SHAPE PRESERVING PIECEWISE CUBIC INTERPOLATION

ＳＨＡＰＥＰＲＥＳＥＲＶＩＮＧＰＩＥＣＥＷＩＳＥＣＵＢＩＣＩＮＴＥＲＰＯＬＡＴＩＯＮＬｕｏＹａｎｇａｎｄＺｈｕＨｕｉｙａｎＡｂｓｔｒａｃｔ．ＴｈｉｓｐａｐｅｒｐｒｅｓｅｎｔｓａＣ１－ｉｎｔｅｒｐｏｌａｔｉｏｎｗｈｉｃｈｐｒｅｓｅｒｖｅｓｃｏｎｖｅｘｉ...     

ON SHAPE PRESERVING EXTENSION AND APPROXIMATION OF FUNCTIONS

We establish the concept of shapes of functions by using partial differential inequalites.Our definition about shapes includes some usual shapes such as convex,subharmonic,etc.,andgives many new shapes of functions.The main results show that the shape preserving approxi-mation has close relation to the shape preserving extension.One of our main results shows thatif f∈C(Ω)has some shape defined by our definition,then f can be uniformly approximatedby polynomials P_n ∈p_n(n∈N)which have the same shape in Ω,and the degree of the ap-proximation is Cω(f,n~(-β))with constants C,β>0.     

THE SHAPE PRESERVING AND HIGHACCURACYAPPROXIMATIONWITHMULTIQUADRIC

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SUPERAPPROXIMATION PROPERTIES OF THE INTERPOLATION OPERATOR OF PROJECTION TYPE AND APPLICATIONS

AbstractSome superapproximation and ultra-approximation properties in function, gradient and two-order derivative approximations are shown for the interpolation operator of projection type on two-dimensional domain. Then, we consider the Ritz projection and Ritz-Volterra projection on finite element spaces, and by means of the superapproximation elementary estimates and Green function methods, derive the superconvergence and ultraconvergence error estimates for both projections, which are also the finite element approximation solutions of the elliptic problems and the Sobolev equations, respectively.     

ON CONVERGENCE OF PAL-TYPE INTERPOLATION POLYNOMIALS

Let {x_k~*}_(k=1)~(n-1) be the zeros of the (n-1) -th Legendre polynomial p_(n-1)(x) and {x_k}_(a=1)~n be the zeros of the polynomial w(x)= (1-x2~)p_(n-1)~1(x). By the theory of the Pal interpolation, for afunction f ∈ C_([-1,1])~1, there exists a unique polynomial Q_n(f, x) of degree 2n-1 satisfying conditions Q_n(f, x_k)=f(x_k), Q'_n(f, x_k~*)=f'(x_k~*), where k=1, 2, …, n and x_n~*=-1. The main result of this paper is that if f ∈ C_([-1,1])~r, thenf(x)-Q_n(f, x)=O(1)W(x)w(f~(r), 1/n)n~((1/2)-r), -1≤x≤1.Hence, if f ∈ C_[-1,1])~1, then Q_n(f, x) converges to the function f(x)uniformly on the interval [-1, 1].     

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

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

UNIFORM CONVERGENCE OF HERMITE INTERPOLATION OF HIGHER ORDER

In this paper the uniform convergence of Hermite-Fejer interpolation and Griinwald type theorem of higher order on an arbitrary system of nodes are presented.     

ON L_p—CONVERGENCE OF GRUNWALD INTERPOLATION

In this paper,the L_p-convergence of Grünwald interpolation G_n(f,x)based on the zerosof Jacobi polynomials J~((α,β))_n(x)(-1<α,β<1)is considered.L_p-convergence(00)of G_n(f,x)is obtained for-1<α,β≤0.Therefore,the results of[1]and [3-5]are improved.     

QUASI—INTERPOLATION AND APPROXIMATION VIA NONSEPARABLE SCALING FUNCTION

Quasi-interpolation has been studied in many papers,e.g.,[5].Here we introduce nonseparable scaling function quasi-interpolation and show that its approximation can provide similar convergence properties as scalar wavelet system.Several equivalent statements of accuracy of nonseparable scaling function are also given.In the numerical experiments,it appears that nonseparable scaling function interpolation has better convergence results than scalar wavelet systems in some cases.     

