**Definition. **A *Schur function*
is 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 , rere is PR. Consequently, is minimum phase which implies all roots of lie in the unit circle. Thus is analytic in . Also,

Conversely, suppose is Schur. Solving Eq.(5) for gives

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.

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