Related papers: Analyticity of functions analytic on circles

The aim of this paper is to give an algebraic characterization of the rings $C(X,\mathbb{Q}_p)$ of all continuous $\mathbb{Q}_p$-valued functions on a compact space $X$. The characterization is similar to that of M. Stone from 1940 for the…

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded…

Let $D, G\subset{\Bbb C}$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup(A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then…

The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues…

We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex…

We describe dynamical properties of a map $\mathfrak{F}$ defined on the space of rational functions. The fixed points of $\mathfrak{F}$ are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.

In this paper we study the function algebra generated by z^2 and g^2 on a small closed disk centered at the origin of the complex plane. We prove, using a biholomorphic change of coordinates and already developed techniques in this area,…

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…

In this paper we investigate the global behavior of proper holomorphic maps f from the unit disc U={|z|<1} to C^2. The fact that U is transcendental imposes certain restrictions on the image f(U). For instance, f(U) cannot be contained in…

We determine the boundedness and compactness of a large class of operators, mapping from general Banach spaces of holomorphic functions into a particular type of spaces of functions determined by the growth of the functions, or the growth…

Generalized analytic functions over generalized analytic manifolds are build from sums of convergent real power series with non-negative real exponents (and some well-ordering condition on the support). In a paper by Mart\'in-Villaverde,…

For an arbitrary function f:\Omega \rightarrow C (where \Omega is a subset of the field C) and a positive integer k let f act on all diagonalizable complex matrices whose all eigenvalues lie in Omega in the following way: f[P…

Let $f = f(z,t)$ be a function holomorphic in $z \in O \subseteq {\mathbb C}^d$ for fixed $t\in \Omega$ and measurable in $t$ for fixed $z$ and such that$z \mapsto f(z,\cdot)$ is bounded with values in$E := L_{p}(\Omega)$, $1\le p \le…

It is proved an inequality - integrated analogue of the Hardy inequality and as application simplified proof of the theorem of S. A. Vinogradov for the bounded Toeplitz operators on the space of functions analytic and bounded in the unit…

In this text we prove that if X is a reduced non-archimedean analytic space and f is a analytic function on a dense Zariski-open subspace of X whose zero-locus is closed in X, then f is a meromorphic function on X. As a corollary, we deduce…

If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if…

We investigate the existence, and lack of unicity, of a holomorphic fibration by discs transversal to a rational curve in a complex surface.

The zero set of the hyperbolic Gaussian analytic function is a random point process in the unit disc whose distribution is invariant under automorphisms of the disc. We study the variance of the number of points in a disc of increasing…

Examples by Poletsky and the author and by Zwonek show the existence nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions and contains…

We prove that the image of a finely holomorphic map on a fine domain in $\mathbb{C}$ is pluripolar subset of $\mathbb{C}^{n}$. We also discuss the relationship between pluripolar hulls and finely holomorphic function.