  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.
