Quasi-Herglotz functions and convex optimization

Author:

Ivanenko Y.1ORCID,Nedic M.2ORCID,Gustafsson M.3ORCID,Jonsson B. L. G.4ORCID,Luger A.2,Nordebo S.1ORCID

Affiliation:

1. Department of Physics and Electrical Engineering, Linnæus University, 351 95 Växjö, Sweden

2. Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

3. Department of Electrical and Information Technology, Lund University, Box 118, 221 00 Lund, Sweden

4. School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden

Abstract

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.

Funder

Stiftelsen för Strategisk Forskning

Publisher

The Royal Society

Subject

Multidisciplinary

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