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On Herglotz-Nevanlinna functions in several variables
Stockholm University, Faculty of Science, Department of Mathematics. (Analysis)ORCID iD: 0000-0001-7867-5874
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 [en]
Herglotz-Nevanlinna functions, several complex variables, integral representations, Nevanlinna measures
Keywords [sv]
Herglotz-Nevanlinnafunktioner, flera komplexa variabler, integralframställningar, Nevanlinnamått
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-162354ISBN: 978-91-7797-510-6 (print)ISBN: 978-91-7797-511-3 (electronic)OAI: oai:DiVA.org:su-162354DiVA, id: diva2:1265871
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
List of papers
1. A characterization of Herglotz–Nevanlinna functions in two variables via integral representations
Open this publication in new window or tab >>A characterization of Herglotz–Nevanlinna functions in two variables via integral representations
2017 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 55, no 1, p. 199-216Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
integral representation, Herglotz-Nevanlinna function, several complex variables
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-151762 (URN)10.4310/ARKIV.2017.v55.n1.a10 (DOI)000424524100010 ()
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-01-18 Created: 2018-01-18 Last updated: 2018-11-26Bibliographically approved
2. Herglotz-Nevanlinna functions in several variables
Open this publication in new window or tab >>Herglotz-Nevanlinna functions in several variables
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

Keywords
integral representations, several complex variables, Herglotz-Nevanlinna functions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162279 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-11-22 Created: 2018-11-22 Last updated: 2018-11-26Bibliographically approved
3. A subclass of boundary measures and the convex combination problem for Herglotz-Nevanlinna functions in several variables
Open this publication in new window or tab >>A subclass of boundary measures and the convex combination problem for Herglotz-Nevanlinna functions in several variables
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we begin by investigating a particular subclass of boundary measures of Herglotz-Nevanlinna functions in two variables. Based on this, we then proceed to solve the convex combination problem for Herglotz-Nevanlinna functions in several variables.

Keywords
integral representation, Herglotz-Nevanlinna function, several complex variables, convex combination
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162246 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-11-20 Created: 2018-11-20 Last updated: 2018-11-26Bibliographically approved
4. Geometric properties of measures related to holomorphic functions having positive imaginary or real part
Open this publication in new window or tab >>Geometric properties of measures related to holomorphic functions having positive imaginary or real part
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

Keywords
analytic functions, Nevanlinna measures, poly torus, measure with vanishing mixed Fourier coefficients, poly-upper half-plane
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162278 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-11-22 Created: 2018-11-22 Last updated: 2018-11-26Bibliographically approved
5. On pseudo-passive causal operators of slow growth
Open this publication in new window or tab >>On pseudo-passive causal operators of slow growth
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we study a class of convolution operators on the space of distributions that enlarge the well-studied class of passive operators. In this larger class, we are able to associate, to each operator, a holomorphic function in the right half-plane with a specific constraint on its range, determined by the operator. Afterwards, we investigate whether the properties of causality and slow growth hold automatically in our larger class of convolution operators. Finally, an alternative class of convolution operators is also considered.

Keywords
convolution operators, pseudo-passivity, causality, slow-growth, Laplace transform
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162251 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-11-20 Created: 2018-11-20 Last updated: 2018-11-26Bibliographically approved
6. Quasi-Herglotz functions and convex optimization
Open this publication in new window or tab >>Quasi-Herglotz functions and convex optimization
Show others...
2018 (English)Report (Other academic)
Abstract [en]

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.

Publisher
p. 24
Keywords
Quasi-Herglotz functions, non-passive systems, approximation, convex optimization, sum rules
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162353 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-11-26 Created: 2018-11-26 Last updated: 2019-01-20Bibliographically approved
7. Herglotz functions and applications in electromagnetics
Open this publication in new window or tab >>Herglotz functions and applications in electromagnetics
Show others...
(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:nbn:se:su:diva-162345 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2018-11-26 Created: 2018-11-26 Last updated: 2018-11-27Bibliographically approved

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Output format
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