Quasi-Herglotz functions and convex optimization

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.

arXiv comment23 pages, 5 figures. Updated Introduction and Sections 2.1 and 2.4. Restructured and updated Section 5. New numerical example in Section 5.3

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Cited by in corpus (4)

• On quasi-Herglotz functions in one variable
• On the structure of Nevanlinna measures
• Characterizations of Herglotz-Nevanlinna functions using positive semi-definite functions and the Nevanlinna kernel in several variables
• Characterizations of the Lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part