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 , 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 n.
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 nth 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 .