  Power moments, modified moments, and optimized moments are powerful tools for
solving microscopic models of macroscopic systems; however the expansion of the
density of states as a continued fraction does not converge to the macroscopic
limit point-wise in energy with increasing numbers of moments. In this work the
moment problem is further constrained by minimal lifetimes or maximal breaking
of time-reversal symmetry, to yield approximate densities of states with
point-wise macroscopic limits. This is applied numerically to models with one
and two finite bands with various singularities, as well as to a model with
infinite band-width, and the results are compared with the maximum entropy
approximation where possible.
