We investigate perturbations of orthonormal bases of $L^2$ via a composition operator $C_h$ induced by a mapping $h$. We provide a comprehensive characterization of the mapping $h$ required for the perturbed sequence to form an orthonormal or Riesz basis. Restricting our analysis to differentiable mappings, we reveal that all Riesz bases of the given form are induced by bi-Lipschitz mappings. In addition, we discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct complete sequences with favorable approximation properties.

翻译：我们研究了通过映射$h$诱导的复合算子$C_h$对$L^2$正交基的扰动。我们全面刻画了使扰动序列构成正交基或Riesz基所需的映射$h$的特征。将分析限制于可微映射时，我们揭示了所有此类形式的Riesz基均由双Lipschitz映射诱导。此外，我们讨论了这些结果在逼近论中的意义，强调了利用双射神经网络构造具有良好逼近性质的完备序列的潜力。