In this chapter we discuss several aspects related to interpolation. In the first section, we derive some simple interpolation properties that can be easily obtained from the properties of the functions of the second kind that were studied earlier. It also turns out that interpolation of the positive real function Ωμ, whose Riesz–Herglotz–Nevanlinna measure μ is the measure that we used for the inner product, will imply that in Ln the measure can be replaced by the rational Riesz–Herglotz–Nevanlinna measure for the interpolant without changing the inner product. Some general theorems in this connection will be proved in Section 6.2. This will be important for the constructive proof of the Favard theorems to be discussed in Chapter 8. We then resume the interpolation results that can be obtained with the reproducing kernels and some functions that are in a sense reproducing kernels of the second kind.

We then show the connection with the algorithm of Nevanlinna–Pick in Section 6.4. This algorithm provides an alternative way to find the coefficients for the recurrence of the reproducing kernels that we gave in Section 3.2, without explicitly generating the kernels themselves. If all the interpolation points are at the origin, then the algorithm reduces to the Schur algorithm. It was designed originally to check whether a given function is in the Schur class. It basically generates a sequence of Schur functions by Möbius transforms and extractions of zeros.

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.