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Nonlinear Riemann-Hilbert Problems and Boundary Interpolation

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Nonlinear Riemann-Hilbert Problems and Boundary Interpolation

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2003
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321034
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Mar 1, 2004

Keywords: Riemann-Hilbert problem; interpolation problem; normal family; boundary value problem; 30E25; 30E05; 35Q15; 41A29

References

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    Cantor, D G; Phelps, R R
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    Cargo, G T
  • Nonlinear Riemann-Hilbert problems without transversality
    Efendiev, M A; Wendland, W L
  • Nonlinear Riemann-Hilbert problems for doubly connected domains and closed boundary data
    Efendiev, M A; Wendland, W L
  • Blaschke product interpolation and its application to the design of digital filters
    Ruscheweyh, S; Jones, W B
  • Nevanlinna-Pick interpolation with boundary data
    Sarason, D
  • The degree of a quasi-linearlike mapping and the nonlinear Hilbert problem (in Russian)
    Shnirel’man, A I
  • Polynomial hulls and H∞ control for a hypoconvex constraint
    Whittlesey, M A