THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|~a

This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the function f(x) =|x|~a(1相似文献   

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α

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.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|α (2＜α＜4) AT EQUIDISTANT NODES

It is a classical result of Bernstein 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(2 ＜α＜ 4) on equidistant nodes in [-1,1] diverges everywhere, except at zero and the end-points.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION IN EQUIDISTANT NODES

It is a classical result of Bernstein 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 this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [- 1,1 ] still diverges every where in the interval except at zero and the end-points.     

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO |x|α

In this paper we present a generalized quantitative version of a result due to M. Revers concerning the exact convergence rate at zero of Lagrange interpolation polynomial to f(x) = |x|α with on equally spaced nodes in [-1, 1].     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α（2〈α〈4）AT EQUIDISTANT NODES

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.     

ON THE ORDER OF APPROXIMATION FOR THE RATIONAL INTERPOLATION TO |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For thespecial case where the interpolation nodes are xi= (i/n)(i= 1,2,…,n;r>0) , it is proved that the exact order of approximation is O(1/n),O(1/nlogn) and O(1/n), respectively, corresponding to O1.     

THE CONVERGENCE ORDER OF THE INTERPOLATION PROCESS BY LAGRANGE POLYNOMIALS

Let f(x) be an arbitrary continuous function on [-1, 1] and letus denote T_n(x)=cos nθ, x=cos θ,T_n(x) is to be known as the first kind of Chebyshev polynomial ofdegree n. The zeros. of T_n(x) are     

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO ｜x｜^α

In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally     

ON THE ORDER OF APPROXIMATION FOR THE RATIONAL INTERPOLATION TO ｜x｜

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are $x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0)$x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0) , it is proved that the exact order of approximation is O( \frac1n ),O( \frac1nlogn ) and O( \frac1n^r )O\left( {\frac{1}{n}} \right),O\left( {\frac{1}{{n\log n}}} \right) and O\left( {\frac{1}{{n^r }}} \right) , respectively, corresponding to 01.     

ON LAGRANGE INTERPOLATION TO |x|α(1 ＜α＜ 2) WITH EQUALLY SPACED NODES

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)..     

ON LAGRANGE INTERPOLATION TO |x|α(1 < α < 2) WITH EQUALLY SPACED NODES

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|α1(1 < α < 2)..     

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.     

COMPARISON OF LAGRANGE INTERPOLATION AND HERMITE－FEJER TYPE INTERPOLATION OF HIGHER ORDER

ＣＯＭＰＡＲＩＳＯＮＯＦＬＡＧＲＡＮＧＥＩＮＴＥＲＰＯＬＡＴＩＯＮＡＮＤＨＥＲＭＩＴＥ－ＦＥＪＥＲＴＹＰＥＩＮＴＥＲＰＯＬＡＴＩＯＮＯＦＨＩＧＨＥＲＯＲＤＥＲＳｈｉＹｉｎｇｇｕａｎｇ（史应光）（ＣｏｍｐｕｔｉｎｇＣｅｎｔｅｒ，ＡｃａｄｅｍｉａＳｉｎｉ...     

ON LAGRANGE INTERPOLATION FOR [X|a (0＜a＜1)

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

APPROXIMATION OF MODIFIED LAGRANGE INTERPOLATION IN ORLICZ SPACES

In this paper, the authors give the Marcinkiewicz-Zygmund inequality based on the zeros of the first kind Chebyshev polynomials in Orlicz norm. As application, the degree of approximation by two kinds of modified Lagrange inter polatory polynomials in Orlicz spaces is studied.     

A NOTE ON LAGRANGE INTERPOLATION OF FUNCTIONS OF GENERALIZED BOUNDED VARIATION

In this paper we solve a remained problem in[2],whether the following estimate approximation for theclass f∈C[-1,1]∩BV by Lagrange interpolation based on the Jacobi abscissasi,L_n~(α,β)(f,x)-f(x)=Ο(1/n)holds,ifα≠β≥-1.     

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 THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.