Extremal solutions of matricial Carathéodory problem and Nevanlinna–Pick-type interpolation for Carathéodory matrix functions in the nondegenerate case

Abstract

This article is devoted to the investigation of remarkable properties of extremal solutions F ^[U] and G ^[U] with a q × q unitary parameter U and their distinguished one-parameter families and with |w| = 1 within the solution sets of Problem (C) (the matricial Carathéodory problem) and Problem (NP) (a certain Nevanlinna–Pick-type interpolation for Carathéodory matrix functions), respectively, in the nondegenerate case. We show connections between the poles of the functions F ^[U] and at the unit circle of the complex plane and their Riesz–Herglotz measures, and discuss closely related matters for the extremal solutions F ^[U] and . These results further serve as a starting point to look for corresponding properties of the functions G ^[U] and based on the so-called modified block Toeplitz vector approach.

Keywords:

• Nevanlinna–Pick-type interpolation
• matricial Carathéodory problem
• matrix-valued Carathéodory function
• Riesz–Herglotz measure
• extremal solution

AMS Subject Classifications::

• 30E05
• 47A56

Acknowledgements

The authors thank the referee for helpful comments and for drawing our attention to the article in 22 by A. Lasarow. This work was supported by the National Natural Science Foundation of China (No. 11071017) and the Program for New Century Excellent Talents in University.