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I'm currently reading a paper by Olevskii on almost integer translates: https://www.sciencedirect.com/science/article/pii/S0764444297878731

In this paper the author considers for a given sequence $\{ \lambda _n \} \subset \mathbb R$ the following two statements:

  1. For any positive $u \in C(\mathbb R)$, there exists a measurable function $w$ such that $0< \omega \leq u$ and such that the system of exponentials $\{ e^{i \lambda _n x} \}$ is complete in the weighted space $L^2_\omega(\mathbb R)$
  2. The system of exponentials $\{ e^{i \lambda _n x} \}$ is complete with respect to convergence almost everywhere in $\mathbb R$

It is claimed that $(1)\implies (2)$ is trivial. However, I don't see how to prove it. Maybe I'm missing something here. Does anyone of you can provide a formal proof of this implication?

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  • $\begingroup$ Does complete mean having a dense linear span? Every $L_\omega^2(\mathbb R)$ convergent sequence has an almost everywhere convergent subsequence. $\endgroup$
    – Jochen Wengenroth
    Commented Sep 27, 2022 at 7:29

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