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Mathematics > Numerical Analysis

arXiv:1812.08319 (math)
[Submitted on 10 Dec 2018 (v1), last revised 27 Aug 2019 (this version, v2)]

Title:Quasi-Herglotz functions and convex optimization

Authors:Yevhen Ivanenko, Mitja Nedic, Mats Gustafsson, B. L. G. Jonsson, Annemarie Luger, Sven Nordebo
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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.
Comments: 23 pages, 5 figures. Updated Introduction and Sections 2.1 and 2.4. Restructured and updated Section 5. New numerical example in Section 5.3
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph)
Cite as: arXiv:1812.08319 [math.NA]
  (or arXiv:1812.08319v2 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.1812.08319
Journal reference: Royal Soc. Open Sci. 7 (2020), 191541
Related DOI: https://doi.org/10.1098/rsos.191541

Submission history

From: Yevhen Ivanenko [view email]
[v1] Mon, 10 Dec 2018 21:02:45 UTC (1,125 KB)
[v2] Tue, 27 Aug 2019 20:37:39 UTC (999 KB)
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