ORDER OF MEAN APPROXIMATION BY MIXED QUASI HERMITE-FEJER INTERPOLATION

In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi-tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be-sides,we also get some other results about the mean approximation.     

THEORY OF INTERPOLATION ON INFINITE INTERVAL—Ⅰ(UNIFORM CONVERGENCE OF STABLE INTERPOLATION PROCESSES)

§1. Preliminaries and Results Let us begin by defining a triangular matrix L whose n-th row consists of n distinct points x_(vn), v=1, n; n=1, 2, … such that 0相似文献   

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     

CHARACTERIZATION OF THE CONVERGENCE DOMAINS OF POLYNOMIAL SERIES AND THE MINIMAL CONVERGENCE DOMAIN

A characterization of the convergence domains of polynomial series is disucssed. the minimalconvergence domain for a kind of polynomial series is shown.     

CONVERGENCE OF CASCADE ALGORITHMS AND SMOOTHNESS OF REFINABLE DISTRIBUTIONS

In this paper, the author at first develops a method to study convergence of the cascade algorithm in a Banach space without stable assumption on the initial (see Theorem 2.1), and then applies the previous result on the convergence to characterizing compactly supported refinable distributions in fractional Sobolev spaces and Holder continuous spaces (see Theorems 3.1, 3.3, and 3.4). Finally the author applies the above characterization to choosing appropriate initial to guarantee the convergence of the cascade algorithm (see Theorem 4.2).     

DIMENSION AND DIFFERENTIABILIIY OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, a new iterated function system consisting of non-linear affinemaps is constructed. We investigate the fractal interpolation functions generated bysuch a system and get its differentiability, its box dimension, its packing dimension,and a lower bound of its Hansdorff dimension.     

THE SMOOTHNESS AND DIMENSION OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, we investigate the smoothness of non-equidistant fractal interpolation functions We obtain the Holder exponents of such fractal interpolation functions by using the technique of operator approximation. At last, We discuss the series expressiong of these functions and give a Box-counting dimension estimation of “critical” fractal interpohltion functions by using our smoothness results.     

QUASI－CONVEX MULTIOBJECTIVE GAME－SOLUTION CONCEPTS，EXISTENCE AND SCALARIZATION

ＱＵＡＳＩ－ＣＯＮＶＥＸＭＵＬＴＩＯＢＪＥＣＴＩＶＥＧＡＭＥ－ＳＯＬＵＴＩＯＮＣＯＮＣＥＰＴＳ，ＥＸＩＳＴＥＮＣＥＡＮＤＳＣＡＬＡＲＩＺＡＴＩＯＮ￥ＬＩＹＵＡＮＸＩＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｓｏｌｕｔｉｏｎｃｏｎｃｅ...     

A MODIFICATION OF POWELL-ZANGWILL'S METHOD AND ITS RATE OF CONVERGENCE

In this paper,a modified version of Powell-Zangwill's method for function minimizationwithout calculating derivatives is proposed.The new method possesses following properties:quadratic termination,global convergence for strictly convex function and Q-linearconvergence rate for uniformly convex function.Furthermore,the main part of this paperis to show that the rate of convergence of the new method is quadratic for every n(2n 1)line searches if the objective function is a uniformly convex and suitably smooth functionon R~n.     

QUASI FOURIER-JACOBI SERIES AND A KIND OF NEW APPROXIMATION

In this paper, we discuss the Quasi Fourier-Jacobi series. Based on it, for a kind of new approximation problem, we give its direct and inverse theorems.     

THE DOMAIN OF ATTRACTION AND LOCAL UNIFORM CONVERGENCE

For independent and identical distributed random variables,weestablish some relations between the three domains of attraction for maximaand some uniform convergences on distribntions of the linear normalized max-ima.Indeed,it is shown that,there are three types of subdomains of attra-ction.