Theory Of Holomorphic Functions Research Articles

Explicit Multi-slit Loewner Flows and Their Geometry

AbstractIn this paper we present explicit solutions to the radial and chordal Loewner PDEs and we make an extensive study of their geometry. Specifically, we study multi-slit Loewner flows, driven by the time-dependent point masses $$\mu _{t}:=\sum _{j=1}^{n}b_j \delta _{\{\zeta _j e^{iat}\}}$$ μ t : = ∑ j = 1 n b j δ { ζ j e iat } in the radial case and $$\nu _t:=\sum _{j=1}^{n}b_j\delta _{\{k_j\sqrt{1-t}\}}$$ ν t : = ∑ j = 1 n b j δ { k j 1 - t } in the chordal case, where all the above parameters are chosen arbitrarily. Furthermore, we investigate their close connection to the semigroup theory of holomorphic functions, which also allows us to map the chordal case to the radial one.

Herglotz's representation and Carathéodory's approximation

AbstractHerglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain.

Function theory in the bfd-norm on an elliptical region

Let E be the open region in the complex plane bounded by an ellipse. The B. and F. Delyon norm ‖⋅‖bfd on the space Hol(E) of holomorphic functions on E is defined by‖f‖bfd=defsupT∈Fbfd(E)⁡‖f(T)‖, where Fbfd(E) is the class of operators T such that the closure of the numerical range of T is contained in E. The name of the norm recognizes a celebrated theorem of the brothers Delyon, which implies that ‖⋅‖bfd is equivalent to the supremum norm ‖⋅‖∞ on Hol(E).The purpose of this paper is to develop the theory of holomorphic functions of bfd-norm less than or equal to one on E. To do so we shall employ a remarkable connection between the bfd norm on Hol(E) and the supremum norm ‖⋅‖∞ on the space H∞(G) of bounded holomorphic functions on the symmetrized bidisc, the domain G in C2 defined byG=def{(z+w,zw):|z|<1,|w|<1}. It transpires that there exists a holomorphic embedding τ:E→G having the property that, for any bounded holomorphic function f on E,‖f‖bfd=inf⁡{‖F‖∞:F∈H∞(G),F∘τ=f}, and moreover, the infimum is attained at some F∈H∞(G). This result allows us to derive, for holomorphic functions of bfd-norm at most one on E, analogs of the well-known model and realization formulae for Schur-class functions. We also give a second derivation of these models and realizations, which exploits the Zhukovskii mapping from an annulus onto E.

ON POSITIVE CONTINUOUS FUNCTIONS DEFINED IN THE UNIT POLYDISC

In theory of holomorphic functions having bounded L-index in a direction b an auxiliary class of positive continuous functions L is important to describe properties of the holomorphic functions by some inequalities andestimates containing the function L. This class is defined by local behavior of the function L. In the simplest one-dimensional case, the functionshould not vary locally too quickly, i.e. L(r+O(1/L(r))) = O(L(r)) for r = |z| → +∞. The paper is devoted an analog of this function class for the unit polydisc, i.e. for the Cartesian product of the unit discs. There is proved an equivalence of three different approaches to define the class. It is described by the local behavior on the slice z+tb for given z fromthe unit polydisc and for a fixed direction b, where the complex variable t belongs to some disc with radius dependent on b and z. These estimates must be fulfilled uniformly in all z. There is indicated a possible explicitform of functions belonging to the class. The form is given as a product of arbitrary positive continuous function defined in the closed unit polydisc and the minimum of the expressions 1/(1−|z j|) in all variables z j.

