
May 17, 1999. Date revised: October 20, 1999. Date accepted: March 17, 2000. 相似文献
3.
Quasi-interpolation of radial basis functions on finite grids is a very useful strategy in approximation theory and its applications. A notable strongpoint of the strategy is to obtain directly the approximants without the need to solve any linear system of equations. For radial basis functions with Gaussian kernel, there have been more studies on the interpolation and quasi-interpolation on infinite grids. This paper investigates the approximation by quasi-interpolation operators with Gaussian kernel on the compact interval. The approximation errors for two classes of function with compact support sets are estimated. Furthermore, the approximation errors of derivatives of the approximants to the corresponding derivatives of the approximated functions are estimated. Finally, the numerical experiments are presented to confirm the accuracy of the approximations. 相似文献
4.
Temlyakov 《Foundations of Computational Mathematics》2008,3(1):33-107
Abstract. Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements
used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated.
While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical
applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on. The standard
problem in this regard is the problem of m -term approximation where one fixes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is
the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation.
Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in
some given norm (or metric). We are also interested in stable algorithms for finding good or near best approximations using
m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more
complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the
basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation
that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms), and adaptive basis
selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the
other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the
current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation
where a theory is only now emerging. 相似文献
5.
Dietrich Braess 《Aequationes Mathematicae》1975,12(1):80-81
Summary A comparison theorem is derived for Chebyshev approximation by spline functions with free knots. This generalizes a result of Bernstein for approximation by polynomials. 相似文献
6.
G. Mastroianni M. G. Russo W. Themistoclakis 《Integral Equations and Operator Theory》2002,42(1):57-89
The mapping properties of the Cauchy singular integral operator with constant coefficients are studied in couples of spaces equipped with weighted uniform norms. Recently weighted Besov type spaces got more and more interest in approximation theory and, in particular, in the numerical analysis of polynomial approximation methods for Cauchy singular integral equations on an interval. In a scale of pairs of weighted Besov spaces the authors state the boundedness and the invertibility of the Cauchy singular integral operator. Such result was not expected for a long time and it will affect further investigations essentially. The technique of the paper is based on properties of the de la Vallée Poussin operator constructed with respect to some Jacobi polynomials. 相似文献
7.
We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian–Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer–Bernstein polynomial operators. 相似文献
8.
Heiner Gonska Jürgen Prestin Gancho Tachev Ding‐xuan Zhou 《Mathematische Nachrichten》2013,286(4):349-359
In this paper we discuss approximation of continuous functions f on [0, 1] in Hölder norms including simultaneous approximation of derivatives of f. 相似文献
9.
In hyperconvex metric spaces we consider best approximation, invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive. 相似文献
10.
In this paper, the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered. The solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented, and the expression of the solution for the optimal approximation problem is obtained. 相似文献
11.
The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659. 相似文献
12.
We study the linear approximation of utility-based hedging strategies for small number of contingent claims. We show that this approximation is actually a mean-variance hedging strategy under an appropriate choice of a numéraire and a risk-neutral probability. In contrast to previous studies, we work in the general framework of a semimartingale financial model and a utility function defined on the positive real line. 相似文献
13.
Dr. Olof Widlund 《Numerische Mathematik》1976,27(3):327-338
Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration. 相似文献
14.
We study the approximation, Gelfand and Kolmogorov numbers of embeddings in function spaces of Besov and Triebel-Lizorkin type. Our aim here is to provide sharp estimates in several cases left open in the literature and give a complete overview of the known results. We also add some historical remarks. 相似文献
15.
The purpose of this paper is to introduce and to discuss the concept of approximation preserving operators on Banach lattices with a strong unit. We show that every lattice isomorphism is an approximation preserving operator. Also we give a necessary and sufficient condition for uniqueness of the best approximation by closed normal subsets of X+, and show that this condition is characterized by some special operators. 相似文献
16.
The approximation problem considered in the paper is to approximate a continuous multivariate function f(x)=f(x1,…,xd) by sums of two ridge functions in the uniform norm. We give a necessary and sufficient condition for a sum of two ridge functions to be a best approximation to f(x). This main result is next used in a special case to obtain an explicit formula for the approximation error and to construct one best approximation. The problem of well approximation by such sums is also considered. 相似文献
17.
Our topic is the uniform approximation ofx
k
by polynomials of degreen (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann
2 and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p
k,n
, which can be identified with a certain probability. The numberp
k,n
is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn. 相似文献
18.
Summary.
This paper presents a method to recover
exponential accuracy at all points (including at the
discontinuities themselves), from the knowledge
of an approximation to the
interpolation polynomial (or trigonometrical polynomial).
We show that if we are given the collocation point values
(or a highly accurate approximation) at the Gauss
or Gauss-Lobatto points,
we can reconstruct an uniform exponentially convergent
approximation to the function in any sub-interval
of analyticity. The proof covers the cases of Fourier,
Chebyshev, Legendre, and more
general Gegenbauer collocation methods.
A numerical example is also provided.
Received
July 17, 1994 / Revised version received December 12, 1994 相似文献
19.
Scale transformations are common in approximation. In surface approximation from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the approximation near steep gradients implied by the data. In that case, scale transformations can be used to give some control over overshoot when the surface has large variations of its gradient. Conversely, in image analysis, scale transformations are used in preprocessing to enhance some features present on the image or to increase jumps of grey levels before segmentation of the image. In this paper, we establish the convergence of an approximation method which allows some control over the behavior of the approximation. More precisely, we study the convergence of an approximation from a data set
of
, while using scale transformations on the
values before and after classical approximation. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples. 相似文献
20.
C.V. Pao 《Applicable analysis》2013,92(4):269-282
Simple direct proofs of some recent results by Kalla, Conde, and Hubbell for a generalized elliptic type integral [Appl. Anal., 22 (1986), pp. 273-287] are presented. Furthermore, a new single term asymptotic approximation for this function is derived, which is superior to the two term approximation given by these authors 相似文献