Abstract theory of universal series and applications.

*(English)*Zbl 1147.30003Inspired by the fact that all – constructive or Baire-category – proofs of existence of universal series (of several kinds, such as Taylor series, Laurent series, Dirichlet series, Faber series, Fourier series, harmonic expansions, etc) share some similarities (in particular, they need to exhibit a function which approximates both a given function in the space where the universal function should live and a given function in the space where the universal property holds), the authors undertake the task of putting the theory of universal series in an appropriate abstract framework, so obtaining in a unified way most of the existing results as well as new and stronger results.

This is carried out by using, in each case, an appropriate approximation theorem: Mergelyan’s theorem, Runge’s theorem, Weierstrass’ theorem, DuBois-Reymond’s lemma, Walsh’s theorem, Lusin’s theorem, etc. In addition, dense lineability, that is, existence of large linear manifolds of universal series, as well as extensions to several dimensions, are obtained.

The following known results, among others, are proved or even improved as special cases of their abstract theory:

Theorem. Let \(X\) be a metrizable topological vector space over the field \(\mathbb K= \mathbb R\) or \(\mathbb C\) whose topology is induced by a translation-invariant metric \(\rho\). Let \(x_0,x_1,x_2, \dots\) be a fixed sequence of elements in \(X\). Fix a subspace \(A\) of the space \(\omega\) of all \(\mathbb K\)-valued sequences, and assume that \(A\) carries a complete metrizable vector space topology, induced by a translation-invariant metric \(d\). Suppose that the coordinate projections \(a = (a_n)_{n \geq 0} \mapsto a_m \in\mathbb K\) are continuous for any \(m \in\mathbb N\), and that the set \(\{a = (a_n)_{n \geq 0} \in \omega: \{n:a_n \neq 0\}\) is finite} is a dense subset of \(A\). Let \(\mu = (\mu_n)_{n \geq 0}\) be an increasing sequence of positive integers, and denote by \(U_A^\mu\) (with \(U_A := U_A^\mu\) when \(\mu =\mathbb N\)) the class of all sequences \(a = (a_n)_{n \geq 0} \in A\) such that, for every \(x \in X\), there exists a subsequence \((\lambda_n)\) of \(\mu\) satisfying \(\sum_{j=0}^{\lambda_n} a_j x_j \to x\) and \(\sum_{j=0}^{\lambda_n} a_j e_j \to a\) as \(n \to \infty\), where \((e_j)\) is the canonical basis of \(\omega\). Then the following are equivalent:

This is carried out by using, in each case, an appropriate approximation theorem: Mergelyan’s theorem, Runge’s theorem, Weierstrass’ theorem, DuBois-Reymond’s lemma, Walsh’s theorem, Lusin’s theorem, etc. In addition, dense lineability, that is, existence of large linear manifolds of universal series, as well as extensions to several dimensions, are obtained.

The following known results, among others, are proved or even improved as special cases of their abstract theory:

- {\(\bullet\)}
- M. Fekete’s theorem [C. R. Acad. Sci. 158, 1256–1258 (1914; JFM 45.0630.03)] on the existence of a power series \(\sum_{n=1}^\infty a_n x^n\) such that, for every continuous function \(f\) on \([-1,1]\) with \(f(0) = 0\), there exists a sequence \((\lambda_n) \subset\mathbb N= \{0,1,2,\dots\}\) such that \(\sum_{j=1}^{\lambda_n} a_j x^j \to f(x)\) uniformly on \([-1,1]\).
- {\(\bullet\)}
- D. Menshov’s theorem [ C. R. (Dokl.) Acad. Sci. URSS, New. Ser. 49, 79–82 (1945; Zbl 0060.18504)] on the existence of a trigonometric series \(\sum_{n=-\infty}^{+\infty} a_n e^{\text{int}}\) with \(a_n \to 0\) \((| n| \to \infty)\) such that every Lebesgue-measurable function \(f:\mathbb T\to\mathbb C\) (\(\mathbb T=\) the unit circle) is a limit almost everywhere of partial sums of this series. Some enhancements were obtained by Chui and Parnes, Luh, Grosse-Erdmann, Kahane, Melas, Nestoridis, Koumoulis, etc. All of them are covered by the results of the present paper.
- {\(\bullet\)}
- A. I. Seleznev’s theorem [Mat. Sb., N. Ser. 28 (70), 453–460 (1951; Zbl 0043.29501)] on the existence of a series \(\sum_{n=0}^\infty a_n z^n\) whose partial sums approximate every entire function in each compact subset \(K \subset\mathbb C\setminus \{0\}\) without holes.

