INTERPOLATION WITH LAGRANGE POLYNOMIALS A SIMPLE PROOF OF MARKOV INEQUALITY AND SOME OF ITS GENERALIZATIONS

The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.

ON SIMULTANEOUS APPROXIMATION BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.

SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).

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.

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|~a

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.

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.

A New Third S.N.Bernstein Interpolation Polynomial

In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number.

Optimization Methods for Box-Constrained Nonlinear Programming Problems Based on Linear Transformation and Lagrange Interpolating Polynomials

In this paper,an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating polynomials.Based on this condition,two new local optimization methods are developed.The solution points obtained by the new local optimization methods can improve the Karush–Kuhn–Tucker(KKT)points in general.Two global optimization methods then are proposed by combining the two new local optimization methods with a filled function method.Some numerical examples are reported to show the effectiveness of the proposed methods.

Generalized and Unified Families of Interpolating Subdivision Schemes

We present generalized and unified families of (2n)-point and (2n − 1)-point p-ary interpolating subdivision schemes originated from Lagrange polynomialfor any integers n ≥ 2 and p ≥ 3. Almost all existing even-point and odd-pointinterpolating schemes of lower and higher arity belong to this family of schemes. Wealso present tensor product version of generalized and unified families of schemes.Moreover error bounds between limit curves and control polygons of schemes arealso calculated. It has been observed that error bounds decrease when complexityof the scheme decrease and vice versa. Furthermore, error bounds decrease withthe increase of arity of the schemes. We also observe that in general the continuityof interpolating scheme do not increase by increasing complexity and arity of thescheme.

Transportation Problem with Multi-choice Cost and Demand and Stochastic Supply

This paper analyzes the multi-choice stochastic transportation problem where the cost coefficients of the objective function and the demand parameters of the constraints follow multi-choice parameters.Assume that the supply parameters of the constraints in a transportation problem(TP)follow logistic distribution.The main objective of this paper is to select an appropriate choice from the multi-choices for the cost coefficients of the objective function and the demand of the constraints in the TP by introducing Lagrange’s interpolating polynomial in such a way that the total cost is minimized and satisfies the required demand.Using stochastic programming,the stochastic supply constraints of the TP are transformed into deterministic constraints.Finally,a non-linear deterministic model is formulated.Using Lingo software,the optimal solution of the proposed problem is derived.To illustrate the methodology,a real-life problem on the TP is considered.

On Ordinary Words of Standard Reed-Solomon Codes over Finite Fields

Reed-Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm.Usually we use the maximum likelihood decoding(MLD)algorithm in the decoding process of Reed-Solomon codes.MLD algorithm relies on determining the error distance of received word.Dür,Guruswami,Wan,Li,Hong,Wu,Yue and Zhu et al.got some results on the error distance.For the Reed-Solomon code C,the received word u is called an ordinary word of C if the error distance d(u,C)=n-deg u(x)with u(x)being the Lagrange interpolation polynomial of u.We introduce a new method of studying the ordinary words.In fact,we make use of the result obtained by Y.C.Xu and S.F.Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed-Solomon codes over the finite field of q elements.This completely answers an open problem raised by Li and Wan in[On the subset sum problem over finite fields,Finite Fields Appl.14(2008)911-929].