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Herglotz functions and applications in electromagnetics
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0001-7867-5874
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

Herglotz functions inevitably appear in pure mathematics, mathematical physics, and engineering with a wide range of applications. In particular, they are the pertinent functions to model passive systems, and thus appear in modeling of electromagnetic phenomena in circuits, antennas, materials, and scattering. In this chapter, we review the basic theory of Herglotz functions and its applications to determine sum rules and physical bounds for passive systems. 

National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering Other Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-162345OAI: oai:DiVA.org:su-162345DiVA, id: diva2:1265707
Funder
Swedish Foundation for Strategic Research , AM13-0011Available from: 2018-11-26 Created: 2018-11-26 Last updated: 2018-11-27Bibliographically approved
In thesis
1. On Herglotz-Nevanlinna functions in several variables
Open this publication in new window or tab >>On Herglotz-Nevanlinna functions in several variables
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we investigate different aspects of the class of Herglotz-Nevanlinna functions in several variables. These are holomorphic functions on the poly-upper half-plane having non-negative imaginary part. Our results are presented in the four research articles A1 - A4, which are included in this thesis.

Articles A1 and A2 establish a characterization of Herglotz-Nevanlinna functions in terms of an integral representation formula. The case of functions of two complex variables is presented in article A1, while the general case is treated in article A2, where different symmetry properties of Herglotz-Nevanlinna functions are also discussed.

Article A3 discusses, in detail, the convex combination problem for Herglotz-Nevanlinna functions. This problem asks us to relate the representing parameters of different Herglotz-Nevanlinna functions under the assumption that these functions are related in a very particular way involving the convex combination of several independent variables. A related class of boundary measures is also discussed.

Article A4 investigates the properties of Nevanlinna measures with respect to restrictions to coordinate orthogonal hyperplanes and the geometry of the support. A related class of measures on the unit poly-torus is also considered.

Furthermore, this thesis is supplemented by three additional publications concerning Herglotz-Nevanlinna functions in one variable, related topics and applications.

Article B1 concerns a particular class of convolution operators on the space of distributions that generalizes the well-studied class of passive operators. Article B2 introduces the class of quasi-Herglotz functions and discusses their integral representations, boundary values and sum-rules, as well as their applications in connection with convex optimization. Finally, the summary book-chapter C1 provides a general overview of the applications of Herglotz-Nevanlinna functions in electromagnetics.

Abstract [sv]

I denna avhandling undersöker vi de olika aspekterna av klassen av Herglotz-Nevanlinnafunktioner i flera variabler. Dessa är holomorfa funktioner definierade i det polyövre halvplanet som har en icke-negativ imaginärdel. Våra resultat presenteras i de fyra vetenskapliga artiklarna A1 – A4 som inkluderas i avhandlingen.

Artiklar A1 och A2 sätter upp en karakterisering av Herglotz-Nevnlinnafunktioner genom en integralframställning. Fallet där vi behandlar funktioner av två komplexa variabler presenteras i artikeln A1 medan det generella fallet diskuteras i artikeln A2. Där diskuteras även egenskaper av den symmetriska utvidgningen av Herglotz-Nevnlinnafunktioner.

I artikeln A3 presenteras en detaljerad undersökning av det konvexa kombinationsproblemet för Herglotz-Nevanlinnafunktioner. Detta problem frågar efter ett samband mellan representationsparametrar av olika Herglotz-Nevanlinnafunktioner under antagandet att dessa funktioner kan förknippas med varandra genom en konvex kombination av flera oberoende variabler. En relaterad klass av randmått undersöks också.

Artikeln A4 undersöker egenskaper av Nevanlinnamått med avseende på deras restriktioner till koordinatortogonala hyperplan och geometrin av stödet. En relaterad klass av mått i enhets polytorusen diskuteras också.

Vidare kompletteras avhandlingen med ytterligare tre publikationer som handlar om Herglotz-Nevanlinnafunktioner i en variabler, relaterade ämnena och deras tillämpningar.

Artikeln B1 behandlar en viss klass av faltningsoperatorer definierade på rummet av distributioner som generalisera den välkända klassen av passiva operatorer. Artikeln B2 introducerar klassen av quasi-Herglotzfunktioner och presenterar deras integralframställningar, randvärdena och summaregler så som deras tillämpningar i sambandet med konvex optimering. Till slut förser sammanfattningsbokkapitel C1 oss en överblick över tillämpningar av Herglotz-Nevanlinnafunktioner inom elektromagnetisk teori.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2019. p. 15
Keywords
Herglotz-Nevanlinna functions, several complex variables, integral representations, Nevanlinna measures, Herglotz-Nevanlinnafunktioner, flera komplexa variabler, integralframställningar, Nevanlinnamått
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162354 (URN)978-91-7797-510-6 (ISBN)978-91-7797-511-3 (ISBN)
Public defence
2019-01-25, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Foundation for Strategic Research , AM13-0011
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 7: Manuscript.

Available from: 2018-12-19 Created: 2018-11-26 Last updated: 2018-12-06Bibliographically approved

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