Abstract We develop a new approach to obtain derivatives of functions on arbitrary interval by extending these functions to the whole real axis based on Hermite function approximation. A modified Tikhonov regularization with a new penalty term derived from the Fourier transform is used to stabilize the extension process. A complete convergence theory is given and the process of numerical implementation is also discussed. Finally, we provide some examples to verify the effectiveness of the method.

中文翻译：

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数值微分的 Hermite 扩展方法

摘要 我们开发了一种新方法，通过基于 Hermite 函数逼近将这些函数扩展到整个实轴来获得任意区间上的函数的导数。使用从傅立叶变换导出的新惩罚项的修改的 Tikhonov 正则化来稳定扩展过程。给出了完整的收敛理论，并讨论了数值实现的过程。最后，我们提供了一些例子来验证该方法的有效性。