First applications: Scalar interpolation and first-order discrete systems

In the present chapter we discuss the Schur algorithm and some interpolation problems in the scalar case. We make use, in particular, of the theory of J-unitary rational functions presented in Chapter 9. We also discuss first-order discrete systems. In the classical case, interpolation problems in the Schur class can be considered in a number of ways, of which we mention: 1. A recursive approach using the Schur algorithm (as in Schurs 1917 paper [261]) or its variant as in Nevanlinna's 1919 paper [244] in the scalar case. 2. The commutant lifting approach, in its various versions; see Sarason's seminal paper [258], the book [254] of Rosenblum and Rovnyak, and the book [179] of Foias and Frazho. 3. The state space method; see [87]. 4. The fundamental matrix inequality method, due to Potapov, see [249], with further far-reaching elaboration due to Katsnelson, Kheifets and Yuditskii, see [226, 227]. 5. The method based on extension of operators and Krein's formula; see the works of Krein and Langer [233, 232] and also [25]. 6. The reproducing kernel method; see [26, 172].