Generic properties of Padé approximants and Padé universal series

Abstract

Abstract. We establish properties concerning the distribution of poles of Padé approximants, which are generic in Baire category sense. We also investigate Padé universal series, an analog of classical universal series, where Taylor partial sums are replaced with Padé approximants. In particular, we complement previous studies on this subject by exhibiting dense or closed infinite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Padé universal series. We also show how Padé universal series can be built from classical universal series with large Ostrowski-gaps.

Applications of Baire’s category theorem in complex analysis in one and several complex variables

“…We now present the main result of this section. Its proof is s close adaptation to that of Theorem 4.17 in [9]. Theorem 4.42.…”

“…Lemma 4.41 (Similar to Lemma 4.4 of [9]; slightly extended, [9]). Let Ω ⊆ C be a simply connected domain and a ∈ C\Ω, ζ ∈ Ω be two fixed elements.…”

“…If such a rational function exists, then its irreducible form is unique, as it is well known (see for example [9]) and it is called the (p, q) -Padé approximant of f (at ζ).…”

“…Then, the class A is spaceable in H(Ω); that is, it contains a closed subspace of infinite dimension (except 0). Definition 4.35 (Basic sequence of a Fréchet space, [9]). Let X be a Fréchet space over a field K and {u n } n≥0 ⊆ X.…”

“…Constant of basicity,[9]). In the previous lemma (Lemma 4.36), the infimum of the constant B satisfying Properties (4.158) and (4.159) is called constant of basicity of the sequence {u n } n≥0 .Lemma 4.38 (Lemma 4.10 of [9], Lemme 2.…”