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Quasi-Herglotz functions and convex optimization

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 functi...

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Autores principales: Ivanenko, Y., Nedic, M., Gustafsson, M., Jonsson, B. L. G., Luger, A., Nordebo, S.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7029951/
https://www.ncbi.nlm.nih.gov/pubmed/32218971
http://dx.doi.org/10.1098/rsos.191541
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author Ivanenko, Y.
Nedic, M.
Gustafsson, M.
Jonsson, B. L. G.
Luger, A.
Nordebo, S.
author_facet Ivanenko, Y.
Nedic, M.
Gustafsson, M.
Jonsson, B. L. G.
Luger, A.
Nordebo, S.
author_sort Ivanenko, Y.
collection PubMed
description 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.
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spelling pubmed-70299512020-03-26 Quasi-Herglotz functions and convex optimization Ivanenko, Y. Nedic, M. Gustafsson, M. Jonsson, B. L. G. Luger, A. Nordebo, S. R Soc Open Sci Mathematics 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. The Royal Society 2020-01-15 /pmc/articles/PMC7029951/ /pubmed/32218971 http://dx.doi.org/10.1098/rsos.191541 Text en © 2020 The Authors. http://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Mathematics
Ivanenko, Y.
Nedic, M.
Gustafsson, M.
Jonsson, B. L. G.
Luger, A.
Nordebo, S.
Quasi-Herglotz functions and convex optimization
title Quasi-Herglotz functions and convex optimization
title_full Quasi-Herglotz functions and convex optimization
title_fullStr Quasi-Herglotz functions and convex optimization
title_full_unstemmed Quasi-Herglotz functions and convex optimization
title_short Quasi-Herglotz functions and convex optimization
title_sort quasi-herglotz functions and convex optimization
topic Mathematics
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7029951/
https://www.ncbi.nlm.nih.gov/pubmed/32218971
http://dx.doi.org/10.1098/rsos.191541
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