Holomorphic Functions Research Articles

Large-N limit of the Segal–Bargmann transforms on the spheres

We study the large-N limit of the Segal–Bargmann transform on SN−1(N), the (N−1)-dimensional sphere of radius N, as a unitary map from the space of square-integrable functions with respect to the normalized spherical measure onto the space of holomorphic square-integrable functions with respect to a certain measure on the quadric. In particular, we give an explicit formulation and describe the geometric models for the limit of the domain, the limit of the range, and the limit of the transform when N tends to infinity. We show that the limiting transform is still a unitary map from the limiting domain onto the limiting range.

Certified Severi dimensions for hyperelliptic and supersymmetric cusps

In a previous paper, the first three authors formulated a precise conjecture about the dimension of the generalized Severi varietyMd,g;S,kn of degree-d holomorphic maps P1→Pn whose images' singularities are singleton cusps with value semigroups S and ramification profiles k. In this paper, we prove that an adjusted form of the conjecture holds for generic profiles k associated with two distinguished (infinite) classes of semigroups S.

Properties of Bounded Holomorphic Functions: A Survey

We discuss interrelations between {mathcal {H}}^infty -convex domains and {mathcal {H}}^infty -domains of holomorphy for various classes of domains in {mathbb {C}}^n.

Two-dimensional integrable systems with position-dependent mass via complex holomorphic functions

We study the relationship between integrable systems with a position-dependent mass (PDM) and complex holomorphic functions and the potential applications of the latter to deduce the former. For a prescribed mass term the associated complex function is derived. The complex function and related plane transformation are used to generate the PDM systems of three integrable Hénon–Heiles systems and a Holt system as well. We also figure out a holomorphic function, which ensures separability of the corresponding PDM systems in the polar-like coordinates. The holomorphic function together with Jacobi method have yielded a variety of generalized separable systems. At last we put forward an example of a family of separable systems to show that not all PDM systems can be deduced through some holomorphic function.

New family of bi-univalent functions with respect to symmetric conjugate points associated with Borel distribution

Abstract In this paper, we introduce and investigate a new family, denoted by 𝒲Σ sc (λ, η, δ, r), of normalized holomorphic and bi-univalent functions with respect to symmetric conjugate points, defined in 𝕌, by making use the Borel distribution series, which is associated with the Horadam polynomials. We derive estimates on the initial Taylor-Maclaurin coefficients and solve the Fekete-Szeg˝o type inequalities for functions in this family.

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

Knots and Their Related $q$-Series

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.

N-Best kernel approximation in reproducing kernel Hilbert spaces

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy space and a recent result of n-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets

On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets

Sandwich Subordinations Imposed by New Generalized Koebe-Type Operator on Holomorphic Function Class

In the complex field, special functions are closely related to geometric holomorphic functions. Koebe function is a notable contribution to the study of the geometric function theory (GFT), which is a univalent function. This sequel introduces a new class that includes a more general Koebe function which is holomorphic in a complex domain. The purpose of this work is to present a new operator correlated with GFT. A new generalized Koebe operator is proposed in terms of the convolution principle. This Koebe operator refers to the generality of a prominent differential operator, namely the Ruscheweyh operator. Theoretical investigations in this effort lead to a number of implementations in the subordination function theory. The tight upper and lower bounds are discussed in the sense of subordinate structure. Consequently, the subordinate sandwich is acquired. Moreover, certain relevant specific cases are examined.

Arithmetic geometric model for the renormalisation of irrationally indifferent attractors

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points. The geometric model incorporates the fine arithmetic properties of the rotation number at the fixed point. Using this model for the renormalisation, we build a topological model for the dynamics of a holomorphic map near an irrationally indifferent fixed point. We also explain the topology of the maximal invariant set for the model, and also explain the dynamics of the map on the maximal invariant set.

The SL(2,R) Group Representations on Spaces of Holomorphic Functions on the Unit Disc

We can realise the representations of the group SL(2,R) on the unit disc. This is due to an isomorphism between the group SL(2,R) and the group SU(1,1). The discrete series representations for the group SL(2,R)given by\pi_{n}(g)\varphi(z)=\varphi (\frac{d z-b}{a-cz} )(a-c z)^{-n}, where n is an integer number,is on the Bergman space where n>2 .Lang studies the discrete series on the group in the upper half-plane and on the unit disc. For n=1, the SL(2,R) representation is called the mock discrete series. The representation space of the mock discrete series is the Hardy space.In this article we describe the SL(2,R) representation on the Dirichlet space.

