Elias Wegert

Optimal approximation by Blaschke forms

We study best approximation of functions in the Hardy space H 2(𝔻) by Blaschke forms, which are finite linear combinations of modified Blaschke products. These functions have poles outside the unit disk which are adapted according to the function to be decomposed. We prove the existence of minimizers and propose an algorithm for their construction.

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.

On Hyperbolic Divided Differences and the Nevanlinna-Pick Problem

Starting from the notion of the complex pseudo-hyperbolic distance and the hyperbolic difference quotient introduced by A. F. Beardon and D. Minda in [1], we define hyperbolic divided differences for unimodularly bounded holomorphic functions in the complex unit disk and investigate their mapping properties. In particular, we show that they operate on Blaschke products in the same way as the ordinary divided differences act on polynomials. As a simple corollary we obtain a multi-point Schwarz-Pick Lemma.Using these concepts we investigate the classical interpolation problem of Pick and Nevanlinna and reformulate the Nevanlinna-Schur algorithm in terms of hyperbolic divided differences. This leads to a scheme that (formally) coincides with Newton’s algorithm for polynomial interpolation.

Blow-Up in a Modified Constantin-Lax-Majda Model for the Vorticity Equation

We propose a one-dimensional model for the vorticity equation involving-viscosity. Complex methods are utilized in order to study finite time blow-up of the solutions. In particular it is shown that the blow up time depends monotoneously on the viscocity.

Nonlinear boundary value problems for elliptic systems in the plane

The solvability of a wide class of nonlinear boundary value problems of Riemann–Hilbert type of generalized analytic functions (in the sense of Vekua) is shown. Applications to nonlinear boundary value problems of Poincare type for elliptic equations of second order are discussed.

The Riemann Zeta Function on Arithmetic Progressions

We prove asymptotic formulas for the first discrete moment of the Riemann zeta function on certain vertical arithmetic progressions inside the critical strip. The results give some heuristic arguments for a stochastic periodicity that we observed in the phase portrait of the zeta function.

Phase Diagrams of Meromorphic Functions

AbstractThe paper demonstrates the use of phase diagrams as tools for visualizing and exploring meromorphic functions. With any such function

Nonlinear Riemann-Hilbert Problems and Boundary Interpolation

f:D\ \rightarrow\ \hat{\rm C}

On Beurling's boundary value problem in circle packing

we associate two mappings