Connections between real Schur polynomials and half order complex Schur polynomials

(where a, , , , = a _ , = 0). Suppose that [possibly after replacement of f ( s ) by f ( s ) l either a,, > 0 for i = 0, 1,. . . , n or a , > 0 , a,, > 0 and a, , , , > 0 for i = 0, 1,. . . , n 1. Then if any of f ( s ) , f , (s) and f,(s) has all zeros in Re[sl < 0, the other two polynomials have this property. In this work, we shall indicate how a parallel result can be obtained for discrete-time polynomials. The above theorem can be modified to cope with odd degree f ( s ) 151. So can the discrete-time theorem. Stability of f (s) , f,(s) and f,(s) can be examined by checking the positive definiteness of the * Supported in part by NSF grant MIP-901170.