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An extension of the universal power series of Seleznev. (English) Zbl 1451.30114

In this very interesting paper the authors expand a classical result due to A. I. Seleznev [Mat. Sb., N. Ser. 28(70), 453–460 (1951; Zbl 0043.29501)]. In particular they prove the generic existence of power series \(\sum_{n=0}^{+\infty}a_nz^n\), such that the partial sums \(\sum_{n=0}^{N}b_n(a_0,a_1,\ldots,a_n)z^n, \ N\in\mathbb{N}\), approximate all polynomials on every compact subset in \(\mathbb{C}\setminus\{0\}\) with connected complement. The functions \(b_n: \mathbb{C}^{n+1}\to\mathbb{C}\) are assumed to be continuous and the functions \(b_n(a_0,a_1,\ldots, a_{n-1}, \cdot):\mathbb{C}\to \mathbb{C}\) are assumed to be onto \(\mathbb{C}\) for every choice of \(a_0,a_1\ldots,a_{n-1}\in\mathbb{C}\). In order to prove the result, the authors follow methods used in Universal Taylor Series. Seleznev’s result corresponds with case \(b_n(a_0,a_1,\ldots,a_n)=a_n, \ n\in\mathbb{N}\). Another interesting result, connected with Cesàro summability, is the case where \(b_n(a_0,a_1,\ldots,a_n)=\dfrac{a_0+a_1+\ldots+a_n}{n+1}, \ n\in\mathbb{N}.\)

MSC:

30K05 Universal Taylor series in one complex variable
30E10 Approximation in the complex plane

Citations:

Zbl 0043.29501
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References:

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[3] Luh, W., Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen, 88, 1-56 (1970) · Zbl 0231.30005
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[5] Seleznev, AI, On universal power series’ (Russian), Mat. Sbornik (N. S.), 28, 453-460 (1951)
[6] Vlachou, V., On some classes of universal functions, Analysis, 22, 149-161 (2002) · Zbl 1001.30004
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