Herglotz Functions Research Articles

On quasi-Herglotz functions in one variable

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 and sum-rule identities.

Extended higher Herglotz function II

Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for ζ(2m+1). We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.

Analytic Continuation of Concrete Realizations and the McCarthy Champagne Conjecture

Abstract In this paper, we give formulas that allow one to move between transfer function type realizations of multi-variate Schur, Herglotz, and Pick functions, without adding additional singularities except perhaps poles coming from the conformal transformation itself. In the two-variable commutative case, we use a canonical de Branges–Rovnyak model theory to obtain concrete realizations that analytically continue through the boundary for inner functions that are rational in one of the variables (so-called quasi-rational functions). We then establish a positive solution to McCarthy’s Champagne conjecture for local to global matrix monotonicity in the settings of both two-variable quasi-rational functions and $d$-variable perspective functions.

The analytical structure of acoustic and elastic material properties

In this paper, we take an in-depth look at the analytical structure of the material transfer functions which govern acoustic and elastic response. These include wavenumber (κ) in such media and refractive index (n′), density (ρ) and its inverse, stiffness (C) and compliance (D) tensors as well as the Bulk modulus (B), and finally the broader generalization of these properties which is now known as the Willis tensor (L). Our goal is to clarify the appropriate dispersion relations applicable to these properties from the perspective of passivity. Under some mild assumptions, causality ensures that these properties are analytical in the upper half but deriving dispersion relations for them requires one to know how they behave in the limit |ω|→∞. Unlike electromagnetism, such a determination cannot be made on physical grounds since in that limit the continuum approximation breaks down. Instead, we can exploit the properties of the Herglotz–Nevanlinna (H–N) functions along with their tensorial counterparts which characterize the transfer functions of certain passive systems and for which the appropriate dispersion relation is known. Our aim, therefore, is to clarify the relationship that these transfer functions have with Herglotz functions, which in turn determines the appropriate dispersion relation for them. Our analysis shows that based upon passivity alone, dispersion relations of minimum order 1 apply to the Fourier transforms of D,ρ,n′, and the inverse of B, order 3 apply to C,B, and the inverse of ρ, and order 2 applies to κ.

The reciprocity gap operator for electromagnetic scattering in chiral media

We define a reciprocity gap functional for electromagnetic scattering in chiral media. We assume that a perfectly conducting scatterer is embedded in a piecewise homogeneous isotropic background chiral medium. The incident field is a spherical electromagnetic wave due to a point source located at a point contained in a chiral environment. We also define the chiral reciprocity gap operator and we prove that it is injective and it has dense range. Using these results and appropriate density properties of chiral Herglotz functions we solve an inverse scattering problem for the determination of the boundary of the scatterer from the knowledge of the tangential components of the electric and magnetic fields.

On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications

This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than π. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in this paper include the results in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632 as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as its interior angle is not π when the conductive transmission eigenfunctions satisfy certain Herglotz functions approximation properties. That means, as long as the corner singularity is not degenerate, the vanishing property holds if the underlying conductive transmission eigenfunctions can be approximated by a sequence of Herglotz functions under mild approximation rates. Third, the regularity requirements on the interior transmission eigenfunctions in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632 are significantly relaxed in the present study for the conductive transmission eigenfunctions. In order to establish the geometric properties for the conductive transmission eigenfunctions, we develop technically new methods and the corresponding analysis is much more complicated than that in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632. Finally, as an interesting and practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter.

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 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.

Herglotz functions: The Fueter variables case

In this paper we provide realizations of Herglotz functions in the quaternionic case where hyperholomorphic functions are meant in the sense of Fueter. We consider the counterparts of the Arveson space and of the Herglotz–Agler functions by introducing suitable positive kernels. Crucial tools are the so-called Fueter variables and the CK-product. Our realization results also show that positivity implies analyticity.

Herglotz functions of several quaternionic variables

We first review realizations of Herglotz functions in the unit ball of CN and provide new insights. Then, we define the corresponding class and prove the extend the results in the case of several quaternionic variables.

Representation of free Herglotz functions

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.

On the uniqueness of inverse spectral problems associated with incomplete spectral data

The inverse spectral problem for a Sturm–Liouville problem which consists of a Sturm–Liouville equation defined on the interval [0,a] and two separated boundary conditions at the endpoints 0 and a, one of which is involved function f(λ), is considered in this paper. When f(λ) is the type of Herglotz functions and is known as a priori, we prove that the potential on [0,a] and the other boundary condition can be uniquely determined in terms of appropriate partial information on the spectrum and partial information on the set of normalizing constants.

Bounds on Herglotz functions and fundamental limits of broadband passive quasistatic cloaking

Using a sum rule, we derive new bounds on Herglotz functions that generalize those given in Bernland et al. [J. Phys. A: Math. Theor. 44(14), 145205 (2011)] and Gustafsson and Sjöberg [New J. Phys. 12(4), 043046 (2010)]. These bounds apply to a wide class of linear passive systems such as electromagnetic passive materials. Among these bounds, we describe the optimal ones and also discuss their meaning in various physical situations like in the case of a transparency window, where we exhibit sharp bounds. Then, we apply these bounds in the context of broadband passive cloaking in the quasistatic regime to refute the following challenging question: is it possible to construct a passive cloaking device that cloaks an object over a whole frequency band? Our rigorous approach, although limited to quasistatics, gives quantitative limitations on the cloaking effect over a finite frequency range by providing inequalities on the polarizability tensor associated with the cloaking device. We emphasize that our results hold for a cloak or object of any geometrical shape.

Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ R n and Solutions of the Helmholtz Equation

The purpose of this article is to extend to \(\mathbb {R}^{n}\) known results in dimension 2 concerning the structure of a Hilbert space with reproducing kernel of the space of Herglotz wave functions. These functions are the solutions of Helmholtz equation in \(\mathbb {R} ^{n}\) that are the Fourier transform of measures supported in the unit sphere with density in \(L^{2}(\mathbb {S}^{n-1})\). As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in \(\mathbb {R}^{n}\) belonging to weighted Sobolev type spaces \(\mathcal {H}^{p}\) having in a non local norm that involves radial derivatives and spherical gradients. We calculate the reproducing kernel of the Herglotz wave functions and study in \(\mathcal {H}^{p}\) and in mixed norm spaces, the continuity of the orthogonal projection \(\mathcal {P}\) of \(\mathcal {H}^{2}\) onto the Herglotz wave functions.

Boolean convolution in the quaternionic setting

In this paper we begin a study of free analysis in the quaternionic setting, and consider Boolean convolution for quaternion-valued measures. To this end we also study Boolean convolution for matrix-valued complex measures, also proving Boolean infinite divisibility and central limit theorems for these measures. Moreover we prove an integral representation for quaternionic Carathéodory and Herglotz functions.

Meromorphic Continuations of Finite Gap Herglotz Functions and Periodic Jacobi Matrices

We find a necessary and sufficient condition for a Herglotz function m to be the Borel transform of the spectral measure of an exponential decaying perturbation of a periodic Jacobi matrix. The condition is in terms of meromorphic continuation of m to a natural Riemann surface and the structure of its zeros and poles.The analogous result is also established for the Borel transform of the spectral measure of eventually periodic Jacobi matrices.This paper generalizes the corresponding result from the author’s (Constr Approx 36(2):267–309, 2012) for exponential perturbations of the free Jacobi matrix.

A remark on the discriminant of Hill’s equation and Herglotz functions

We establish a link between the basic properties of the discriminant of periodic second-order differential equations and an elementary analysis of Herglotz functions. Some generalizations are presented using the language of self-adjoint extensions.

Asymptotic generalized value distribution of solutions of the Schrödinger equation

We develop the theory of generalized value distribution for the class of functions defined as boundary values of Herglotz functions. One of the main results is an estimate of asymptotic generalized value distribution for Herglotz functions translated by an increment iδ off the real axis. We also present precise relations between the employed measures, and the link with compositions of Herglotz functions. A case of particular interest which we study is that of the Weyl-Titchmarsh m-function associated with Sturm-Liouville differential operators. In this context, this generalized theory allows for the possibility to describe the asymptotic value distribution of solutions of the Schrodinger equation in terms of other measures than Lebesgue measure.

Optimal Realizations of Passive Structures

This paper presents a convex optimization approach to study optimal realizations of passive electromagnetic structures. The optimization approach complements recently developed theory and techniques to derive sum rules and physical limitations for passive systems operating over a given bandwidth. The sum rules are based solely on the analytical properties of the corresponding Herglotz functions. However, the application of sum rules is limited by certain assumptions regarding the low- and high-frequency asymptotic behavior of the system, and the sum rules typically do not give much information towards an optimal realization of the passive system at hand. In contrast, the corresponding convex optimization problem is formulated to explicitly generate a Herglotz function as an optimal realization of the passive structure. The procedure does not require any additional assumptions on the low- and high frequency asymptotic behavior, but additional convex constraints can straightforwardly be incorporated in the formulation. Typical application areas are concerned with antennas, periodic structures, material responses, scattering, absorption, reflection, and extinction. In this paper, we consider three concrete examples regarding dispersion compensation for waveguides, passive metamaterials and passive radar absorbers.

A general scattering theorem for chiral Herglotz functions

A chiral electric Herglotz pair is an incident upon a scatterer that is a perfect conductor. First, we define appropriate tangential subspaces of that satisfy the conditions of the left circularly and right circularly polarization, and we prove that they are complete and orthogonal spaces. Then, we define integral operators on these spaces that compose the chiral far‐field operator. We prove a general scattering theorem that connects chiral Herglotz functions with the chiral far‐field operator, which covers known results in the achiral case. Copyright © 2013 John Wiley & Sons, Ltd.

Optical Theorem and Forward Scattering Sum Rule for Periodic Structures

Based on energy conservation, an optical theorem is constructed for a slab having an arbitrary periodic microstructure in a plane. A sum rule for low pass structures is derived using analytic properties of Herglotz functions based on causality and passivity. The sum rule relates the total cross section to the static polarizability per unit cell, and quantifies the interaction between the slab and electromagnetic fields possible over all wavelengths. The results are illustrated with several numerical and experimental examples.