A free boundary problem of elasticity in an angle and a vector Riemann–Hilbert problem on a torus

An inverse problem of elasticity of reconstructing the shape of an inclusion placed in a wedge, when the whole inclusion is subjected to a prescribed antiplane uniform stress, is considered. The doubly connected physical domain is conformally mapped onto a parametric plane cut along two finite segments on the real axis. Determination of such a map requires solving a vector Riemann–Hilbert problem of the theory of holomorphic functions on a symmetric genus-1 Riemann surface. The main steps of the method include factorization of a discontinuous function on a torus, solution of the associated Jacobi inversion problem, representation of the unknown vector-functions in terms of Weierstrass integrals, and reduction of the vector Riemann–Hilbert problem to a single integral kernel of the second kind with a discontinuous kernel on the elliptic surface. An integral representation of the conformal map in terms of the solution to the integral equation is given. In addition to four dimensionless problem parameters, the conformal map possesses five free parameters. Numerical results, which reconstruct the inclusion shape for sample sets of the parameters, are reported and discussed.

Extension theorem and representation formula in non-axially-symmetric domains for slice regular functions

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of f(q)=\sum_{n\in\mathbb{N}}(q-p)^{*n} a_n , given by a \sigma -ball \Sigma(p,r) , is not open in \mathbb{H} unless p\in\mathbb{R} . This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along these lines, we can establish an extension theorem and a representation formula in a slice domain.

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ^*-algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on {mathbb {R}}^{2^n}. One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set.

Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain $C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. In addition, we discuss the theory of holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results.

Singularities of bicomplex holomorphic functions

In the theory of bicomplex holomorphic functions, there is not concept of isolated singularities; that is, such functions do not have singularities just at a point like holomorphic functions in one complex variable. However, there are other type of singularities that behave similarly to the isolated singularities in one complex variable. In this work, we describe how they can be classified in such a way that it resembles the classification made for the complex analysis case. It turns out that to singularities there corresponds their orders which are hyperbolic numbers with integer components, not real integers. We give also the Residue Theorem in the bicomplex analysis setting.

Slice quaternionic analysis in two variables

Slice quaternionic analysis in two variables is a generalization of the theory of several complex variables to quaternions. This study relies on the theory of stem functions and the theory of holomorphic functions in two complex variables. Our approach is to introduce holomorphicity for stem functions in terms of two commutative complex structures. It turns out that, locally, a function which is slice regular corresponds exactly to the Taylor series of two ordered quaternions, with the coefficients on the right. The Hartogs phenomenon holds in our setting; however, its proof is subtle due to some topological obstacles. We overcome them by showing that holomorphic stem functions preserve the property of being intrinsic after extension.

The Cauchy Integral Formula in Hermitian, Quaternionic and $$\mathfrak {osp}(4|2)$$ Clifford Analysis

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as to how hermitian Clifford analysis in euclidean space $${\mathbb {R}}^{2n}$$ of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on $${\mathbb {R}}^{2n}$$ , quaternionic Clifford analysis arose as a further refinement by introducing a so-called hypercomplex structure $${\mathbb {Q}}$$ , i.e. three complex structures ( $${\mathbb {I}}$$ , $${\mathbb {J}}$$ , $${\mathbb {K}}$$ ) which follow the quaternionic multiplication rules, on $${\mathbb {R}}^{4p}$$ , the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Cauchy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy integral formulae for $$\mathfrak {osp}(4|2)$$ -monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.

Toeplitz Operators on the Symmetrized Bidisc

Abstract The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

The current paper proposes a parametric eigenvalue problem for the Helmholtz equation on the plane based on two well-known physical models of emission from two-dimensional (2-D) microcavity lasers. The first model has been developed for passive, either lossy or lossless, open cavities and is usually referred to as the Complex-Frequency Eigenvalue Problem. The second model, named the Lasing Eigenvalue Problem (LEP), has been tailored to characterize emission from open cavities, filled in with gain material, on the threshold of nonattenuating in time radiation. Our generalized model contains both of them as special cases. We reduce the original problem to a nonlinear eigenvalue problem for a set of boundary integral equations with weakly singular kernels and formulate it as a parametric eigenvalue problem for a holomorphic Fredholm operator-valued function, which is convenient for mathematical and numerical analysis. Using this formulation and fundamental results of the theory of holomorphic operator-valued functions, we study the properties of the spectrum. For numerical solution of the problem, we propose a Nyström method, prove its convergence, and derive error estimates in the eigenvalue approximation. Combining this discretization with the residual inverse iteration technique, we compute approximate solutions of LEP and compare them with the exact ones and with the results obtained using other numerical methods.

