Related papers: Abel universal functions: boundary behaviour and T…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…

Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $\Delta$. It is well-known that every complex-valued harmonic function in the unit disc $\Delta$ can be uniquely…

Let ${\mathcal A}$ be the class of functions analytic in the unit disk ${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class $\mathcal{U}(\lambda)$,…

We prove simultaneous Universal Approximation of a certain type of Pade Approximants and of Taylor series with the same indexes. This is a generic phenomenon in the space of holomorphic functions in any simply connected domain, as well as…

Inner functions are the backbone of holomorphic function theory. This paper studies the inner functions on quotient domains of the open unit polydisc, $\bD^d$, arising from the group action of finite pseudo-reflection groups. Such quotient…

We establish generic existence of Universal Taylor Series on products $\Omega = \prod \Omega_i$ of planar simply connected domains $\Omega_i$ where the universal approximation holds on products $K$ of planar compact sets with connected…

We describe a generalization of the notion of a Hilbert space model of a function $\varphi$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $\varphi$ at a mild…

The universal approximation theorem, in one of its most general versions, says that if we consider only continuous activation functions $\sigma$, then a standard feedforward neural network with one hidden layer is able to approximate any…

We study in detail the one-variable local theory of functions holomorphic over a finite-dimensional commutative associative unital $\mathbb{C}$-algebra $\mathcal{A}$, showing that it shares a multitude of features with the classical…

We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the…

We investigate asymptotic polynomial approximation for a class of weighted Bloch functions in the unit disc. Our main result is a structural theorem on asymptotic polynomial approximation in the unit disc, in the flavor of the classical…

A holomorphic function f on a simply connected domain $\Omega$ belongs to a subclass of universal Taylor series if prescribed and infinite number of partial sums of the Taylor expansion of f around a given center $\zeta_0$ realize…

Let $\mathcal{H}$ be the space of all functions that are analytic in $\mathbb{D}$. Let $\mathcal{A}$ denote the family of all functions $f\in\mathcal{H}$ and normalized by the conditions $f(0)=0=f'(0)-1$. Obradovi\'{c} and Ponnusamy have…

The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc $\mathbb{D}$ under a holomorphic function $f$ (such that $f(0)=0$ and $f'(0)=1$)…

In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk $\mathbb{U}$. Further, these results are extended to a…

The main goal of this article is to bring together the theories of holomorphic iteration in the unit disc and semigroups of holomorphic functions. We develop a technique that allows us to partially embed the orbit of a holomorphic self-map…

Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$…

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…

Let $\mathbb{U}$ be a Banach Lie group and $S\subseteq \mathbb{U}$ an ad-bounded subset thereof, in the sense that there is a uniform bound on the adjoint operators induced by elements of $S$ on the Lie algebra of $\mathbb{U}$. We prove…

A toral algebraic set $A$ is an algebraic set in $\C^n$ whose intersection with $\T^n$ is sufficiently large to determine the holomorphic functions on $A$. We develop the theory of these sets, and give a number of applications to function…