TY - JOUR

T1 - Convex cones of generalized positive rational functions and the Nevanlinna-Pick interpolation

AU - Alpay, Daniel

AU - Lewkowicz, Izchak

N1 - Funding Information:
This research is partially supported by BSF Grant No. 2010117. D. Alpay thanks the Earl Katz family for endowing the chair which

PY - 2013/5/15

Y1 - 2013/5/15

N2 - 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 (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. The sets of even generalized positive and odd functions form subcics. It is well known that the Nevanlinna-Pick interpolation problem is not always solvable by positive functions. Unfortunately, there is no computationally simple procedure to carry out this interpolation in the framework of generalized positive functions. Through examples it is illustrated how the two above partitionings of generalized positive functions can be exploited to introduce simple ways to carry out the Nevanlinna-Pick interpolation. Finally we show that only some of these properties are carried over to rational generalized bounded functions, mapping the imaginary axis to the unit disk.

AB - 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 (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. The sets of even generalized positive and odd functions form subcics. It is well known that the Nevanlinna-Pick interpolation problem is not always solvable by positive functions. Unfortunately, there is no computationally simple procedure to carry out this interpolation in the framework of generalized positive functions. Through examples it is illustrated how the two above partitionings of generalized positive functions can be exploited to introduce simple ways to carry out the Nevanlinna-Pick interpolation. Finally we show that only some of these properties are carried over to rational generalized bounded functions, mapping the imaginary axis to the unit disk.

KW - Carathéodory functions

KW - Cauer classes

KW - Convex invertible cones of rational functions

KW - Nevanlinna functions

KW - Nevanlinna-Pick interpolation

KW - Positive real functions

UR - http://www.scopus.com/inward/record.url?scp=84875461776&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.01.023

DO - 10.1016/j.laa.2012.01.023

M3 - Article

AN - SCOPUS:84875461776

VL - 438

SP - 3949

EP - 3966

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 10

ER -