Related papers: Universal Taylor series on specific compact sets

We give necessary and sufficient condition so that we have d-hypercyclicity for operators who map a holomorphic function to a partial sum of the Taylor expansion. This problem is connected with doubly universal Taylors series and this is an…

We consider a formal power series in one variable whose coefficients are holomorphic functions in a given multidimensional complex domain. Assume the following two conditions on the series. (C1) The restriction of the series at each point…

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U…

Let $A$ be an algebra of bounded smooth functions on the interior of a compact set in the plane. We study the following problem: if $f,f_1,\dots,f_n\in A$ satisfy $|f|\leq \sum_{j=1}^n |f_j|$, does there exist $g_j\in A$ and a constant…

Let $K$ be a compact set in $\mathbb{C}$, $f$ a function analytic in $\overline{\mathbb{C}}\smallsetminus K$ vanishing at $\infty $. Let $% f\left( z\right) =\sum_{k=0}^{\infty }a_{k}\ z^{-k-1}$ be its Taylor expansion at $\infty $, and…

We prove that, for every rational $d\ne 0,\pm 1$ and every compact set $K\subset\{s\in\mathbb{C}:1/2<\Re(s)<1\}$ with connected complement, any analytic non-vanishing functions $f_1,f_2$ on $K$ can be approximated, uniformly on $K$, by the…

This text surveys cohomological properties of pairs $(U,f)$ consisting of a smooth complex quasi-projective variety $U$ together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function…

It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality…

A correspondence between arbitrary Fourier series and certain analytic functions on the unit disk of the complex plane is established. The expression of the Fourier coefficients is derived from the structure of complex analysis. The…

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete…

We identify all uniform limits of polynomials on the closed unit disc with respect to the chordal metric \c{hi} . One such limit is f=oo. The other limits are holomorphic functions f:-->C so that for every {\zeta} in the boundary of unit…

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all…

We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $\varphi$ be a holomorphic…

The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of…

A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that…

Let $D$ be a closed disk in the complex plane centered at the origin, $f, g$ complex valued continuous function on $D$. Let $P[f,g; D]$ (res. $R[f, g; D])$) be the uniform closure on $D$ of polynomials (res. rational functions) in variables…

Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint…

We consider "Taylor domination" property for an analytic function $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \]…

The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which…

We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form $\sum_{k=0}^n a_k x_{n,k}$ for given sequences of vectors $(x_{n,k})_{n\geq k\geq 0}$ in a topological vector…