基于Chebyshev节点组的多元张量积多项式插值在布朗片测度下的平均误差

利用一元函数的Lagrange多项式插值构造了一种线性张量积多项式插值逼近多元函数。对于加权L2范数, 在布朗片测度下讨论了其平均误差,得到了相应量的强渐近阶。同过去利用线性泛函信息构造算法相比, 本文的算法利用的是标准信息, 且算法是构造性的, 可以直接解决实际问题。而且在平均误差方面, 结果显示该算法在一维情形下是阶最优的, 且在高维情形下与利用线性泛函信息得到的最优算子具有类似的逼近阶。

The average error of linear tensor product multivariate polynomial interpolation based on Chebyshev nodes on the Brownian sheet measure

Based on univariate Lagrange polynomial interpolation, a kind of linear tensor product polynomial interpolation is constructed to approximate multivariate functions. For the weighted L2-norm,their average errors is studied on the Brownian sheet measure and obtained the corresponding stronger asymptotic order. Compared with the past algorithms based on linear functional information, our algorithms are based on standard information and it is constructive. It can also be applied to solve practical problems. From the perspective of average error, it is showed that algorithms are order optimal to the univariate function case setting, and have a similar approximation order to the optimal algorithms based on linear functional information to the multivariate function case setting.