The positive definite case and applications

  • Adhemar Bultheel
Part of the Operator Theory: Advances and Applications book series (OT, volume 27)


In this chapter we shall relate our results to some classical functional problems that were studied by Carathéodory and Schur and some related moment problems. A survey can be found in [AKH]. In this context F(z) will have a Laurent series expansion
$$F(z) = \sum\nolimits_{ - \infty }^\infty {{f_k}{z^k}} $$
in the neighborhood of |z|= p = 1 and
$${f_{ - k}} = {{\bar f}_k},k\, \in \,\mathbb{Z}$$
.This means that our meromorphic function F(z) will take real and positive values on the unit circle. The symmetry in the problem that is caused by
$${f_{ - k}} = {{\bar f}_k},k\, \in \,\mathbb{Z}$$
will essentially reduce the complexity of the problem to half the complexity for the general case. The quantities with and without hat will contain the same information, so that we can drop one of them. Many of the algorithms that we have seen before will reduce to well known classical algorithms.


Unit Circle Unit Disc Laurent Series Outer Function Transmission Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Birkhäuser Verlag Basel 1987

Authors and Affiliations

  • Adhemar Bultheel
    • 1
  1. 1.Dept. Computer ScienceK. U. LeuvenLeuven-HeverleeBelgium

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