It was recently shown that on a large class of important Banach spaces there exist no linear methods which are able to approximate the Hilbert transform from samples of the given function. This implies that there is no linear algorithm for calculating the Hilbert transform which can be implemented on a digital computer and which converges for all functions from the corresponding Banach spaces. The present paper develops a much more general framework which also includes non-linear approximation methods. All algorithms within this framework have only to satisfy an axiom which guarantees the computability of the algorithm based on given samples of the function. The paper investigates whether there exists an algorithm within this general framework which converges to the Hilbert transform for all functions in these Banach spaces. It is shown that non-linear methods give actually no improvement over linear methods. Moreover, the paper discusses some consequences regarding the Turing computability of the Hilbert transform and the existence of computational bases in Banach spaces.

最近显示，在一大类重要的Banach空间上，不存在能够从给定函数的样本中近似希尔伯特变换的线性方法。这意味着没有线性算法可以计算希尔伯特变换，该算法可以在数字计算机上实现，并且可以收敛来自相应Banach空间的所有函数。本文开发了一个更通用的框架，其中还包括非线性逼近方法。该框架内的所有算法仅需满足一个公理，该公理可基于给定的函数样本来保证算法的可计算性。本文研究了在这种通用框架内是否存在一种算法，该算法可以收敛到这些Banach空间中所有函数的希尔伯特变换。结果表明，非线性方法实际上没有对线性方法的改进。此外，本文还讨论了有关希尔伯特变换的图灵可计算性以及Banach空间中计算基础的存在的一些后果。