The Average Errors for Hermite Interpolation on the 1-Fold Integrated Wiener Space

For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space.By this result,it is known that in the sense of information-based complexity,if permissible information functionals are Hermite data,then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.     

ON NEWMAN-TYPE RATIONAL INTERPOLATION TO ｜x｜ AT THE CHEBYSHEV NODES OF THE SECOND KIND

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）     

ON LAGRANGE INTERPOLATION FOR |X|~α (0 < α < 1)

We study the optimal order of approximation for |x|α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.     

GLOBAL SMOOTHNESS PRESERVATION BY BIVARIATE INTERPOLATION OPERATORS

1　IntroductionIn the case when Pn(f,x) represents the univariate interpolation polynomial of Her-mite-Fejér based on Chebyshev nodesof the firstkind or the univariate interpolation polyno-mials of Lagrange based on Chebyshev nodes of the second kind and± 1 ,or the univariaterational Shepard operators,the following result of partial preservation of global smoothnessis proved in[4] :If f∈Lip M(α;[-1 ,1 ] ) ,0 <α≤ 1 ,then there existsβ=β(α) <α and M′>such thatω(Pn(f ) ;h)≤ M′h…     

基于修正的第二类Chebyshev结点的Newman型有理插值函数对|x|的逼近

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.     

ASYMPTOTIC REPRESENTATION FOR REMAINDER OF QUASI-HERMITE-FEJER INTERPOLATION POLYNOMIAL

Let Q_(2n+1)(f,x)be the quasi-Hermite-Fejer interpolation polynomial of functionf(x)∈C_[-1,1]based on the zeros of the Chebyshev polynomial of the second kind U_n(x)=sin((n+l)arccosx)/sin(arc cosx). In this paper, the uniform asymptotic representation for thequantity| Q_(2n+l)(f, x) -f(x) |is given. A similar result for the Hermite-Fejer interpolationpolynomial based on the zeros of the Chebyshev polynomial of the first kind is alsoestablished.     

ON LAGRANGE INTERPOLATION FOR |X|~α (0 &lt; α &lt; 1)

We study the optimal order of approximation for |x|α (0 &lt; α &lt; 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.     

基于切比雪夫节点的一类带显式系数的广义高斯求积公式(英文)

The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the（n-1）st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].     

A Note on a Theorem of J. Szabados

In this note, we establish a companion result to the theorem of J. Szabados on the maximum of fundamental functions of Lagrange interpolation based on Chebyshev nodes.     

球面上的Lagrange插值

In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.     

On the Negative Extremums of Fundamental Functions of Lagrange Interpolation Based on Chebyshev Nodes

In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.     

Approximation to Continuous Functions by a Kind of Interpolation Polynomials

In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function. f(x)∈ C^b[-1,1] (0≤b≤1) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number.     

样条型矩阵有理插值

The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula.     

Approximation of functions on the Sobolev space with a Gaussian measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the L q space for 1≤q ∞, and the Fourier partial summation operators and the Valle-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the L q space for 1≤q ∞.     

APPROXIMATION PROPERTIES OF LAGRANGE INTERPOLATION POLYNOMIAL BASED ON THE ZEROS OF （1- x^2）cosnarccosx

We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.     

On approximation and generalized type of analytic functions of several complex variables

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.     

ON BIRKHOFF INTERPOLATION (0;2) CASE

P. Turan and his associates considered in detail the problem of (0.2) interpolation based on the zeros of πn(x). Motivated by these results and an earlier result of Szabados and Varma[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial Rn (x) satisfying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turan.     

UNIFORM APPROXIMATION OF ENTIRE FUNCTIONS OF SLOW GROWTH ON COMPACT SETS

In the present paper,we study the polynomial approximation of entire functions of several complex variables.The characterizations of generalized order and generalized type of entire functions of slow growth are obtained in terms of approximation and interpolation errors.     

Efficiency of weak greedy algorithms for m-term approximations

We investigate the efficiency of weak greedy algorithms for m-term expansional approximation with respect to quasi-greedy bases in general Banach spaces.We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm(WTGA) and weak Chebyshev thresholding greedy algorithm.Then we discuss the greedy approximation on some function classes.For some sparse classes induced by uniformly bounded quasi-greedy bases of L_p,1p∞,we show that the WTGA realizes the order of the best m-term approximation.Finally,we compare the efficiency of the weak Chebyshev greedy algorithm(WCGA) with the thresholding greedy algorithm(TGA) when applying them to quasi-greedy bases in L_p,1≤p∞,by establishing the corresponding Lebesgue-type inequalities.It seems that when p2 the WCGA is better than the TGA.     

Block based Newton-like blending osculatory rational interpolation

With Newton’s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the expansive Newton’s polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.