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## Schur polynomials.

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 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 [5].

Greg Ammar
Thu Sep 18 20:40:30 CDT 1997