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Convergence to zero of exponential sums with positive integer coefficients and approximation by sums of shifts of a single function on the line. (English) Zbl 1413.42002
This paper solves two problems arising from a result by the first author P. A. Borodin, [Izv. Math. 78, No. 6, 1079–1104 (2014; Zbl 1316.46015); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 6, 21–48 (2014)]. The main results are
Theorem 1. There is a sequence of trigonometric polynomials \[ Q_{\nu}(x)=\sum_{s=1}^{s_{\nu}}\,n_s^{(\nu)}e^{ik_s^{(\nu)}x} \tag{(1)} \] converging to zero almost everywhere; \(k_s^{(\nu)}\) are (distinct) integers and \(n_s^{(\nu)}\) are positive integers.
Theorem 2. There is a function \(f:\mathbb{R}\rightarrow\mathbb{R}\) such that the sums \[ \sum_{k=1}^n\,f(x-a_k),\;a_k\in\mathbb{R},\;n\in\mathbb{N} \tag{(2)} \] of its shifts are dense in all real spaces \(L_p(\mathbb{R})\) for \(2\leq p<\infty\) and also in the real space \(C_0(\mathbb{R})\) of continuous functions tending to zero at \(\pm\infty\).
The layout of the paper is as follows:
1. Introduction: Definitions, historical information and the main tool for the proof of Theorem 1 given in Lemma 1:
For any \(\delta>0\) there is a sum \[ f_{\delta}(x)=\sum_{j=1}^J\, g(m_jx) \] with positive integers \(m_1,\ldots,m_J\) and a set \(E\subset\mathbb{T}\), such that \(\text{mes}\,E\geq 2\pi -\delta\) and \(f_{\delta}(x)>\delta\) for \(x\in E\). (\(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z};\;\text{mes}\) is the Lebesgue measure; \(g(x)=\ln{| 1-e^{ix}|}\))
2. Deduction of Theorem from Lemma 1.
The paper concludes with a list of remarks and problems.

MSC:
42A05 Trigonometric polynomials, inequalities, extremal problems
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
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