The well-known Wiener's lemma is a valuable statement in harmonic analysis; in the Banach space of functions with absolutely convergent Fourier series, the lemma proposes a sufficient condition for the existence of a pointwise multiplicative inverse. We call the functions that admit an inverse as \emph{reversible}. In this paper, we introduce a simple and efficient method for approximating the inverse of functions, which are not necessarily reversible, with elements from the space. We term this process \emph{pseudo-reversing}. In addition, we define a condition number to measure the reversibility of functions and study the reversibility under pseudo-reversing. Then, we exploit pseudo-reversing to construct a multiscale pyramid transform based on a refinement operator and its pseudo-reverse for analyzing real and manifold-valued data. Finally, we present the properties of the resulting multiscale methods and numerically illustrate different aspects of pseudo-reversing, including the applications of its resulting multiscale transform to data compression and contrast enhancement of manifold-valued sequence.

翻译：著名的 Wiener 引理是调和分析中的一个重要结论；在具有绝对收敛傅里叶级数的函数构成的巴拿赫空间中，该引理提出了存在逐点乘法逆元的充分条件。我们把存在逆元的函数称为"可逆的"。本文提出一种简单高效的方法，利用空间中的元素来近似逼近未必可逆的函数的逆元，我们将这一过程称为"伪逆化"。此外，我们定义条件数来衡量函数的可逆性，并研究伪逆化下的可逆性。进而利用伪逆化，基于细化算子及其伪逆构造多尺度金字塔变换，用于分析实值数据和流形值数据。最后，我们给出所得到的多尺度方法的性质，并通过数值实验展示伪逆化的多个方面，包括其生成的多尺度变换在数据压缩和流形值序列对比度增强中的应用。