Laurent Series and their Padé Approximations pp 207-232 | Cite as

# The positive definite case and applications

Chapter

## Abstract

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 in the neighborhood of |z|= p = 1 and .This means that our meromorphic function 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.

*F*(*z*) will have a Laurent series expansion$$F(z) = \sum\nolimits_{ - \infty }^\infty {{f_k}{z^k}} $$

$${f_{ - k}} = {{\bar f}_k},k\, \in \,\mathbb{Z}$$

*F*(*z*) will take real and positive values on the unit circle. The symmetry in the problem that is caused by$${f_{ - k}} = {{\bar f}_k},k\, \in \,\mathbb{Z}$$

## Keywords

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