Concurrent universal Padé approximation

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

“…The sets B(s, k, n, j) have been proven to be open in [20] for q (n) j ≥ 1 and in [29] for q (n) j = 0. We will now prove that each A(i, s, m, n, j) is an open set in H(Ω) for q (n) j ≥ 1 (see also [24]). We fix the parameters i, s, m, n ≥ 1 and j ∈ {1, • • • , N (n)} and we consider a function f ∈ A(i, s, m, n, j).…”

“…All results concerning Padé approximants are proved to be generic by using Baire's theorem and are derived mainly from [27]. Moreover, they can be altered in order to achieve simultaneous Padé -Taylor approximation with the same indices ( [24]).…”

“…All results concerning Padé approximants are proved to be generic by using Baire's theorem and are derived mainly from [27]. Moreover, they can be altered in order to achieve simultaneous Padé -Taylor approximation with the same indices ( [24]). However, many results of approximation presented in [27] are omitted, mainly those of Seleznev type (concerning formal power series) and those referring to the space X ∞ (Ω) which is a closed subspace of A ∞ (Ω) (where Ω ⊆ C is an open set).…”

“…Thus, X ∞ (Ω) is a closed subset of a complete metric space and therefore is a complete metric space itself. We refer to [27] and [37] for the further results and definitions.…”

“…Each chapter of this work is carefully presented so that the reader could study each chapter independently, apart, of course, from the necessary cross -references between chapters. Naturally, many results were omitted, although we present most of the results in [17], [23], [25] and [27] and we refer to these works (and also in [24]) for further results, examples and information in general.…”