Study on the Criteria for Starlikeness in Integral Operators Involving Bessel Functions

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present research focuses on the Bessel function of the first kind and order ν unveiling the conditions for this function to be univalent and further using its univalent form in order to define a new integral operator on the space of holomorphic functions. For particular values of the parameters implicated in the definition of the new integral operator involving the Bessel function of the first kind, the well-known Alexander, Libera, and Bernardi integral operators can be obtained. In the first part of the study, necessary and sufficient conditions are obtained for the Bessel function of the first kind and order ν to be a starlike function or starlike of order α∈[0,1). The renowned prolific method of differential subordination due to Sanford S. Miller and Petru T. Mocanu is employed in the reasoning. In the second part of the study, the outcome of the first part is applied in order to introduce the new integral operator involving the form of the Bessel function of the first kind, which is starlike. Further investigations disclose the necessary and sufficient conditions for this new integral operator to be starlike or starlike of order 12.

Normal families and quasiregular mappings

AbstractBeardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.

Boundedness and Compactness of Weighted Composition Operators from α-Bloch Spaces to Bers-Type Spaces on Generalized Hua Domains of the First Kind

We address weighted composition operators ψCϕ from α-Bloch spaces to Bers-type spaces of bounded holomorphic functions on Y, where Y is a generalized Hua domain of the first kind, and obtain some necessary and sufficient conditions for the boundedness and compactness of those operators.

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

We analyzed the classical problem of decomposing the Hilbert space of holomorphic functions, especially their splitting into the product or sum of domain-separated components. For the Bergman space of analytical functions, we obtained a special decomposition satisfying the assigned growth degree properties. Concerning a general Hilbert space of analytical functions on a connected domain, we studied its α-invariant decomposition and related ergodic consequences. As an interesting consequence, we obtained the decomposition theorem for an ergodic α-mapping on the Bergman space of holomorphic functions.

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

The General Theory of Superoscillations and Supershifts in Several Variables

In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators acting on holomorphic functions with growth conditions of exponential type. Additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for superoscillating sequences in several variables are extended to sequences of supershifts in several variables.

Inner functions as strongly extreme points: stability properties

Given a Banach space \(\mathcal X\), let \(x\) be a point in ball\((\mathcal X)\), the closed unit ball of \(\mathcal X\). We say that \(x\) is a strongly extreme point of ball\((\mathcal X)\) if it has the following property: for every \(\varepsilon>0\) there is \(\delta>0\) such that the inequalities \(\|x\pm y\|<1+\delta\) imply, for \(y\in\mathcal X\), that \(\|y\|<\varepsilon\). We are concerned with certain subspaces of \(H^\infty\), the space of bounded holomorphic functions on the disk, that arise upon imposing finitely many linear constraints and can be viewed as small perturbations of \(H^\infty\). It is well known that the strongly extreme points of ball\((H^\infty)\) are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions \(f\in H^\infty\) with \(\log(1-|f|)\) non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed \(H^\infty\)-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point.

Exceptional values of entire functions of finite order in one of the variables

Let F(z,w) be a holomorphic function in Cn×C of finite order in w with n≥2. Let Ω be the set of points z∈Cn where F(z,w) is a non-constant function omitting a value π(z). Near a finite accumulation point z0 of Ω, we prove in the main result (Theorem 1) that Ω is a local analytic set and π(z) is holomorphic, and show the existence of a proper globally analytic set Δ of Cn such that either Ω⊂Δ or Ω=Cn∖Δ, being possible in the last case to also determine F(z,w) in terms of π(z). We apply this result to several problems. First, we extend a Theorem due to Nishino about exceptional values when near z0 dimension of Ω is n and assure the existence of a meromorphic function α(z) in Cn such that π(z)=α(z) except at points where α(z) has poles or F(z,w) is constant (also being F(z,w) a polynomial in w if α(z) is ∞). After, we prove that Ω is a local analytic set in Cn and the existence of a proper analytic subset E of Cn such that Ω⊂E or Ω=Cn∖E. Finally, we generalize a Lelong-Gruman Theorem about the set of points z where π(z)=0.