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 1 1 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 .

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