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1.
《Quaestiones Mathematicae》2013,36(2):157-165
Abstract

The purpose of this paper is to relate the continuity and selection properties of the one-sided best uniform approximation operator to similar properties of the metric projection. Let M be a closed subspace of C(T) which contains constants. Then the one-sided best uniform approximation operator is Hausdorff continuous (resp. Lipschitz continuous) on C(T) if and only if the metric projection PM is Haudorff continuous (resp. Lipschitz continuous) on C(T). Also, the metric projection PM admits a continuous (resp. Lipschitz continuous) selection if and only if the one-sided best uniform approximation operator admits a continuous (resp. Lipschitz continuous) selection.  相似文献   

2.
Summary Forn=1, 2, 3, ..., let n denote the Lebesgue constant for Lagrange interpolation based on the equidistant nodesx k, n =k, k=0, 1, 2, ...,n. In this paper an asymptotic expansion for log n is obtained, thereby improving a result of A. Schönhage.  相似文献   

3.
Given a function f on a bounded open subset Ω of with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of . This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.  相似文献   

4.
We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.  相似文献   

5.
The paper deals with the approximation of functions f on (0,+), where f can be singular at the origin, by means of Bernstein-type sequences. Error estimates in weighted uniform spaces with some converse results are given.Research was supported in part by grant INDAM-GNIM Progetto Equazioni Integrali e Problemi di Algebra Lineare Connessi.  相似文献   

6.
Let Bn (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, Bn (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {Bn (f, qn; x)} with qn ↓ 1 is not an approximating sequence for fC[0, 1], in contrast to the standard case qn ↓ 1. At the same time, there exists a sequence 0 < δn ↓ 0 such that the condition implies the approximation of f by {Bn (f, qn; x)} for all fC[0, 1]. Received: 15 March 2005  相似文献   

7.
LetX={x 1,x 2,..., n }I=[–1, 1] and . ForfC 1(I) definef* byfp f =f*, wherep f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT n (x) and obtain x m p x m8eE n–1(x m ), whereE n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE n (f) that omits theassumptionf (n+1)>0 in a similar estimate of Meinardus.  相似文献   

8.
The Padua points are a family of points on the square [−1, 1]2 given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The L p convergence of the interpolation polynomials is also studied. S. De Marchi and M. Vianello were supported by the “ex-60%” funds of the University of Padua and by the INdAM GNCS (Italian National Group for Scientific Computing). Y. Xu was partially supported by NSF Grant DMS-0604056.  相似文献   

9.
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   

10.
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   

11.
In 1985, Khan, R. A. established the asymptotic formulas of operators of probabilistic type inL1, space by introducing a newLp-norm. The purpose of this paper is to study the asymptotic rate of these operators, inLp (p>1) spaces. Project supported by the National Natural Science Foundation of China  相似文献   

12.
It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL 1 metric in the optimal order {n –1} if and only if 2 f is of bounded variation where and , respectively.  相似文献   

13.
Summary LetF be an approximating function with parameter setP, a subset ofn-space, on an intervalI satisfying the hypotheses of Meinardus and Schwedt (the local Haar condition and propertyZ) which result in an alternating theory. Consider uniform approximation to functionf onX, a finite subset ofI. A sufficient condition is given, involving best parameters being in a closed set, on which the degree isn that a family of functionsf have a uniform (parameterwise) local strong uniqueness constant > 0. The necessity of the condition in this and related problems, in particular ordinary rational approximation on an interval, is examined.  相似文献   

14.
An asymptotic expansion is obtained which provides upper and lower bounds for the error of the bestL 2 polynomial approximation of degreen forx n+1 on [–1, 1]. Because the expansion proceeds in only even powers of the reciprocal of the large variable, and the error made by truncating the expansion is numerically less than, and has the same sign as the first neglected term, very good bounds can be obtained. Via a result of Phillips, these results can be extended fromx n+1 to anyfC n+1[–1, 1], provided upper and lower bounds for the modulus off (n+1) are available.  相似文献   

15.
Summary. In this paper, we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of Kergin and Hakopian. This formula involves only derivatives of order one higher than the degree of the interpolating polynomial space, and from it we can obtain sharp -estimates. These -estimates are precisely those that numerical analysts want, to guarantee that a scheme based on such an interpolation has the maximum possible order. Received July 11, 1994 / Revised version received February 12, 1996  相似文献   

16.
LetS N k (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst i determined from a fixed functiont by the rulet i=t(i/N). In this paper we introduce sequences of operators {Q N } N =1 fromC k [0, 1] toS N k (t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL 2[0, 1]. Precisely, we show thatN k–1((f–Q N f) i –dist2(f (1),S N k–1 (t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816  相似文献   

17.
In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.  相似文献   

18.
A family of Hermite interpolants by bisection algorithms   总被引:9,自引:0,他引:9  
A two point subdivision scheme with two parameters is proposed to draw curves corresponding to functions that satisfy Hermite conditions on [a, b]. We build two functionsf andf 1 on dyadic numbers and for some values of the parameters,f is in 1 withf 1=f. Examples are provided which show how different the curves can be.  相似文献   

19.
Summary. The main result of this paper is an abstract version of the KowalewskiCiarletWagschal multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding –error bounds. Another application is the construction of a family of error formul? for linear interpolation (indexed by real measures of unit mass) which includes some recently obtained formul?. It is also shown how the problem of constructing an error formula for Lagrange interpolation from a D–invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a ‘simpler’ one–point interpolation map. Received March 29, 1996 / Revised version received November 22, 1996  相似文献   

20.
Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L 2 together with its directional derivatives mentioned above. Moreover, for data sequences inl p ( d ), 1p2, there is a spline function inL p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst  相似文献   

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