共查询到20条相似文献,搜索用时 0 毫秒
1.
Z. Ditzian 《Acta Mathematica Hungarica》2014,143(2):378-421
A new measure of smoothness is defined and related to best approximation by polynomials with respect to weighted L p (R) with Freud-type weights. Other related norms are also discussed. Comparisons with the known measure of smoothness on weighted L p spaces are obtained. Related K-functionals and realization functionals are introduced. The new measure of smoothness allows us to consider a more general class of function spaces, to achieve Marchaud, Jackson and Bernstein-type inequalities, and to relate it to expressions involving the coefficients of the expansion by orthogonal polynomials with respect to Freud-type weights. Some of the results are new for approximation by Hermite polynomials in the weighted L p space with the weight \({e^{-x^{2}}}\) . 相似文献
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Vilmos Totik 《Journal of Approximation Theory》1999,99(2):571
Polynomial approximation by weighted polynomials of the form wn(x) Pn(x) is investigated on closed subsets of the real line. It is known that the possibility of approximation is closely related to the density of an extremal measure associated with w via a weighted energy problem. It is also known that if in a neighborhood of a point x0 this density is continuous and positive, then, in that neighborhood, any continuous function can be approximated. The aim of the present paper is twofold. On the one hand it is shown that the same approximation theorem is true if in a neighborhood of x0 the density is slowly varying and is bounded away from 0. This allows singularities of logarithmic types. On the other hand, we also show that under some mild conditions, if the density at x0 is slowly varying, then approximation is still possible even if the density vanishes at x0 . This is the first positive result for approximation with a vanishing density. 相似文献
3.
Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure
P. C. Simeonov 《Acta Mathematica Hungarica》1999,82(4):265-296
Let w : [0, ) be a weight function on a set R. We assume that the associated extremal measure has density function v(t) with finitely many singularities of logarithmic type. We show that any continuous function f on which vanishes outside the set where v is positive or has a logarithmic singularity, is the uniform limit on of a sequence of weighted polynomials of the form wn Pn, where Pn is of degree n. This extends previous results for continuous densities to densities having logarithmic singularities. 相似文献
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Yu. S. Volkov 《Proceedings of the Steklov Institute of Mathematics》2018,300(1):187-198
We study the general problem of interpolation by polynomial splines and consider the construction of such splines using the coefficients of expansion of a certain derivative in B-splines. We analyze the properties of the obtained systems of equations and estimate the interpolation error. 相似文献
6.
For the approximation in $L_p$-norm, we determine the weakly asymptotic
orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For $p = 1$, $∞$, we obtain its values.
By these results we know that for the Sobolev classes, the approximation errors by
piecewise cubic Hermite interpolation are weakly equivalent to the corresponding
infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional
Kolmogorov widths. 相似文献
7.
The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is: Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C(-∞,∞) satisfy |y|= O(e^(1/2-δo)xk^2,) and |f(x)|t= O(e^(1-εo )x2^). Then for any given point x ∈ R, we have limn→Hn,(f, x) = f(x). 相似文献
8.
将经典“试探函数组”1,x,x^2应用于扩展乘数法,建立了一个判别线性正算子能否改造为逼近任何无界连续函数的充要条件。利用该条件给出了一类变形的插值多项式算子的收敛性定理,得到了具有一般性的结论。 相似文献
9.
The present paper investigates the convergence of Hermite interpolation operators on the real line.The main result is: Given 0 <δ0 < 1/2,0 < ε0 < 1.Let f ∈ C(-∞,∞) satisfy |yk| = O(e(1/2-δ0)x2k) and |f(x)| = O(e(1-ε0)x2).Then for any given point x ∈R,we have limn→∞ Hn(f,x) = f(x). 相似文献
10.
Inequalities of Jackson and Bernstein type are derived for polynomial approximation on simplices with respect to Sobolev norms. Although we cannot use orthogonal polynomials, sharp estimates are obtained from a decomposition into orthogonal subspaces. The formulas reflect the symmetries of simplices, but analogous estimates on rectangles show that we cannot expect rotational invariance of the terms with derivatives. 相似文献
11.
As is well known the kernel of the orthogonal projector onto the polynomials of
degree n in L2(wα,β, [−1, 1]), wα,β(t) = (1 − t)α(1 + t)β, can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling
a compactly supported C∞ function then the resulting function has nearly exponential (faster than any polynomial) rate of decay away from the main
diagonal. This result is used for the construction of tight polynomial frames for L2(wα,β) with elements having almost exponential localization. 相似文献
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Rostam Sabeti 《代数通讯》2013,41(10):4054-4069
Let I ? K[x 1,…, x n ] be an ideal and G be the reduced Gröbner basis of I with respect to lexicographic monomial order. We introduce the index of an expression of f ∈ K[x 1,…, x n ] with respect to G. A minimal expression is characterized as the one with zero G-index. In case where I is a binomial prime ideal, a new division algorithm with minimal and unique expression is presented. The application of our new method on benchmark polynomial systems cyclic-9 and cyclic-12 shows its superiority in comparison with the existing division algorithm. 相似文献
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Ukrainian Mathematical Journal - The title of the article should read: Approximation by Interpolation Trigonometric Polynomials in Metrics of the Space Lp on the Classes of Periodic Entire... 相似文献
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An algorithm for the approximate evaluation of the Hilbert transform has been proposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
17.
This paper considers filtered polynomial approximations on the unit sphere \(\mathbb {S}^d\subset \mathbb {R}^{d+1}\), obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on \([0,\infty )\) such that \(h(t)=1\) for \(t\in [0,1]\) and \(h(t)=0\) for \(t\ge 2\). The resulting “filtered polynomial approximation” (a spherical polynomial of degree \(2L-1\)) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to \(\lfloor \tfrac{d-1}{2}\rfloor \) are absolutely continuous, while its right and left derivatives of order \(\lfloor \tfrac{d+1}{2}\rfloor \) exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space \(W^s_p(\mathbb {S}^d),\ 1\le p\le \infty \), both approximations are of the optimal order \( L^{-s}\), in the first case for \(s>0\) and in the second fully discrete case for \(s>d/p\), conditions which in both cases cannot be weakened. 相似文献
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Vladimir V. Andrievskii 《Constructive Approximation》2012,36(1):117-143
We establish L p ,1??p<?? Markov?CBernstein-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) arc in the complex plane. 相似文献
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Ukrainian Mathematical Journal - We establish asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with equidistant distribution of... 相似文献