By Dahmen W., Micchelli C.A.

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**Extra resources for Banded matrices with banded inverses, II.. Locally finite decomposition**

**Sample text**

Substitute this into (4) and use the definitions of Ti to get z =(T1-TzQT3)w. [] For computations itis useful to have explicit realizations of the transfer matrices Ti (i=1-3). Start with a minimal realization of G: G(s)=[A, B, C, D]. Since the input and output of G are partitioned as [:] the matrices B, C, and D have corresponding partitions: B = [ B I Bz] C = C2 F ,I D= LD21D22 " Then Gij(s) = [A, Bj, Ci, Di)], i , j = l , 2 . Note that D22=0 because G2z is strictly proper. It can be proved that stabilizability of G (an assumption from Section 3) implies that (A,B2) is stabilizable and (C2,A) is detectable.

1 Hankel Operators Let X and Y be Hilbert spaces and let qb be a linear function from X to Y. The function ~ is bounded if there exists a real number a such that II@xII

Our interest is in Hankel operators with real-rational symbols, Example 7 being a special case. Theorem 1. If F ~ RL,~, then F F has finite rank. Proof. e. F2~ RH~. Since FF=FF1, we might as well assume at the start that F is strictly proper and analytic in Re s <0. Introduce a minimal realization: F(s)=[a, B, C, 0]. The operator FF and its time-domain analog have equal ranks. As in Example 7 the latter operator equals WoWc. By controllability and observability Wc is surjective and Wo is injective.

### Banded matrices with banded inverses, II.. Locally finite decomposition by Dahmen W., Micchelli C.A.

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