Determination of inner functions by their value sets on the circle
This paper is devoted to the question of reconstructing an inner function from its values at a discrete set of points on the unit circle. More precisely, by establishing an explicit formula, the authors show that the set of points at which such a function takes two values of modulus one determines an inner function. If the inner function has only one singularity, it is unique up to a hyperbolic automorphism. They also obtain a full characterization of the sets at which the prescribed values can be taken. The authors first study interpolation by finite Blaschke products where each unimodular value is taken finitely many times on the circle. They link this question with the study of functions mapping the upper half-plane into itself and the real axis to the extended real axis. They make heavy use of the Hermite-Biehler criterion (also attributed to M. G. Krein) which characterizes the real meromorphic functions (that is, those meromorphic functions in the plane which take real values or the infinite value on the real line) mapping the upper half-plane to itself. The paper contains relevant examples, as well as applications to isometric composition operators and to the numerical range of the restricted shift operator on the model space of an inner function.
scientific article; zbMATH DE number 5965046
scientific article