The Operators' Approximation Problem Between L_nf And B_nf

The approximation problem researchers usually focus on the Bernstein operator's uniform convergence and the Lagrange operator's interpolating effect. However, both of them have itself weakness. The B_n^k operators and αB_n operators are between B_n and αB_n operators, and they have the virtues of B_n and αB_n operators and are short of their shortcomings.This paper study the approximation properties of B_n^k and αB_n, and give the formula and matrix formal of the B₃^k (k = 0,1,2,3), B₄^k (k = 0,1,2,3,4), aB₃ (α= 0,1,2,3,4). Then we draws a general conclusion by illustrating the approximation effect. This paper also study how to make use of these operators to conduct design of polynomial curve satisfying certain given conditions. Because three orders polynomials are the best fit for applying, this paper has some practical values.