Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

The Average Error for Lagrange Interpolation and Hermite-Fejér Interpolation on the Wiener Space

在L_q-范数逼近的意义下,确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶.从我们的结果可以看出,当2≤q<∞,1≤p<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差.在信息基计算复杂性的意义下,如果可允许信息泛函为计算函数在固定点的值,那么当1≤p,q<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径.

For the Lq-norm approximation, we determine the weakly asymptoticl order for the p-average errors of the Lagrange interpolation sequence and the Hermite-Fejér interpolation sequence based on the Chebyshev nodes on the Wiener space. By these results we know that for 2 ≤ q 〈 ∞, 1 ≤ p 〈 ∞, the p-average errors of Lagrange interpolation sequence and Hermite-Fejér interpolation sequence based on the Chebyshev nodes are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence. In the sense of Information-Based Complexity, if permissible information functionals are function evaluations at fixed points, then the p-average errors of Lagrange interpolation sequence and Hermite-Fejér interpolation sequence based on the Chebyshev nodes are weakly equivalent to the corresponding sequence of minimal p-average radii of nonadaptive information.