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.

On generalized Vietoris’ number sequences

Recently, by using methods of hypercomplex function theory, the authors have shown that a certain sequence S of rational numbers (Vietoris’ sequence) combines seemingly disperse subjects in real, complex and hypercomplex analysis. This sequence appeared for the first time in a theorem by Vietoris (1958) with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/Salinas, 2004). A non-standard application of Clifford algebra tools for defining Clifford-holomorphic sequences of Appell polynomials was the hypercomplex context in which a one-parametric generalization S(n),n≥1, of S (corresponding to n=2) surprisingly showed up. Without relying on hypercomplex methods this paper demonstrates how purely real methods also lead to S(n). For arbitrary n≥1 the generating function is determined and for n=2 a particular case of a recurrence relation similar to that known for Catalan numbers is proved.

Quaternionic &lt;i&gt;G&lt;/i&gt;-Monogenic Mappings in &lt;i&gt;E&lt;sub&gt;m&lt;/sub&gt;&lt;/i&gt;

We consider a class of so-called quaternionic G-monogenic mappings associatedwith m-dimensional (m 2 f2; 3; 4g) partial differential equations and propose a description of allmappings from this class by using four analytic functions of complex variable. For G-monogenicmappings we generalize some analogues of classical integral theorems of the holomorphic functiontheory of the complex variable (the surface and the curvilinear Cauchy integral theorems,the Cauchy integral formula, the Morera theorem), and Taylor’s and Laurent’s expansions.Moreover, we investigated the relation between G-monogenic and H-monogenic (differentiablein the sense of Hausdorff) quaternionic mappings.

Free Holomorphic Functions on the Noncommutative Polydomains and Universal Models

In this paper, we study free holomorphic functions on the noncommutative polydomains \(\mathbf{D_f^m}(\mathcal {H})\), which were recently introduced by Popescu (J Funct Anal 265(10):2500–2552, 2013; Adv Math 279:104–158, 2015; Trans Am Math 368:4357–4416, 2016). We obtain some characterizations of free holomorphic functions by noncommutative Berezin transforms and the universal models associated with the abstract noncommutative polydomains \(\mathbf{D_f^m}\). As consequences, we provide noncommutative multivariable analogues of several results from the classical holomorphic function theory such as: Wiener inequality, Weierstrass theorem, Montel theorem, Vitali theorem. Moreover, we also prove that the universal models are essentially unique.

Free Holomorphic Functions on the Regular Polyball

In this paper, we study free holomorphic functions on the regular polyball, which was recently introduced by Popescu (Adv Math 279:104–158, 2015, Trans Am Math Soc 368:4357–4416, 2016). The purpose of this paper is to continue the line of Popescu to develop a theory of free holomorphic functions. Some results from the classical holomorphic function theory have free analogues in this noncommutative setting. In particular, we prove the maximum principle, Weierstrass and Montel type theorems for free holomorphic functions. As an application, we construct a metric on $$\mathrm{Hol}(\mathbf{B_n}(\mathcal {H}))$$ such that it becomes a complete space.

Poincaré–Bertrand and Hilbert formulas for the Cauchy–Cimmino singular integrals

The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincare–Bertrand formula related to the Cauchy–Cimmino singular integrals over piecewise Lyapunov surfaces in $$\mathbb {R}^4$$ is derived with recourse to arguments involving quaternionic analysis. Furthermore, this paper obtains some analogues of the Hilbert formulas on the unit 3-sphere and on the 3-dimensional space for the theory of Cimmino system.

Generalized integral theorems for quaternionic G-monogenic mappings

For G-monogenic mappings taking values in the algebra of complex quaternions, we generalize some analogues of classical integral theorems of the holomorphic function theory of complex variable (the surface and curvilinear Cauchy integral theorems).