  The purpose of this paper is to present new classes of function systems as
part of multiresolution analyses. Our approach is representation theoretic, and
it makes use of generalized multiresolution function systems (MRSs). It further
entails new ideas from measurable endomorphisms-dynamics. Our results yield
applications that are not amenable to more traditional techniques used on
metric spaces. As the main tool in our approach, we make precise new classes of
generalized MRSs which arise directly from a dynamical theory approach to the
study of surjective endomorphisms on measure spaces. In particular, we give the
necessary and sufficient conditions for a family of functions to define
generators of Cuntz relations. We find an explicit description of the set of
generalized wavelet filters. Our results are motivated in part by analyses of
sub-band filters in signal/image processing. But our paper goes further, and it
applies to such wider contexts as measurable dynamical systems, and complex
dynamics.
  A unifying theme in our results is a new analysis of endomorphisms in general
measure space, and its connection to multi-resolutions, to representation
theory, and generalized wavelet systems.
