# Quasi-Herglotz functions and convex optimization

## TL;DR

This paper introduces quasi-Herglotz functions, extending Herglotz functions, and demonstrates their usefulness in modeling non-passive systems through convex optimization and numerical examples.

## Contribution

It defines the set of quasi-Herglotz functions, explores their properties, and applies them to model non-passive media via convex optimization techniques.

## Key findings

- Quasi-Herglotz functions form a linear space extending Herglotz functions.
- Properties like integral representations and boundary values are inherited.
- Numerical examples show effective modeling of non-passive gain media.

## Abstract

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modeling 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 modeling 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 modeling of a non-passive gain media formulated as a convex optimization problem, where the generating measure is modeled by using a finite expansion of B-splines and point masses.