  In [19] there is an approach to the investigation of the pseudocontinuability
of Schur functions in terms of Schur parameters. In particular, there was
obtained a criterion for the pseudocontinuability of Schur functions and the
Schur parameters of rational Schur functions were described. This approach is
based on the description in terms of the Schur parameters of the relative
position of the largest shift and the largest coshift in a completely
nonunitary contraction. It should be mentioned that these results received a
further development in [8, 21-24].
  This paper is aimed to give a survey about essential results on this
direction. The main object in the approach is based on considering a Schur
function as characteristic function of a contraction (see Section 1.2). This
enables us outgoing from Schur parameters to construct a model of the
corresponding contraction (see Section 2). In this model, the relative position
of the largest shift and the largest coshift in a completely nonunitary
contraction is described in Section 3 and then, based on this model, to find
characteristics which are responsible for the pseudocontinuability of Schur
functions (see Sections 4 and 5). The further parts of this paper (see Sections
6-8) admit applications of the above results to the study of properties of
Schur functions and questions related with them.

Complex Variables;Analysis of PDEs;Classical Analysis and ODEs