Concurrent universal Padé approximation (Q1996322)

The authors consider holomorphic functions \(f\in H(\Omega)\) where \(\Omega\subseteq\mathbb{C}\) is a simply connected domain. Let \(\zeta\in\Omega\) and \(\mu\) be an infinite subset of \(\mathbb{N}\), the holomorphic function \(f\) is called \textit{universal Taylor series} if for every compact set \(K\in\mathbb{C}\setminus\Omega\) with connected complement and every function \(h\in A(K)\) there exists a sequence \((\lambda_n)_{n\in\mathbb{N}}\subseteq\mu\) satisfying (i) \(\sup_{z\in K}\,|S_{\lambda_n}(f,\zeta)(z)-h(z)|\rightarrow 0\hbox{ as }n\rightarrow\infty,\) (ii) \(\sup_{z\in J}\,|S_{\lambda_n}(f,\zeta)(z)-f(z)|\rightarrow 0\hbox{ as }n\rightarrow\infty\) for every compact set \(J\subseteq\Omega.\) Here the \(S_N(f,\zeta)\) are the partial sums of the Taylor expansion of \(f\) centered at \(\zeta\in\Omega\). They then introduce the recent approach that replaces the partial sums by some rational functions: the well-known Padé approximants of \(f\). If \(f(z)=\sum_{n=0}^{\infty}\,a_n (z-\zeta)^n\) is a formal power series with center \(\zeta\in\Omega\) and \(p,q\in\mathbb{N}\), then the \((p,q)-\)Padé approximant with center \(\zeta\in\Omega\) is a rational function \([f;p/q]_{\zeta}(z)=A(z)/B(z)\), where the polynomials \(A,B\) satisfy \(\hbox{deg}\,A(z)\leq p,\hbox{deg}\,B(z)\leq q, B(\zeta)=1\) and the Taylor expansion of the function \(A(z)/B(z)=\sum_{n=0}^{\infty}\,b_n(z-\zeta)^n\) satisfies \(a_n=b_n\) for \(n\leq p+q\). If this approximant exists, the authors introduce two different types of \textit{universal Padé approximants}. It is outside the usual length of a review to state the two definitions explicitly, but `grosso modo' it boils down to -- Type I: replace the partial sums of the Taylor expansion of the function \(f\) by its Padé approximant in the definition given above (see [\textit{N. Daras} and \textit{V. Nestoridis}, ``Universal Padé approximation'', Preprint, \url{arXiv:1102.4782}; \textit{N. Daras} et al., ``Universal Padé approximants on simply connected domains'', Preprint, \url{arXiv:1501.02381})]), -- Type II: replace the absolute value used in the defition by the \textit{chordal metric} of the absolute value in the formula with the Padé approximant (see [\textit{V. Nestoridis}, J. Contemp. Math. Anal., Armen. Acad. Sci. 47, No. 4, 168--181 (2012; Zbl 1302.41021)]). The reader should be aware that the terminology `type I' and `type II' is not the same as used in the main part of the literature: there it is connected with rational approximation. The layout of the paper is as follows: \S1. Introduction (2\ 1/2 pages) Also states the main results in Theorem 1.1 (= Theorem 4.1 from \S4) and Theorem 1.2 (= Theorem 3.5 in \S3). \S2. Preliminaries (3 pages) \S3. Arbitrary open sets (6\ 1/2 pages) \S4. Simply connected domains (6 pages) \S5. Affine genericity of a class of functions (5 pages) References (20 items) To show the form of the main results, one of the theorems will be given below: Theorem 1.1. Let \(\Omega\subseteq\mathbb{C}\) be a simply connected domain and \(L\subseteq\Omega\) a compact set. We consider a sequence \((p_n)_{n\geq 1}\) with \(p_n\rightarrow\infty\). Now, for every \(n\in\mathbb{N}\) let \(q_1^{(n)}, q_2^{(n)},\ldots, q_{N(n)}^{(n)}\in\mathbb{N}\), where \(N(n)\) is another natural number. Then there exists a function \(f\in H(\Omega)\) satisfying the following. For every compact set \(K\subseteq\mathbb{C}\setminus\Omega\) with connected complement and for every function \(h\in A(K)\), there exists a subsequence \((p_{k_n})_{n\geq 1}\) of the sequence \((p_n)_{n\geq 1}\) such that (1) \(f\in D_{p_{k_n},q_j^{k_n}}(\zeta)\) for every \(\zeta\in L\), for every \(n\in\mathbb{N}\) and for every \(j\in\{1,\ldots,N(k_n)\}\). (2) \( \max_{j=1,\ldots,N(k_n)}\,\sup_{\zeta\in L}\,\sup_{z\in K}\, |[f;p_{k_n}/q_j^{(k_n)}]_{\zeta}(z)-h(z)|\rightarrow n \hbox{ as }n\rightarrow\infty.\) (3) For every compact set \(J\subseteq\Omega\) it holds \[ \max_{j=1,\ldots,N(k_n)}\,\sup_{\zeta\in J}\,\sup_{z\in K}\, |[f;p_{k_n}/q_j^{(k_n)}]_{\zeta}(z)-f(z)|\rightarrow n \hbox{ as }n\rightarrow\infty.\] Moreover, the set of all functions \(f\) satisfying properties (1)--(3) is \(G_{\delta}\) -- dense in \(H(\Omega)\).