  Multiscale transforms for real-valued data, based on interpolatory
subdivision operators have been studied in recent year. They are easy to
define, and can be extended to other types of data, for example to
manifold-valued data. In this paper we define linear multiscale transforms,
based on certain linear, non-interpolatory subdivision operators, termed
"even-reversible". For such operators, we prove, using Wiener's lemma, the
existence of an inverse to the linear operator defined by the even part of the
subdivision mask, and termed it "even-inverse". We show that the
non-interpolatory subdivision operators, with spline or with pseudo-spline
masks, are even-reversible, and derive explicitly, for the quadratic and cubic
spline subdivision operators, the symbols of the corresponding even-inverse
operators. We also analyze properties of the multiscale transforms based on
even-reversible subdivision operators, in particular, their stability and the
rate of decay of the details.
