  There is a well-known series expansion (Neumann series) in functional
analysis for perturbative inversion of specific operators on Banach spaces.
However, operators that appear in signal processing (e.g. folding and
convolution of probability density functions), in general, do not satisfy the
usual convergence condition of that series expansion. This article provides
some theorems on the convergence criteria of a similar series expansion for
this more general case, which is not covered yet by the literature.
  The main result is that a series expansion provides a robust unbiased
unfolding and deconvolution method. For the case of the deconvolution, such a
series expansion can always be applied, and the method always recovers the
maximum possible information about the initial probability density function,
thus the method is optimal in this sense. A very significant advantage of the
presented method is that one does not have to introduce ad hoc frequency
regulations etc., as in the case of usual naive deconvolution methods. For the
case of general unfolding problems, we present a computer-testable sufficient
condition for the convergence of the series expansion in question.
  Some test examples and physics applications are also given. The most
important physics example shall be (which originally motivated our survey on
this topic) the case of pi^0 --> gamma+gamma particle decay: we show that one
can recover the initial pi^0 momentum density function form the measured single
gamma momentum density function by our series expansion.