- {\(\bullet\)}
- Corresponding universality theorems for multiply connected domains, via Taylor series, or Laurent series or Faber series, due to Gehlen, Luh, Vlachou, Costakis, Nestoridis, Papadoperakis, Kariofillis, Mouratides, Diamantopoulos, Müller, Yavrian, etc.
- {\(\bullet\)}
- Armitage’s theorem (2006) on the existence of an harmonic function \(f:\Omega \to\mathbb R\) – where \(\Omega\) is a domain in \(\mathbb R^N\), \(N \geq 2\), with \((\mathbb R^N \cup \{\infty\}) \setminus \overline{\Omega}\) connected – such that all its partial derivatives extend continuously to \(\overline{\Omega}\) and approximate any harmonic function \(\mathbb R^N \to\mathbb R\) on any compact set \(K \subset\mathbb R^N \setminus \overline{\Omega}\) without holes.
- {\(\bullet\)}
- F. Bayart’s theorem [Rev. Mat. Complut. 19, No. 1, 235–247 (2006; Zbl 1103.30003)] on the existence of universal Dirichlet series \(\sum_{n=1}^\infty a_n n^{-z}\).
- {\(\bullet\)}
- Costakis-Marias-Nestoridis’ theorem G. Costakis, M. Marias, V. Nestoridis, Analysis, München 26, No. 3, 401–409 (2006; Zbl 1148.41303)] on the existence of a real \(C^\infty\)-function on an open subset \(\Omega \subset\mathbb R^N\), whose partial sums \(S_n(f,x)\) of Taylor series at a point \(x \in \Omega\) approximate any \(C^\infty\)-function \(\mathbb R^N \to\mathbb R\) on some open set containing \(K\), where \(K \subset\mathbb R^N \setminus \Omega\) is a compact set.

Theorem. Let \(X\) be a metrizable topological vector space over the field \(\mathbb K= \mathbb R\) or \(\mathbb C\) whose topology is induced by a translation-invariant metric \(\rho\). Let \(x_0,x_1,x_2, \dots\) be a fixed sequence of elements in \(X\). Fix a subspace \(A\) of the space \(\omega\) of all \(\mathbb K\)-valued sequences, and assume that \(A\) carries a complete metrizable vector space topology, induced by a translation-invariant metric \(d\). Suppose that the coordinate projections \(a = (a_n)_{n \geq 0} \mapsto a_m \in\mathbb K\) are continuous for any \(m \in\mathbb N\), and that the set \(\{a = (a_n)_{n \geq 0} \in \omega: \{n:a_n \neq 0\}\) is finite} is a dense subset of \(A\). Let \(\mu = (\mu_n)_{n \geq 0}\) be an increasing sequence of positive integers, and denote by \(U_A^\mu\) (with \(U_A := U_A^\mu\) when \(\mu =\mathbb N\)) the class of all sequences \(a = (a_n)_{n \geq 0} \in A\) such that, for every \(x \in X\), there exists a subsequence \((\lambda_n)\) of \(\mu\) satisfying \(\sum_{j=0}^{\lambda_n} a_j x_j \to x\) and \(\sum_{j=0}^{\lambda_n} a_j e_j \to a\) as \(n \to \infty\), where \((e_j)\) is the canonical basis of \(\omega\). Then the following are equivalent:

- (1)
- \(U_A \neq \emptyset\).
- (2)
- For every \(p \in\mathbb N\), \(x \in X\) and \(\varepsilon > 0\), there exist \(n \geq p\) and \(a_p,a_{p+1}, \dots a_n \in\mathbb K\) such that \(\rho (\sum_{j=p}^n a_jx_j,x) < \varepsilon\) and \(d(\sum_{j=p}^n a_je_j,0) < \varepsilon\).
- (3)
- For every \(x \in X\) and \(\varepsilon > 0\), there exist \(n \geq 0\) and \(a_0,a_1, \dots a_n \in\mathbb K\) such that \(\rho (\sum_{j=0}^n a_j x_j,x) < \varepsilon\) and \(d(\sum_{j=0}^n a_je_j,0) < \varepsilon\).
- (4)
- For every increasing sequence \(\mu\) of positive integers, \(U_A^\mu\) is a dense \(G_\delta\) subset of \(A\).
- (5)
- For every increasing sequence \(\mu\) of positive integers, \(U_A^\mu \cup \{0\}\) contains a dense subspace of \(A\).

Reviewer: Luis Bernal González (Sevilla)