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 ${\mathcal{T}}_{\Lambda }$. 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 $P\in {\mathcal{T}}_{\Lambda }$ and every $y\in {ℝ}^n$ one has ${\left({\int \; <!--nolimits-->}_{|x-y|\le 1}|P\left(x\right){|}^{p}dx\right)}^{1∕p}\le \omega \left(y\right){\left({\int \; <!--nolimits-->}_K|P\left(x\right){|}^{p}dx\right)}^{1∕p}$. Several properties of $p$-coherent sets are proved in this essay.

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Mathematical Subject Classification 2010

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