  The paper contains several theoretical results related to the weighted
nonlinear least-squares problem for low-rank signal estimation, which can be
considered as a Hankel structured low-rank approximation problem. A
parameterization of the subspace of low-rank time series connected with
generalized linear recurrence relations (GLRRs) is described and its features
are investigated. It is shown how the obtained results help to describe the
tangent plane, prove optimization problem features and construct stable
algorithms for solving low-rank approximation problems. For the latter, a
stable algorithm for constructing the projection onto a subspace of time series
that satisfy a given GLRR is proposed and justified. This algorithm is used for
a new implementation of the known Gauss-Newton method using the variable
projection approach. The comparison by stability and computational cost is
performed theoretically and with the help of an example.
