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arXiv · 1109.3805

On the existence of universal series by trigonometric system

Sergo A. Episkoposian
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Abstract

In this paper we prove the following: let $ω(t)$ be a continuous function, increasing in $[0,\infty)$ and $ω(+0)=0$. Then there exists a series of the form$\sum_{k=-\infty}^\infty C_ke^{ikx}$ with $\sum_{k=-\infty}^\infty C^2_k ω(|C_k|)<\infty$, $C_{-k}=\bar{C}_k$, with the following property: for each $ε>0$ a weighted function $μ(x), 0<μ(x) \le1,| \{x\in[0,2π]: μ(x)\not =1 \}| <ε$ can be constructed, so that the series is universal in the weighted space $L_μ^1[0,2π]$ with respect to rearrangements.

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Sergo A. Episkoposian. 2011-09-17. On the existence of universal series by trigonometric system. https://arxiv.org/abs/1109.3805

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