Vassili Nestoridis

On various types of universal taylor series

We denote by SN (f, ζ)(z) the Nth partial sum of the Taylor development of a holomorphic function f on a domain Ω with center ζ ε Ω. Let Ω be a Jordan domain with rectifiable boundary, for example the unit disc. We prove the existence of a holomorphic function f εH(Ω) with the following property: For every compact set with and K c connected and every function h: , continuous on K and holomorphic in K 0, there exists a sequence λ n ε {0,1,2,…}n =1,2,…, such that . We also show that the set of functions fεH(Ω) with the previous property is contained in A ∞(Ω) and is a Gρ-dense subset under the natural topology of A ∞(Ω).

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 Taylor Series in Simply Connected Domains

Let Ω be a simply connected domain in ℂ For a function f holomorphic in Ω let Sn (f, ζ) denote the partial sum of the Taylor development of f with center ζ ∈ Ω. We show that generically overconvergence phenomena of Sn(f, ζ) and their derivatives can occur on the boundary ∂Ω or in parts of it. In the rest of the boundary ∂Ω, the universal function f may be smooth. These parts of the boundary do not have to be connected.

Limit points of partial sums of Taylor series

Fifty years ago Marcinkiewicz and Zygmund studied the circular structure of the limit points of the partial sums for (C, 1) summable Taylor series. More specifically, let be a power series with complex coefficients, let be the partial sums, and let be the Cesaro averages. When the sequence σ n (z) converges to a finite limit σ(Z) , we say that the Taylor series is (C, 1) summable and σ(z) is the (C, 1) sum of the series. Concerning (C, 1) summable Taylor series Marcinkiewicz and Zygmund ([5], [6] Vol. II, p. 178) established the following theorem.

An extension of the disc algebra, II

We identify the complex plane with the open unit disc by the homeomorphism . This leads to a compactification of , homeomorphic to . The Euclidean metric on induces a metric on . We identify all uniform limits of polynomials on with respect to the metric . The class of the above limits is an extension of the disc algebra and it is denoted by . We study properties of the elements of and topological properties of the class endowed with its natural topology. The class is different and, from the geometric point of view, richer than the class introduced, on the basis of the chordal metric .

Universality and ultradifferentiable functions: Fekete’s theorem

The purpose of this article is to establish extensions of Feketes Theorem concerning the existence of universal power series of functions defined by estimates on successive derivatives.

Determination of a Universal Series

The known proofs for universal Taylor series do not determine a specific universal Taylor series. In the present paper, we isolate a specific universal Taylor series by modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific universal Taylor series on the disc or on a polygonal domain. Furthermore in non simply connected domains, when universal Taylor series exist, we can construct a sequence of specific rational functions converging to a universal function, provided the boundary is good enough. The solution uses an infinite denumerable procedure and a finite number of steps is not sufficient. However we solve a Runge’s type problem in a finite number of steps.

Universal approximation by translates of fundamental solutions of elliptic equations

Abstract In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of –Ω, known as the pseudo-boundary. The domains under consideration satisfy a rather mild boundary regularity requirement, namely, the segment condition. We study approximations with respect to the norms of the spaces Cℓ(–Ω)and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Hölder spaces Cℓ,ν(–Ω). The sequence a = {an}n ∈ ℕ of coefficients of the universal series may be chosen in ∩ p > 1lp(ℕ) but it can not be chosen in l1(ℕ).

Universal Laurent series on domains of infinite connectivity

We establish the existence of universal Laurent series on some domains of infinite connectivity. This phenomenon is topologically and algebraically generic.

On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions

Abstract We consider Bergman spaces and variations of them on domains in one or several complex variables. For certain domains , we show that the generic function in these spaces is totally unbounded in and hence non-extendable. We also show that generically these functions do not belong – not even locally – to Bergman spaces of higher order. Finally, in certain domains , we give examples of bounded non-extendable holomorphic functions – a generic result in the spaces of holomorphic functions in whose derivatives of order extend continuously to .