Bernstein算子和Grünwald算子的线性组合

On Linear Combination of Bernstein and Grünwald Operators

以第一类n阶Chebyshev多项式的零点作为插值节点,通过Bernstein算子和Gr櫣nwald算子的线性组合构造一个新算子Gn(f;x).如果f(x)∈Cj[-1,1](0≤j≤9),则Gn(f;x)在区间[-1,1]上一致收敛于f(x)∈Cj[-1,1](0≤j≤9),并且其收敛阶达到最佳,饱和阶为1/n10.

In this paper a new operator G_n(f;x) is constrcucted by means of the linear combination of Bernstein and Grünwald operators on the basis of taking the zeros of the first kind of Chebyshev polynomial of degree n as the interpolating nodes. G_n(f;x) converges to the f(x)∈C^j_([-1,1]), 0≤j≤9 on [-1,1] uniformly and has the best approximation order if the function f(x)∈C^j_([-1,1]), 0≤j≤9. The saturation order is 1/n^(10).