This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor.

It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.

本文考虑了基于有限数量的这些光谱密度样本，用有限的狄利克雷能量逼近连续光谱密度的光谱因子的问题。尽管对于频谱因子存在一个封闭形式的表达式，但是由于频谱因子与频谱密度的非线性相关性以及该表达式中的奇异积分，因此该公式显示出非常复杂的行为。因此，通常采用近似方法来计算光谱因子。

结果表明，不存在基于采样的方法，该方法连续依赖于样本，并且能够近似计算该集中所有密度的光谱因子。取而代之的是，对于任何基于采样的近似方法，都存在大量的光谱密度，因此随着可用采样点数量的增加，该近似方法不会收敛到该组中每个光谱密度的光谱因子。本文还将显示在Hardy空间理论中，基于采样的外部函数计算算法也具有相同的结果。