一类线性插值算子的导数逼近

为了改善Lagrange插播算子的一致收敛性并提高算子最佳收敛阶,我们以一类Ja cobi多项式的零点作为插值结点,通过对插值结点处函数值的线性组合,构造了一类线性插值算子,给出了该类算子的最佳收敛阶定理;进而研究了此类算子的导数逼近问题,利用对算子进行分项估计的方法,不仅证明了该算子的导数一致收敛于具有连续导数的函数,而且给出了算子的一阶导数逼近函数导数的最佳收敛阶.

Derivative-approximation of a Class of Linear Interpolation Operator

In order to improve the uniform convergence property of the Lagrange interpolation operator and to increase the best convergence order, we construct a linear interpolation operator based on the zero node of a kind of Jacobi polynomials as the interpolation nodes, by combining the function values at the interpolation nodes linearly, and then present the theorem of the best convergence order of the operator. Further, we study the problem of derivative-approximation of this class of linear interpolation operators. By using the method of estimation of each term of the operator, we not only prove that the derivative of the operator converges to that of the continuous function uniformly, but also carry out the best convergence order of the derivative of the operator.