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1.
We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case
and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for
standard information, as well as the equivalence of tractability and strong tractability for the two classes.
April 11, 2001. Final version received: May 29, 2001. 相似文献
2.
Ambroladze and Wallin have posed several problems, about balayage of measures, which arose from work on approximation by
polynomial and rational interpolation in the complex plane. These problems concern the possible coincidence of measures swept
onto a Jordan curve from the inner and outer domains. This paper describes when the desired balayage properties hold. 相似文献
3.
4.
D. S. Lubinsky 《Constructive Approximation》2002,18(2):285-308
Let a≥ 0 , ɛ >0 . We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set Here P is an arbitrary polynomial of degree ≤ n . We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example,
we show that the diagonal \left{ [n/n]\right}
n=1
∞
sequence provides good approximation on almost one-eighth of the circles centre 0 , and the \left{ [2n/n]\right}
n=1
∞
sequence on almost one-quarter of such circles.
July 18, 2000. Date revised: . Date accepted: April 19, 2001. 相似文献
5.
Dedicated to Professor M. J. D. Powell on the occasion
of his sixty-fifth birthday and his retirement.
In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact
support. They are given in closed form, and provide stable, orthogonal decompositions of L
2
(R
2
) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain
all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting
to a box spline of higher polynomial degree. 相似文献
6.
Abstract. Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P
n
, n≥ 0 , be polynomials of respective degrees n=0,1,··· that are orthonormal in G with respect to the area measure (the so-called Bergman polynomials). Let ϕ be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P
n
's, the case when ϕ has a singularity on L . To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we
show that the limiting distribution of the zeros of the P
n
's for such lens domains is supported on a Jordan arc joining the two vertices of G . We determine this arc along with the distribution function. 相似文献
7.
We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given. 相似文献
8.
It has been known for a long time that any real sequence y
1
, . . . ,y
n-1
is the sequence of critical values of some real polynomial. Here we show that any complex sequence w
1
, . . . ,w
n-1
is the sequence of critical values of some complex polynomial. 相似文献
9.
Abstract. We show that on the curves γ:=(x,e
t(x)
) , x∈ [a,b] , where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials P
N
(x,y) of degree at most N in each variable x,y: ||(P
N
(x,e
t(x)
))'||
[a,b]
≤ CN
4
||P
N
||
γ
, and the exponent four is sharp. On the other hand, the corresponding tangential Markov factors on curves y=x
α
with irrational α grow exponentially in the degree of the polynomials. 相似文献
10.
We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex
curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex
domain. 相似文献
11.
This paper is concerned with a family of nonstationary, interpolatory subdivision schemes that have the capability of reproducing
functions in a finite-dimensional subspace of exponential polynomials. We give conditions for the existence and uniqueness
of such schemes, and analyze their convergence and smoothness. It is shown that the refinement rules of an even-order exponentials
reproducing scheme converge to the Dubuc—Deslauriers interpolatory scheme of the same order, and that both schemes have the
same smoothness. Unlike the stationary case, the application of a nonstationary scheme requires the computation of a different
rule for each refinement level. We show that the rules of an exponentials reproducing scheme can be efficiently derived by
means of an auxiliary orthogonal scheme , using only linear operations. The orthogonal schemes are also very useful tools in fitting an appropriate space of exponential
polynomials to a given data sequence. 相似文献
12.
In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer. 相似文献
13.
Abstract. Let K be a compact subset of {\bf C}, and let c denote logarithmic capacity. We prove that if and only if K is countable. As an application, we obtain a short proof of the scarcity theorem for countable analytic multifunctions.
Received: 13 November 2000 / Published online: 18 January 2002 相似文献
14.
Given a pair (G,W) of an open bounded set G in the complex plane and a weight function W(z) which is analytic and different from zero in G , we consider the problem of the locally uniform approximation of any function f(z) , which is analytic in G , by weighted polynomials of the form {W
n
(z)P
n
(z) }
$\infinity$
n=0
, where deg Pn
n. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider
a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations.
May 1, 1996. Date revised: October 8, 1996. 相似文献
15.
S. Mabizela 《Constructive Approximation》1998,14(2):301-310
We examine the existence of continuous selections for the parametric projection onto weak Chebyshev subspaces. In particular, we show that if is the class of polynomial splines of degree n with the k fixed knots then the parametric projection admits a continuous selection if and only if the number of knots does not exceed the degree of splines plus one.
February 15, 1996. Date revised: September 16, 1996. 相似文献
16.
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
17.
Upper and lower estimates for the Bergman kernel and the Bergman metric in
dimension one are given in terms of the logarithmic capacity; some of their applications are added.The second named author started the work on the paper while he was a guest at the Max Planck Institut für Mathematik in Bonn, Germany.The research was partially supported by the KBN grant No. 5 P03A 033 21 and by the Niedersächsisches
Ministerium für Wissenschaften und Kunst, AZ. 15.3-50 113(55) PL. 相似文献
18.
An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the
control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H
1
0
(Ω) so that the state is close to the desired profile while the H
1
(Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal
pairs are established.
Accepted 11 September 1996 相似文献
19.
M. Reimer 《Constructive Approximation》2002,18(2):183-204
Hyperinterpolation on the sphere, as introduced by Sloan in 1995, is a constructive approximation method which is favorable
in comparison with interpolation, but still lacking in pointwise convergence in the uniform norm. For this reason we combine
the idea of hyperinterpolation and of summation in a concept of generalized hyperinterpolation. This is no longer projectory,
but convergent if a matrix method A is used which satisfies some assumptions. Especially we study A partial sums which are defined by some singular integral used by Newman and Shapiro in 1964 to derive a Jackson-type inequality
on the sphere. We could prove in 1999 that this inequality is realized even by the corresponding discrete operators, which
are generalized hyperinterpolation operators. In view of this result the Newman—Shapiro operators themselves gain new attention.
Actually, in their case, A furnishes second-order approximation, which is best possible for positive operators. As an application we discuss a method
for tomography, which reconstructs smooth and nonsmooth components at their adequate rate of convergence. However, it is an
open question how second-order results can be obtained in the discrete case, this means in generalized hyperinterpolation
itself, if results of this kind are possible at all.
March 9, 2000. Date revised: October 2, 2000. Date accepted: March 8, 2001. 相似文献
20.
In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method),
where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a
scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a
projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that
if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal
face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system
of the inequality constrained VI problem. \keywords{Variational inequality, Projection and contraction method, Predictor-corrector
stepsize, Convergence property.} \amsclass{90C30, 90C33, 65K05.}
Accepted 5 September 2000. Online publication 16 January 2001. 相似文献