Contents

Preface vi i

Part I . Engineerin g motivatio n

1. Engineerin g backgroun d 3

What i s the transfe r function—ho w analyti c function s aris e in

engineering-connection laws.

2. Engineerin g problem s 11

Engineering problems (vs . analysis); a control paradigm.

Part II . Analyti c functio n theor y

How much classical analysi s can you get by staring a t pair s of

invariant subspaces ? Ther e i s a unifie d wa y o f obtainin g th e

theories of classical interpolation , H°° approximation , Coron a

= Bezou t identities , commutant lifting , Wiener-Hop f factoriza -

tion, integrabl e system s (Tod a Lattice , KdV) , matri x L U de-

compositions, interpolation , uppe r triangula r approximation .

The method give s excellent result s when al l functions induce d

are differentiable an d in these cases frequently extend s existing

results i n variou s ways , e.g. , treat s adde d symmetries , allow s

poles in H°° functions .

3. Fractiona l map s an d Grassmannian s 2 1

Basic projective lore. Th e correspondences between linear frac-

tional actions on operators and linear actions on subspaces.

4. Representin g shif t invarian t subspace s 3 5

5. Application s t o factorization , interpolation , an d

approximation 4 3

6. Furthe r application s 5 7

7. Matri x analog s an d generalization s 7 3

A general theor y whic h contains bot h th e matrix an d analyti c

function case .