什么样的连续函数能由整系数多项式逼近

什么样的连续函数能由整系数多项式逼近

摘要：本文主要译自[1]，它系统地讨论了什么样的连续函数能由整系数多项式逼近，有以下两种情形：

（1）区间长度大于等于4的区间上，能ว由整系数多项式逼近的连续函数是这些整系数多项式本身；

（2）在区间长度严格小于4的区间 上，连续函数 能被整系数多项式1致逼近当且仅当 在 插值多项式为整系数多项式.

关键词：逼近;整系数多项式;插值多项式;代数整数.

What Can Be Approximated by Polynomials with Integer Coefficients

Abstract: This article is mainly translated from [1].What can be approximated by polynomials with integer coefficients is given in the following two aspects:

If the interval has length four or more than ♪the only functions that are approximable on the interval by polynomials with integer coefficients areก these polynomials themselves.

If is a continuous function on an interval I of length strictly less then 4,then is approximable on the interval by polynomials with integer coefficients if and only if its interpolating polynomi☮al on has integer coefficients.

Keywords: approximation; polynomial with integer coefficients; interpolating polynomial; algebraic integer.

目 录

中文摘要 …………………………………………………………………………………………1

英文摘要 …………………………………………………………………………………………1

前言 ………………………………………………………………………………………………2

1 1些符号与专用术语…………………………………………………………………………2

2 区间长度大于等于4时整系数多项式逼近问题………………………………………………2

3 区间 与 上的整系数多项式逼近问题 ………………………………………………………4

4 区间长度小于4时的整系数多项式逼近问题…………………………………………………9

4.1代数整数及其相关概念 ………………………………………………………………………10

4.2插值多项式 ……………………………………………………………………………………10

5 结束语♚ …………………………………………………………………………………………15

6 参考文献 ………………………………………………………………………………………16

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