Makridis, K.; Nestoridis, V. Concurrent universal Padé approximation. (English) Zbl 1460.41008 J. Math. Anal. Appl. 497, No. 2, Article ID 124911, 25 p. (2021). 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 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 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 [N. Daras and V. Nestoridis, “Universal Padé approximation”, Preprint, arXiv:1102.4782; N. Daras et al., “Universal Padé approximants on simply connected domains”, Preprint, arXiv:1501.02381)]),– Type II: replace the absolute value used in the defition by the chordal metric of the absolute value in the formula with the Padé approximant (see [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:§1. Introduction (21/2 pages)Also states the main results in Theorem 1.1 (= Theorem 4.1 from §4) and Theorem 1.2 (=Theorem 3.5 in §3).§2. Preliminaries (3 pages)§3. Arbitrary open sets (61/2 pages)§4. Simply connected domains (6 pages)§5. 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 \(H(\Omega)\). Reviewer: Marcel G. de Bruin (Heemstede) MSC: 41A21 Padé approximation 30K05 Universal Taylor series in one complex variable Keywords:power series; holomorphic function; Padé approximant; Baire’s theorem; chordal distance; affine genericity Citations:Zbl 1302.41021 PDFBibTeX XMLCite \textit{K. Makridis} and \textit{V. Nestoridis}, J. Math. Anal. Appl. 497, No. 2, Article ID 124911, 25 p. (2021; Zbl 1460.41008) Full Text: DOI arXiv References: [1] Baker, G. A.; Graves-Morris, P. R., Padé Approximants (1996), Cambridge University Press · Zbl 0923.41001 [2] Charpentier, S.; Mouze, A., Universal Taylor series and summability, Rev. Mat. Complut., 28, 153-167 (2015) · Zbl 1322.30020 [3] Charpentier, S.; Nestoridis, V.; Wielonsky, F., Generic properties of Padé approximants and Padé universal series, Math. Z., 281, 1-2, 427-455 (2015) · Zbl 1323.41019 [4] Chui, C.; Parnes, M. N., Approximation by overconvergence of power series, J. Math. Anal. 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