一类算子的逼近与拟合

Approximation and Fitting of a Kind of Operators

Lagrange算子(L_n)与Bernstein(B_n)算子是用于处理多项式逼近与拟合问题的两个重要算子,这两种算子各有优缺点。对于这两种算子如何扬长避短,学者们做了不懈努力,其中最为著名的是法国数学家SablonniereP,他于1992年引入并研究了一种新的拟Bernstein插值算子B_n^((k)),这是一类介于Lagrange算子与Bernstein算子之间的拟插值算子,这类算子兼顾了Lagrange算子与Bernstein算子的优点,克服了二者的不足。在给出了当n=3时B_n^((k))算子的表达式之后,提出了如何利用这种算子来完成满足某些给定条件的多项式曲线的设计。

Lagrange operator（L_n）and Bernstein operator（B_n）are two important operators used to deal with polynomial approximation and fitting problems,and they have advantages and disadvantages of their own.As to how to avoid weaknesses of these two operators,scholars have made unremitting efforts.One of the most famous is the French mathematician Sablonniere P,who introduced and studied a kind of new quasi--Bernstein interpolation operators in 1992.This kind of operators have given dual attention to the Lagrange operator and the Bernstein operator merit,and have avoided the two＇s deficiency.After giving the expression of operators B_n^（（k）） in detail when n= 3.How to make use of these operators to conduct design of polynomial curve satisfying certain given conditions was proposed.