Convex Cones of Generalized Positive Rational Functions and Nevanlinna-Pick Interpolation

Daniel Alpay; Izchak Lewkowicz

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arXiv · 1010.0546

Convex Cones of Generalized Positive Rational Functions and Nevanlinna-Pick Interpolation

Daniel Alpay·Izchak Lewkowicz

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Abstract

Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into a pair of even and odd subcics and (ii) to (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane. It is then shown that a positive function can always be written as a sum of even and odd part, only over the larger set of generalized positive. It is well known that over positive functions Nevanlinna-Pick interpolation is not always feasible. Over generalized positive, there is no easy way to carry out this interpolation. The second partitioning is subsequently exploited to introduce a simple procedure for Nevanlinna-Pick interpolation. Finally we show that only some of these properties are carried over to generalized bounded functions, mapping the imaginary axis to the unit disk.

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BibTeX RIS

Daniel Alpay, Izchak Lewkowicz. 2012-02-06. Convex Cones of Generalized Positive Rational Functions and Nevanlinna-Pick Interpolation. https://arxiv.org/abs/1010.0546

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