Gunter Semmler

Boundary Interpolation with Blaschke Products of Minimal Degree

We study a version of the classical Nevanlinna-Pick interpolation problem for Blaschke products where all interpolation points are located on the boundary of the unit disc. It turns out that all problems fall into three classes which are distinguished by the minimal degree of the interpolating function. The properties of the classes are studied in some detail. In particular it is shown that exactly one class consists of well-posed problems. An algorithm for classifying a given boundary interpolation problem is developed and a procedure for finding a solution with minimal degree for the generic class of regular problems is described.

Nonlinear Riemann-Hilbert Problems and Boundary Interpolation

We study properties of solutions to non-linear Riemann-Hilbert problems with smooth compact regularly traceable target manifold. The investigations focus on the dependence of solutions with positive winding numbers on additional parameters. While previous results investigated the behavior of the solutions inside the unit disk, we also pay attention to the boundary functions.As an application we consider boundary interpolation problems involving solutions to Riemann-Hilbert problems. We generalize a result by Ruscheweyh and Jones about boundary interpolation with Blaschke products. It is also shown how solutions with winding number 1 can be determined by three given points on the target manifold.Finally we formulate the problem of boundary interpolation with minimal winding number. It turns out that these problems form three subclasses, the well-posed problems constituting exactly one of them.

Separation Principles and Riemann-Hilbert Problems

Restrictions imposed on the boundary values of holomorphic functions induce restrictions on their values at interior points. The paper is devoted to the following related questionLet A be a subclass of the Hardy space H1 in the complex unit disk D and for each t ∈ ∂D let the complex plane be divided into an upper und a lower domain by some curve Mt. If we know that the boundary values of holomorphic functions ω+ = u+ + iv+ and w− = u− + iv− in A lie in the upper and the lower domain respectively, what can we conclude about the relative position of w+(0) and w−(0)?The problem is studied for several classes A and with different assumptions on the separating curves Mt. A number of counterexamples illustrates the limitations of the results.One main tool in the investigations is a non-linear boundary value problem of Riemann-Hilbert type, which is also of independent interest. Using an approximation procedure and an argument based on normal families we extend earlier results on existence and uniqueness of solutions for smooth problems to the case of piecewise continuous boundary conditions.

Entire Functions in Generalized Bernstein Spaces and Their Growth Behavior

For an \(L^{2}(\mathbb{R})\) function, the famous theorem by Paley and Wiener gives a beautiful relation between extensibility to an entire function of exponential type and the line support of its Fourier transform. However, there is a huge class of entire functions of exponential type which are not square integrable on an axis, but do have integrability properties on certain half lines. In this chapter we investigate such functions, their growth behavior, and their integrability properties in L p -norms. We show generalizations of a theorem of J. Korevaar and the Paley-Wiener theorem.

On the Normality of Topological Target Manifolds for Riemann-Hilbert Problems

In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problems with non-smooth target manifold. His main result is a characterization of solutions to Riemann-Hilbert problems as extremal functions in certain function classes. However, a complete analogy to corresponding results for problems with smooth target manifold holds only for a subclass of the toplogical target manifolds introduced by Belch, which are called normal. The conjecture that this subclass coincides with the whole class of topological target manifolds was left unproved. In the present paper we give a (counter-)example of a topological target manifold for which the solution set of the Riemann-Hilbert problem is in some sense bigger than in the smooth case. The problem to characterize normal topological target manifolds in geometric terms arises now as a challenging question of ongoing research.