Abstract The aim of this paper is to develop a method based on Legendre expansion to compute numerical derivatives of a function from its perturbed data. The Tikhonov regularization combined with a new penalty term is used to deal with the ill posed-ness of the problem. It has been shown that the solution process is uniform for various smoothness of functions. Moreover, the convergence rates can be obtained self-adaptively when we choose the regularization parameter by a discrepancy principle. Numerical tests show that the method gives good results.

中文翻译：

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基于超阶Tikhonov正则化的勒让德展开的数值微分方法

摘要 本文的目的是开发一种基于勒让德展开的方法，从扰动数据计算函数的数值导数。Tikhonov 正则化结合新的惩罚项用于处理问题的不适定性。已经表明，对于函数的各种平滑度，求解过程是一致的。此外，当我们通过差异原则选择正则化参数时，可以自适应地获得收敛速度。数值试验表明，该方法取得了良好的效果。