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Relation to Schur Functions

**Definition.**A

*Schur function*is defined as a complex function analytic and of modulus not exceeding unity in .

**Property.**The function

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

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

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