| 1. |
- Ivanenko, Yevhen, et al.
(author)
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Quasi-Herglotz functions and convex optimization
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Other publication (other academic/artistic)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 of Herglotz functions 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.
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| 2. |
- Ivanenko, Yevhen, et al.
(author)
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Quasi-Herglotz functions and convex optimization
- 2020
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In: Royal Society Open Science. - : The Royal Society Publishing. - 2054-5703. ; 7:1, s. 1-15
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Journal article (peer-reviewed)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.
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| 3. |
- Ivanenko, Yevhen, et al.
(author)
-
Quasi-Herglotz functions and convex optimization
- 2018
-
Reports (other academic/artistic)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 of Herglotz functions 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.
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| 4. |
- Luger, Annemarie, et al.
(author)
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A characterization of Herglotz–Nevanlinna functions in two variables via integral representations
- 2017
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In: Arkiv för matematik. - 0004-2080 .- 1871-2487. ; 55:1, s. 199-216
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Journal article (peer-reviewed)abstract
- We derive an integral representation for Herglotz-Nevanlinna functions in two variables, which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the class of representing measures are also discussed.
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| 5. |
- Luger, Annemarie, et al.
(author)
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An integral representation for Herglotz-Nevanlinna functions in several variables
- 2017
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Reports (other academic/artistic)abstract
- In this article, a characterization of the class of Herglotz-Nevanlinna functions in n variables is given in terms of an integral representation. Furthermore, the properties of the class of representing measures are discussed in detail. Symmetry properties induced by the integral representation are also investigated.
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| 6. |
- Luger, Annemarie, et al.
(author)
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Geometric properties of measures related to holomorphic functions having positive imaginary or real part
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Other publication (other academic/artistic)abstract
- In this paper, we study the properties of a certain class of Borel measures on R^n that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures can not lie within particular geometric regions, \eg strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into the right half-plane.
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| 7. |
- Luger, Annemarie, et al.
(author)
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Geometric Properties of Measures Related to Holomorphic Functions Having Positive Imaginary or Real Part
- 2021
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 31:4, s. 2611-2638
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Journal article (peer-reviewed)abstract
- In this paper, we study the properties of a certain class of Borel measures on Rnthat arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures cannot lie within particular geometric regions, e.g., strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into the right half-plane.
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| 8. |
- Luger, Annemarie, et al.
(author)
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Herglotz-Nevanlinna functions in several variables
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Other publication (other academic/artistic)abstract
- In this article, a characterization of the class of Herglotz-Nevan\-linna functions in several variables is given in terms of an integral representation. The conditions on the representing measure are discussed in detail, and, furthermore, the properties of the symmetric extension of a Herglotz-Nevanlinna function are also investigated.
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| 9. |
- Luger, Annemarie, et al.
(author)
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Herglotz-Nevanlinna functions in several variables
- 2019
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 472:1, s. 1189-1219
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Journal article (peer-reviewed)abstract
- In this article, a characterization of the class of Herglotz-Nevanlinna functions in several variables is given in terms of an integral representation. The conditions on the representing measure are discussed in detail, and, furthermore, the properties of the symmetric extension of a Herglotz-Nevanlinna function are also investigated.
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| 10. |
- Luger, Annemarie, et al.
(author)
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On quasi-Herglotz functions in one variable : [Sur les fonctions quasi-Herglotz d’une variable]
- 2022
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In: Comptes rendus. Mathematique. - : Cellule MathDoc/CEDRAM. - 1631-073X .- 1778-3569. ; 360, s. 937-970
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Journal article (peer-reviewed)abstract
- In this paper, the class of (complex) quasi-Herglotz functions is introduced as the complex vector space generated by the convex cone of ordinary Herglotz functions. We prove characterization theorems, in particular, an analytic characterization. The subclasses of quasi-Herglotz functions that are identically zero in one half-plane as well as rational quasi-Herglotz functions are investigated in detail. Moreover, we relate to other areas such as weighted Hardy spaces, definitizable functions, the Cauchy transform on the unit circle, sum-rule identities and matrix-valued Herglotz functions.
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