Be'zier—Lupas算子的点态逼近度

On the Rate of Convergence of the Be′zier-Lupas Operator

本文利用一种概率分布定义Be'zier-Lupas算子作为新的逼近工具,并讨论它对有界变差函数类BV[0,∞)的点态逼近,给出精确的逼近阶。

For f∈B [0, ∞), x≥0, letB_n^((a))(f, x)=sum from k=0 f(k/n)Q_(n k)^((■))(x)Where a≥1, Q_(n k)^((■))(x)=(sum from j=k b_(n j)^((3))(x))~n -(sum from j=k+1 b(■)(x)) and Which is called Be zier-Lupas operator. In this paper, we study the rate of convergence of the Be'zier-Lupas operator for functions of bounded variation, the following result is obtainedTheorem for (■)f∈BV[0, ∞),f(t)=O(t^p)(p>0,t→∞), x≥0 and n sufficiently large, we have|B_n^((a))(f,x)-[1/2 f(x_x)+(1-1/2)f(x_-)]|/(2α(1+βx)+x)/nx(■)(g)+O(1)(α(2+βx)(3+4(1+β)x)/(nx(1+βx))(1/2))(4x)+[(α+2α(1+β)x)]×|f(x_+)-f(x_)|+(3α+4α(1+β)x)ε(■)(x)| f(x)-f(x_)|]1/nx^(1/2)(1+βx).The estimation is sharp.