Trigonometric series with a given spectrum

Abstract

Let $\mathrm{\Lambda }\subset {ℝ}^n$ be a closed and discrete set. The vector space consisting of all trigonometric sums whose frequencies belong to $\mathrm{\Lambda }$ is denoted by ${\mathcal{T}}_{\mathrm{\Lambda }}$. Given an exponent $p\in \left[1,\infty \right]$ we say that $\mathrm{\Lambda }$ 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 $P\in {\mathcal{T}}_{\mathrm{\Lambda }}$ and every $y\in {ℝ}^n$ one has ${\left({\int }_{|x-y|\le 1}|P\left(x\right){|}^{p}dx\right)}^{1∕p}\le \omega \left(y\right){\left({\int }_K|P\left(x\right){|}^{p}dx\right)}^{1∕p}$. Several properties of $p$-coherent sets are proved in this essay.