**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Introduction to Digital
Filters with Audio Applications", by Julius O. Smith III,
Copyright © *2017-11-26* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Relation to Schur Functions

Definition.ASchur functionis defined as a complex function analytic and of modulus not exceeding unity in .

Theorem.The function

is a Schur function if and only if is positive real.

Proof.Suppose is positive real. Then for , is PR. Consequently, is minimum phase which implies all roots of lie in the unit circle. Thus is analytic in . Also,

By the maxmimum modulus theorem, takes on its maximum value in on the boundary. Thus is Schur.Conversely, suppose is Schur. Solving Eq. (1.2) for and taking the real part on the unit circle yields

(33) (34) (35) (36) (37)

If is constant, then is PR. If is not constant, then by the maximum principle, for . By Rouche's theorem applied on a circle of radius , , on which , the function has the same number of zeros as the function in . Hence, is minimum phase which implies is analytic for . Thus is PR.