The theory of universal Taylor series can be extended to the case of Padé approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the ...Padé approximants of some power series. We present the first generic result in this direction, for Padé approximants corresponding to Taylor developments of holomorphic functions in simply connected domains. The universal approximation is required only on compact sets \(K\) which lie outside the domain of definition and have connected complement. If the sets \(K\) are additionally disjoint from the boundary of the domain of definition, then the universal functions can be smooth on the boundary.
Universal series in ∩ p>1 p Koumandos, S; Nestoridis, V; Smyrlis, Y.-S ...
The Bulletin of the London Mathematical Society,
02/2010, Volume:
42, Issue:
1
Journal Article
In this paper an abstract condition is given yielding universal series defined by sequences
a
= {a
j
}∞
j=1 in ∩
p>1
p
but not in 1. We obtain a unification of some known results related to ...approximation by translates of specific functions including the Riemann -function, or a fundamental solution of a given elliptic operator in with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in simultaneously with respect to all -finite Borel measures in . Stronger results are obtained by using universal Dirichlet series.
Sur les séries de Taylor universelles Kahane, Jean-Pierre; Melas, Antonios; Nestoridis, Vassili
Comptes rendus de l'Académie des sciences. Série I. Mathématique,
6/2000, Volume:
330, Issue:
11
Journal Article
Les résultats concernent les séries trigonométriques au sens de Menchoff et les séries de Taylor universelles au sens de Nestoridis ou de Seleznev. Le théorème 1 dit qu'il existe des séries ...trigonométriques du type de Taylor, à coefficients tendant vers 0 , et qui sont universelles au sens de Menchoff. Les autres théorèmes traitent de l'approximation uniforme par une suite de sommes partielles d'une série de Taylor.
The authors considered universal trigonometric series in the sense of Mens̆ov and universal Taylor series in the sense of Nestoridis or Seleznev. Theorem 1 says that there exist trigonometric series of the Taylor type, with coefficients tending to zero, and universal in the sense of Mens̆ov. The other theorems are related to uniform approximation by a sequence of partial sums of a Taylor series.
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