Wolfgang Luh

On the existence of O-universal functions

In this article we consider functions φ which are holomorphic exactly on a domain and whose power series are universal with respect to overconvergence. Our main purpose is to solve a problem of Nestoridis. In addition some properties of universal power series are proved.

Approximation by antiderivatives

Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ(−n)} ofn-fold antiderivatives (i.e., we haveφ(0)(z)∶=φ(z) andφ(−n)(z)=dφ(−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold:(1)For any compact setB ⊂Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {nk} such that {φ−nk} converges tof uniformly onB.(2)For any open setU ⊂Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {mk} such that {φ−mk} converges tof compactly onU.(3)For any measurable setE ⊂Ω and any functionf that is measurable onE, there exists a sequence {pk} such that {φ(-Pk)} converges tof almost everywhere onE.

Universal functions are automatically universal in the sense of Menchoff

We prove that universal approximation (uniform approximation on compact subsets with connected complement) implies almost everywhere approximation in the sense of Menchoff with respect to any given σ-finite Borel measure on (d≥2).

Universal Transforms of the Geometric Series under Generalized Riesz Methods

In this paper generalized Riesz methods (R, p, M) of summability are considered. We prove that, to each open set O ⊂ ℂ with adequate topological properties and to each sequence {n} ⊂ ℂ tending to infinity, we can associate a corresponding P-regular (R, p, M)-method so that the geometric series and a certain trigonometric series become universal in the sense that its (R, p, M)-transforms approximate any member of certain spaces of holomorphic functions or measurable functions.

Universal entire functions with gap power series

Abstract Let M be the family of all compact sets in C which have connected complement. For K ϵ M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior. Suppose that {zn} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of N 0. If Q has density Δ(Q) = 1 then there exists a universal entire function ϑ with lacunary power series 1. (1) ϑ(z) = ϵ∞v = 0 ϑvZv, ϑv = 0 for v ∉ Q, which has for all K ϵ M the following properties simultaneously 2. (2) the sequence {ϑ(Z + Zn)} is dense in A(K) 3. (3) the sequence {ϑ (ZZn)} is dense in A(K) if 0 ∉ K. Also a converse result is proved: If ϑ is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δmax(Q) = 1.

Entire functions with various universal properties

Let M be the family of all compact sets in which have connected complements. For K∊ M let A(K) be the Banach space of all functions which are continuous on K and holomorphic in its interior. If φ is an entire function then we denote by φ(j) its derivative of order j if and a “normalized” antiderivative of order −j . Suppose that {z n} is an unbounded sequence in We prove the existence of an entire function φ, which has for all K ∊ M the following properties simultaneously: for any fixed the sequence is dense in A(K); for any fixed the sequence is dense in A(K) if 0 ∉ K; the sequence of derivatives is dense in A(K).

On the localization of singularities of Lacunar power series

The main result of this paper is Theorem 4 on the existence of singularities of Lacunary power series on prescribed open boundary arcs of the circle of convergence. The influence of the gaps on the length of these arcs is expressed in terms of newly introduced integral densities. Theorem 4 contains the known Fabry–Pólya theorem on gaps, describing the closed arcs having singularities, and suggests its extensions, using additional information on the coefficients of the power series. An essential step in its proof is Theorem 2, providing necessary and sufficient conditions (in terms of the so called “Coefficient Functions”) for the analytic continuation of power series across fixed open boundary arcs.

UNIVERSALITY AND SUMMABILITY OF TRIGONOMETRIC POLYNOMIALS AND TRIGONOMETRIC SERIES

In 1945 Mensov proved the existence of a so-called universal trigonometric series ∞ ν=0 {aν cos νt+ bν sin νt}

Summierbarkeit der geometrischen Reihe auf vorgeschriebenen Mengen

AbstractIn this paper we are concerned with the summability of the geometric series