Thierry Meyrath

Universal Rational Expansions of Meromorphic Functions

Motivated by known results about universal Taylor series, we show that every function meromorphic on a domain G can be expanded into a series of rational functions, whose partial sums have universal approximation properties on arbitrary compact sets K ⊂ Gc.

Universal meromorphic approximation on Vitushkin sets

AbstractThe paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λn} ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {nk} of ℕ such that

Limit functions of discrete dynamical systems

\left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\}

An assessment of degree-2 Stokes coefficients from Earth rotation data

converges to f(z) uniformly on K.A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λn}={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3].

Mixing Taylor shifts and universal Taylor series

Abstract We show that for functions that are universal in the sense of Voronin’s theorem, some derived functions automatically share a similar universality property. In particular, this holds for the Riemann zeta-function ζ and we are thus able to state universal functions of the form F ( ζ ) .

Universality properties of Taylor series inside the domain of holomorphy

In the theory of dynamical systems, the notion of ω-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in the case of topologically mixing systems on appropriate metric spaces (X, d), the existence of at least one limit function on a compact subset A of X implies the existence of plenty of them on many supersets of A. On the other hand, such sets necessarily have to be small in various respects. The results for general discrete systems are applied in the case of Julia sets of rational functions and in particular in the case of the existence of Siegel disks. 1. Limit functions on small sets Let (X, d) be a complete metric space and let f : X → X be continuous. If f◦n := f ◦ · · · ◦ f denotes the n-th iterate of f and if L is an arbitrary subset of X we write Ωp(L, f) for the collection of all functions g : L → X that are pointwise limits of some subsequence of (f)n on L. Necessarily, such functions have to be of Baire class 1 (cf. [8, p. 192]). Moreover, let K(X) := {E ⊂ X : E nonempty and compact} . For E ∈ K(X), the set of continuous functions from E to X is denoted by C(E,X). We endow C(E,X) with the (complete) uniform metric du,E(f, g) := sup x∈E d(f(x), g(x)) and define Ωu(E, f) to be the set of all functions that are limits of some subsequence of (f)n in C(E,X). We recall some definitions from topological dynamics. A continuous function f : X → X is called topologically transitive if for all nonempty open sets U, V in X, an integer n exists which satisfies f◦n(U) ∩ V = ∅. If this holds true for all sufficiently large n, then f is called topologically mixing. Finally, f is said to be topologically weak-mixing, if f × f is topologically transitive on the product space X ×X. For basic results on topological transitivity and topological (weak-) mixing we refer to [7]. In particular, if (X, d) is separable without isolated points, the Birkhoff transitivity theorem implies that f is topologically weak-mixing if and only if there is a pair (x1, x2) ∈ X×X so that the orbit {(f ×f)(x1, x2) : n ∈ N} is dense in X × X. Thus, f is topologically weak-mixing if and only if there is a Received by the editors May 22, 2013 and, in revised form, November 20, 2013, December 30, 2013, and December 31, 2013. 2010 Mathematics Subject Classification. Primary 37A25, 37F10, 30K99.

GRACE era variability in the Earth's oblateness: a comparison of estimates from six different sources

We consider two classes of meromorphic functions, which have universal approximation properties with respect to translations, and prove that both are residual subsets of the space of all meromorphic functions. Furthermore, we show that the two classes do not coincide.