Another extension of the disc algebra

We identify the complex plane C with the open unit disc D={z:|z|<1} by the homeomorphism z --> z/(1+|z|). This leads to a compactification $\bar{C}$ of C, homeomorphic to the closed unit disc. The Euclidean metric on the closed unit disc induces a metric d on $\bar{C}$. We identify all uniform limits of polynomials on $\bar{D}$ with respect to the metric d. The class of the above limits is an extension of the disc algebra and it is denoted by $\bar{A}(D)$. We study properties of the elements of $\bar{A}(D)$ and topological properties of the class $\bar{A}(D)$ endowed with its natural topology. The class $\bar{A}(D)$ is different and, from the geometric point of view, richer than the class $\tilde{A}(D)$ introduced in Nestoridis (2010), Arxiv:1009.5364, on the basis of the chordal metric.Comment: 14 page

Relations of the spaces Ap (Ω) and C p (∂Ω)

This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Springer.Let Ω be a Jordan domain in C, J an open arc of ∂Ω and φ : D → Ω a Riemann map from the open unit disk D onto Ω. Under certain assumptions on φ we prove that if a holomorphic function f ∈ H(Ω) extends continuously on Ω ∪ J and p ∈ {1, 2, . . . } ∪ {∞}, then the following equivalence holds: the derivatives f (l) , 1 ≤ l ≤ p, l ∈ N, extend continuously on Ω ∪ J if and only if the function f|J has continuous derivatives on J with respect to the position of orders l, 1 ≤ l ≤ p, l ∈ N. Moreover, we show that for the relevant function spaces, the topology induced by the l−derivatives on Ω, 0 ≤ l ≤ p, l ∈ N, coincides with the topology induced by the same derivatives taken with respect to the position on J

Relations of the Spaces Ap(Ω) and Cp(∂Ω)

Let Ω be a Jordan domain in C, J an open arc of ∂Ω and ϕ: D→ Ω a Riemann map from the open unit disk D onto Ω. Under certain assumptions on ϕ we prove that if a holomorphic function f∈ H(Ω) extends continuously on Ω∪ J and p∈ { 1 , 2 , ⋯ } ∪ { ∞} , then the following equivalence holds: the derivatives f(l), 1 ≤ l≤ p, l∈ N, extend continuously on Ω∪ J if and only if the function f| J has continuous derivatives on J with respect to the position of orders l, 1 ≤ l≤ p, l∈ N. Moreover, we show that for the relevant function spaces, the topology induced by the l-derivatives on Ω, 0 ≤ l≤ p, l∈ N, coincides with the topology induced by the same derivatives taken with respect to the position on J. © 2018, The Author(s)

Algebraic genericity of frequently universal harmonic functions on trees

We show that the set of frequently universal harmonic functions on a tree T contains a vector space except 0 which is dense in the space of harmonic functions on T seen as subset of CT. In order to prove this we replace the complex plane C by any separable Fréchet space E and we repeat all the theory. © 2020 Elsevier Inc

Generalized harmonic functions on trees: Universality and frequent universality

Recently, harmonic functions and frequently universal harmonic functions on a tree T have been studied, taking values on a separable Fréchet space E over the field C or R. In the present paper, we allow the functions to take values in a vector space E over a rather general field F. The metric of the separable topological vector space E is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in F. We don&apos;t assume that E is complete and therefore we present an argument avoiding Baire&apos;s theorem. © 2021 Elsevier Inc