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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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