#### Vol. 2, No. 4, 2020

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Trigonometric series with a given spectrum

### Yves Meyer

Vol. 2 (2020), No. 4, 881–906
DOI: 10.2140/tunis.2020.2.881
##### Abstract

Let $\Lambda \subset {ℝ}^{n}$ be a closed and discrete set. The vector space consisting of all trigonometric sums whose frequencies belong to $\Lambda$ is denoted by ${\mathsc{T}}_{\Lambda }$. Given an exponent $p\in \left[1,\infty \right]$ we say that $\Lambda$ is $p$-coherent if there exist a compact set $K\subset {ℝ}^{n}$ and a continuous function $\omega$ defined on ${ℝ}^{n}$ with values in $\left[1,\infty \right)$ such that for every $P\in {\mathsc{T}}_{\Lambda }$ and every $y\in {ℝ}^{n}$ one has ${\left({\int }_{|x-y|\le 1}|P\left(x\right){|}^{p}\phantom{\rule{0.3em}{0ex}}dx\right)}^{1∕p}\le \omega \left(y\right){\left({\int }_{K}|P\left(x\right){|}^{p}\phantom{\rule{0.3em}{0ex}}dx\right)}^{1∕p}$. Several properties of $p$-coherent sets are proved in this essay.

 To the memory of Salah Baouendi
##### Keywords
mean periodic functions, almost periodic functions, trigonometric sums
Primary: 42A32
Secondary: 42B10