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  • 1. Ivanenko, Yevhen
    et al.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Gustafsson, Mats
    Jonsson, B. L. G.
    Luger, Annemarie
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Nordebo, Sven
    Quasi-Herglotz functions and convex optimization2018Rapport (Annet vitenskapelig)
    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.

  • 2.
    Luger, Annemarie
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    A characterization of Herglotz–Nevanlinna functions in two variables via integral representations2017Inngår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 55, nr 1, s. 199-216Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 3.
    Luger, Annemarie
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    An integral representation for Herglotz-Nevanlinna functions in several variables2017Rapport (Annet vitenskapelig)
    Abstract [en]

    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.

  • 4.
    Luger, Annemarie
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Geometric properties of measures related to holomorphic functions having positive imaginary or real partManuskript (preprint) (Annet vitenskapelig)
    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.

  • 5.
    Luger, Annemarie
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Herglotz-Nevanlinna functions in several variablesManuskript (preprint) (Annet vitenskapelig)
    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.

  • 6.
    Luger, Annemarie
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Herglotz-Nevanlinna functions in several variables2019Inngår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 472, nr 1, s. 1189-1219Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 7.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    A subclass of boundary measures and the convex combination problem for Herglotz-Nevanlinna functions in several variablesManuskript (preprint) (Annet vitenskapelig)
    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.

  • 8.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Integral representations of Herglotz-Nevanlinna functions2017Licentiatavhandling, monografi (Annet vitenskapelig)
    Abstract [en]

    In this thesis, we study integral representations of Herglotz-Nevanlinna functions, that is to say holomorphic functions defined on a product of several copies of the complex upper half-plane having non-negative imaginary part. The manuscript is divided into three parts, beginning with a general introduction followed by two papers.

    In the general introduction, we familiarize ourselves with the concept of a Herglotz-Nevanlinna function as well as providing a comprehensive introduction into the theory of integral representations for this particular class of functions.

    Paper I treats exclusively the two-variable case and presents an integral representation of Herglotz-Nevanlinna functions in two complex variables in terms of a real number, two non-negative numbers and a positive Borel measure satisfying two properties. Three properties that hold for the class of measures appearing in such integral representations are also proven.

    In Paper II, we provide an integral representation for the class of Herglotz-Nevanlinna functions in arbitrarily many complex variables in terms of a real number, a linear term and a positive Borel measure satisfying two properties. Properties of the class of measures appearing in this representation are then discussed in detail as well as alternative descriptions of said class. Finally, a symmetry formula satisfied by Herglotz-Nevanlinna functions is proved at the end.

  • 9.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    On Herglotz-Nevanlinna functions in several variables2019Doktoravhandling, med artikler (Annet vitenskapelig)
    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.

  • 10.
    Nedic, Mitja
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    On pseudo-passive causal operators of slow growthManuskript (preprint) (Annet vitenskapelig)
    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.

  • 11.
    Nedic, Mitja
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Ehrenborg, Casimir
    Ivanenko, Yevhen
    Ludvig-Osipov, Andrei
    Nordebo, Sven
    Luger, Annemarie
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Jonsson, B. L. G.
    Sjöberg, Daniel
    Gustafsson, Mats
    Herglotz functions and applications in electromagneticsManuskript (preprint) (Annet vitenskapelig)
    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. 

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