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1.
On the Zero-Divergence of Equidistant Lagrange Interpolation 总被引:1,自引:0,他引:1
Michael Revers 《Monatshefte für Mathematik》2000,131(3):215-221
In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials
converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0.
Moreover, we show that the rate of divergence attains almost the maximal possible rate.
(Received 2 February 2000) 相似文献
2.
S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant
nodes in [−1,1], In 2000, M. Rever generalized S.M. Lozinskii’s result to |x|α(0≤α≤1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α(1<α<2). 相似文献
3.
Zhikang Lu Xifang Ge 《分析论及其应用》2005,21(4):385-394
This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f(z) =|x|^α(1〈α〈2) on [-1,1] can diverge everywhere in the interval except at zero and the end-points. 相似文献
4.
Michael Revers 《Journal of Approximation Theory》2000,103(2):385
In 1918 S. N. Bernstein published the surprising result that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|α (0<α1) on equidistant nodes in [−1, 1] diverges everywhere in the interval except at zero and the end-points. 相似文献
5.
Petar Petrov 《Monatshefte für Mathematik》2006,147(2):165-171
In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials
of fixed degree n with n real zeros in [−1, 1]. 相似文献
6.
Petar Petrov 《Monatshefte für Mathematik》2006,151(1):165-171
In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials
of fixed degree n with n real zeros in [−1, 1]. 相似文献
7.
Hui Su Shusheng Xu 《分析论及其应用》2006,22(2):146-154
It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to |x| at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (2 〈 α 〈 4) on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points. 相似文献
8.
In a previous paper [2] we studied the zeros of hypergeometric polynomials F(−n, b; 2b; z), where b is a real parameter. Making connections with ultraspherical polynomials, we showed that for b > − 1/2 all zeros of F(−n, b; 2b; z) lie on the circle |z − 1| = 1, while for b < 1 − n all zeros are real and greater than 1. Our purpose now is to describe the trajectories of the zeros as b descends below the critical value − 1/2 to 1 − n. The results have counterparts for ultraspherical polynomials and may be said to “explain” the classical formulas of Hilbert
and Klein for the number of zeros of Jacobi polynomials in various intervals of the real axis. These applications and others
are discussed in a further paper [3]. 相似文献
9.
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
10.
V. D. Zalizko 《Ukrainian Mathematical Journal》2007,59(1):28-44
The Jackson inequality
relates the value of the best uniform approximation E
n
(f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω
3(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity
on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials
coconvex to them.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007. 相似文献
11.
F. Peherstorfer 《Constructive Approximation》1997,13(2):261-269
We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect
to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1]. 相似文献
12.
Xu Shusheng 《分析论及其应用》1989,5(1):33-45
We denote En(f) and E
k
n
(f) the best uniform approximations to a continuous function f defined on [a,b] by the sets of algebraic polynomials of degree
≤n and algebraic polynomials of degree ≤n with the coefficients of xk (k≤n) being zero. In this paper, in cases of r<k and r≥k while [a, b]=[−1,1] (or r<k,k≤r<2k and r>2k while [a,b]=[0, 1]),
we separately discuss the condtions for r-times continuously differentiable function f which enables
. 相似文献
13.
A. Kwiatkowska 《Acta Mathematica Hungarica》2008,121(3):229-242
We give necessary and sufficient conditions for a function f: [0, 1] → {1,2,...,w, c} under which there exists a continuous function F: [0, 1] → [0, 1] such that for every y ɛ [0, 1], |F
−1 (y)| = f(y).
相似文献
14.
We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best
uniform polynomial approximation of |x|
p
on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization.
Also, we solve a problem on asymptotics of the best approximation of sgn(x) on [−1,−a]∪[a,1] by Laurent polynomials.
相似文献
15.
Constance van Eeden 《Annals of the Institute of Statistical Mathematics》1985,37(1):461-472
Summary Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sampleX
1, …,X
n, were obtained by Rosenblatt [4] and Epanechnikov [1] for the case when the densityf and its derivativef′ are continuous. They found, under certain additional regularity conditions, that the optimal choiceh
n0 for the scale factorh
n=Kn−α is given byh
n0=K0n−1/5 withK
0 depending onf and the kernel; they also showed that MISE(h
n0)=O(n−4/5) and Epanechnikov [1] found the optimal kernel.
In this paper we investigate the robustness of these results to departures from the assumptions concerning the smoothness
of the density function. In particular it is shown, under certain regularity conditions, that whenf is continuous but its derivativef′ is not, the optimal value of α in the scale factor becomes 1/4 and MISE(h
n0)=O(n−3/4); for the case whenf is not continuous the optimal value of α becomes 1/2 and MISE(h
n0)=O(n−1/2). For this last case the optimal kernel is shown to be the double exponential density.
Supported by the Natural Sciences and Engineering Research Council of Canada under Grant Nr. A 3114 and by the Gouvernement
du Québec, Programme de formation de chercheurs et d'action concertée. 相似文献
16.
A. A. Privalov 《Mathematical Notes》1976,20(2):679-685
Let >–1 and > –1. Then a function f(x), continuous on the segment [–1; 1], exists such that the sequence of Lagrange interpolation polynomials constructed from the roots of Jacobi polynomials diverges almost everywhere on [–1; 1], and, at the same time, the Fourier-Jacobi series of function f(x) converges uniformly to f(x) on any segment [a; b] (1; 1).Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 215–226, August, 1976. 相似文献
17.
We describe polynomials of best uniform approximation to sgn(x) on the union of two intervals [−A,−1] ⊂ [1, B] in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error in this approximation. 相似文献
18.
YU JIARONG 《高校应用数学学报(英文版)》1995,10(4):361-366
WEIGHTEDAPPROXIMATIONOFRANDOMFUNCTIONSYUJIARONGAbstract:Let(Ω,A,P)beaprobabilityspace,X(t,ω)arandomfunctioncontinuousinprobab... 相似文献
19.
Sun Xiehua 《分析论及其应用》1993,9(4):1-8
In this paper we solve a remained problem in [2], whether the following estimate approximation for the classf∈[-1, 1]∩BV by Lagrange interpolation based on the Jacobi abscissas: L
n
(a,d)
(f,x)−f(x)=O(1/n) holds, if α≠β α,β≥−1.
The project is supported by the Natural Science Foundation of Zhejiang Province. 相似文献
20.
TianLiang Tu 《中国科学A辑(英文版)》2009,52(3):493-506
Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions
of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) ε C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order
of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation
to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are
obtained.
相似文献