The functions generated by Schur's algorithm are related by

where denotes the elementary linear fractional transformation (LFT)

We therefore have , where is the composition

In this way we obtain the * Schur continued fraction representation*
of ,

The function is referred to as the * nth approximant*
of , and is called the * nth tail*
of .

It is shown in [5], and easily verified, that the LFT can be expressed as

where the polynomials and satisfy the recurrence relations

It follows by induction that
and have degree * less than n*,
and
,
,
, and for all

The polynomials and are generalizations of the
Schur parameters in the sense that they determine a composition of
the elementary linear fractional transformations determined by
the Schur parameters.
We will refer to and as the **n**th
* Schur polynomials*
associated with the Schur function .

The following result shows how the Schur polynomials provide the Szego polynomial , and hence provide for the first phase of a Toeplitz solver.

The proof follows by comparison of the recurrence relations for with those for and [5].

Thu Sep 18 20:40:30 CDT 1997