In this chapter we shall collect the necessary preliminaries from complex analysis that we shall use frequently. Most of these results are classical and we shall give them mostly without proof.

We start with some elements from Hardy functions in the disk and the half plane in Section 1.1.

The important classes of analytic functions in the unit disk and half plane and with positive real part are called positive real for short and are often named after Carathéodory. By a Cayley transform, they can be mapped onto the class of analytic functions of the disk or half plane, bounded by one. This is the so-called Schur class. These classes are briefly discussed in Section 1.2.

Inner–outer factorizations and spectral factors are discussed in Section 1.3.

The reproducing kernels are, since the work of Szegő, intimately related to the theory of orthogonal polynomials and they will be even more important for the case of orthogonal rational functions. Some of their elementary properties will be recalled in Section 1.4.

The 2 × 2 J-unitary and J-contractive matrix functions with entries in the Nevanlinna class will be important when we develop the recurrence relations for the kernels and the orthogonal rational functions. Some of their properties are introduced in Section 1.5.

Hardy classes

We shall be concerned with complex function theory on the unit circle and the upper half plane. The complex number field is denoted by C.